Metaphysics, Meaning, and Modality: Themes from Kit Fine 0199652627, 9780199652624

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Metaphysics, Meaning, and Modality

OUP CORRECTED PROOF – FINAL, 24/9/2020, SPi

OUP CORRECTED PROOF – FINAL, 24/9/2020, SPi

Metaphysics, Meaning, and Modality Themes from Kit Fine

 

Mircea Dumitru

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Great Clarendon Street, Oxford,  , United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors  The moral rights of the authors have been asserted First Edition published in  Impression:  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  Madison Avenue, New York, NY , United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number:  ISBN –––– 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.

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Contents Acknowledgements List of Contributors Introduction: Kit Fine—A Philosopher’s Philosopher Mircea Dumitru

ix xi 

I. Metaphysics . Ontology: What’s the (Real) Question? Fred Kroon and Jonathan McKeown-Green



. Beyond Reality? Philip Percival



. One Absolutely Infinite Universe to Rule Them All: Reverse Reflection, Reverse Metaphysics Joseph Almog



. Fine on Arbitrary Objects Alasdair Urquhart



. Indefinites, Skolem Functions, and Arbitrary Objects Gabriel Sandu



. Essence and Identity Kathrin Koslicki



. Indeterminate Identity, Personal Identity, and Fission Kit Fine



. Fine’s New Semantics of Vagueness Graeme Forbes



II. Modality . Necessary, Transcendental, and Universal Truth Steven T. Kuhn



. What is Normative Necessity? Gideon Rosen



. The Problem of de re Modality Bob Hale



. Can Metaphysical Modality Be Based on Essence? Penelope Mackie



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. More on the Reduction of Necessity to Essence Fabrice Correia



. Essence and Dependence Jessica Wilson



. Essence and Nominalism Scott A. Shalkowski



. Fine’s Theorem on First-Order Complete Modal Logics Robert Goldblatt



III. Language . Fine on Frege’s Puzzle Gary Ostertag



. Coordination, Understanding, and Semantic Requirements Paolo Bonardi



. Variable Objects and Truthmaking Friederike Moltmann



IV. Kit Fine’s Responses Comments on Fred Kroon and Jonathan McKeown-Green’s “Ontology: What’s the (Real) Question?” Kit Fine Comment’s on Philip Percival’s “Beyond Reality?” Kit Fine Comments on Joseph Almog’s “One Absolutely Infinite Universe to Rule Them All: Reverse Reflection, Reverse Metaphysics” Kit Fine Comments on Alasdair Urquhart’s “Fine on Arbitrary Objects” Kit Fine Comments on Gabriel Sandu’s “Indefinites, Skolem Functions, and Arbitrary Objects” Kit Fine

 

 



Comments on Kathrin Koslicki’s “Essence and Identity” Kit Fine



Comments on Graeme Forbes’s “Fine’s New Semantics of Vagueness” Kit Fine



Comments on Steven T. Kuhn’s “Necessary, Transcendental, and Universal Truth” Kit Fine



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Comments on Gideon Rosen’s “What is Normative Necessity?” Kit Fine



Comments on Bob Hale’s “The Problem of de re Modality” Kit Fine



Comments on Penelope Mackie’s “Can Metaphysical Modality Be Based on Essence?” Kit Fine



Comments on Fabrice Correia’s “More on the Reduction of Necessity to Essence” Kit Fine



Comments on Jessica Wilson’s “Essence and Dependence” Kit Fine



Comments on Scott Shalkowski’s “Essence and Nominalism” Kit Fine



Comments on Robert Goldblatt’s “Fine’s Theorem on First-Order Complete Modal Logics” Kit Fine Comments on Gary Ostertag’s “Fine on Frege’s Puzzle” Kit Fine

 

Comments on Paolo Bonardi’s “Coordination, Understanding, and Semantic Requirements” Kit Fine



Comments on Friederike Moltmann’s “Variable Objects and Truthmaking” Kit Fine



Bibliography of the Publications of Kit Fine Index

 

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Acknowledgements This book has been long in the making. A book of such length and with multiple authorship would not have been possible, let alone actual, without the strong and constant support of people to whom I owe a great debt of gratitude, namely friends, colleagues, and family. They all have shared with me a strong faith in the value and fulfilment of this project. First, of course, my profoundest gratitude goes to Professor Kit Fine, who assisted me personally with the intricate tasks required for the completion of this book. His well-known professionalism and profound originality are unmatched except by his generosity and encouragement for those who undertake difficult philosophical tasks. Then, over the years, and especially during the period of my intellectual and professional growth, I accrued a special debt to Professor Graeme Forbes, my Doktor Vater, from whom I learned almost everything that I know in my field of expertise. As a graduate student at Tulane University, Professor Forbes’s philosophical and pedagogical skills left an indelible mark on my philosophical development. The contributors to this volume deserve a special thanks. Each one understood that such a project would take time to complete and did not put any pressure on me during the long elaborate process that finally resulted in the book. A particular word of gratitude must go to Peter Momtchiloff, OUP Philosophy editor, who always behaved in the most professional and patient manner, providing me with all the freedom and guidance that I needed. I am also grateful to Stefania Olea, librarian for the Faculty of Philosophy, University of Bucharest, who helped me acquire a number of important sources necessary for the preparation of this book. Last, but not least, my deepest gratitude and love go to my family who have supported me throughout this entire process, understanding my need to devote long hours to my work at the university as well as to my work on this book. There is no doubt that without their understanding nothing would have been possible for me, by which I literally mean nothing. Through her unlimited support and patience, my wife, Professor Daniela Dumitru, has been a source of real help and inspiration; my daughter Dr. Ioana A. Dumitru helped me by providing stylistic recommendations and revising my materials; and our daughter Ilinca-Aurora Dumitru has always been a marvelous source of happiness and well-being. Each and every one deserves my deepest and sincerest gratitude and appreciation. Of course, I am the only one responsible for conceiving and executing the project of this book. Mircea Dumitru Bucharest, March 

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List of Contributors J A, University of Turku, University of California, Los Angeles P B, University of Tokyo and eidos, the Centre for Metaphysics F C, University of Geneva M D, University of Bucharest K F, New York University G F, University of Colorado at Boulder R G, Victoria University of Wellington B H{, The University of Sheffield K K, University of Alberta F K, The University of Auckland S T. K, Georgetown University P M, The University of Nottingham J MK-G{, The University of Auckland F M, French National Center for Scientific Research G O, The Graduate Center, CUNY P P, The University of Nottingham G R, Princeton University G S, University of Helsinki S S, University of Leeds A U, University of Toronto J W, University of Toronto

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Introduction Kit Fine—A Philosopher’s Philosopher Mircea Dumitru

This book is the first edited volume on the philosophy of one of the most seminal and profound contemporary philosophers. Over the last forty-odd years, Kit Fine has been one of the most influential and original analytic philosophers. He has made provocative and innovative contributions to several areas of systematic philosophy, including philosophy of language, metaphysics, and the philosophy of mathematics, as well as a number of topics in philosophical logic, such as modal logic, relevance logic, the logic of essence, and the logic of vagueness. These contributions have helped reshape the agendas of those fields and have given fresh impetus to a number of perennial debates. Fine originally focused on mathematical logic, specifically addressing issues in modal and relevance logic, with the publication of “An Incomplete Logic Containing S” in , “Failures of the Interpolation Lemma in Quantified Modal Logic” in , and “Models for Entailment” in , and “Semantics for Quantified Relevance Logic” in , within which he showcased his independent discovery of the ternary relation semantics for the system R. Fine also contributed to issues in the field of modal theories, where, in particular, he delved into the de re–de dicto distinction, analyzed the formal properties of modal theories of set theory, as well as the reduction of possibilist to actualist discourse (see “Model Theory for Modal Logic,” in three parts in , and , and Worlds, Times, and Selves, with A. N. Prior, ). In a series of consecutive publications from  and , Fine argued for a revision of the standard modal account of individual essence, according to which “P is essential to x” on condition that “x has P in every world where x exists.” Contending that such a definition is incomplete and lacks nuance, Fine resurrected an Aristotelian thesis, arguing that real definitions as opposed to de re modalities give a better account of essence (see “Essence and Modality,”  and “The Logic of Essence,” ). Fine’s work has embraced many philosophical subdisciplines, with significant contributions to topics in: () philosophy of vagueness, where he argued for supervaluationism (see “Vagueness, Truth, and Logic,” ); () philosophy of logic, where he developed an account of reasoning with existential statements wherein an Mircea Dumitru, Introduction In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Mircea Dumitru. DOI: 10.1093/oso/9780199652624.003.0001

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instance is taken to follow from an existential statement in virtue of its containing reference to an appropriate arbitrary object (see Reasoning with Arbitrary Objects, ); () philosophy of mathematics, where he contributed to discussions on abstraction and postulation (see The Limits of Abstraction, , “Our Knowledge of Mathematical Objects,” ); and () philosophy of language, where he developed the doctrine of semantic relationism (see Semantic Relationism, ). More recently, Fine’s work focuses on topics such as: essence and ground, realism, material constitution, truthmaking, response dependence, and vagueness (see “The Question of Realism,” , “The Pure Logic of Ground,” , Vagueness. A Global Approach, ). He has, in addition, contributed to several other areas in the course of his career, including the history of philosophy (“Aristotle on Matter,” , “A Puzzle Concerning Matter and Form,” , “Husserl’s Theory of Part-Whole,” , which is a formal reconstruction of Husserl’s third Logical Investigation), formal language theory (“Transparency,” ), the semantics of programming languages (“The Justification of Negation as Failure,” ), and social choice theory (“Social Choice and Individual Ranking,” , with B. Fine). Often eschewing mainstream topics and trajectories in analytical philosophy, Fine’s work, encompassing six books and over one hundred papers and critical reviews, has been original, profound, and prolific. The impact of Fine’s work on the field is constantly growing, as can be seen in the steady stream of papers and books written in reaction to Fine’s own work. Recently, philosophical engagements with Fine’s work have resulted in a conference entitled “Fine Philosophy—The Philosophy of Kit Fine.” Papers presented at this conference were collected and published in the journal Dialectica, , , under the editorship of Professor Kevin Mulligan. Fine’s work is greatly appreciated by analytic philosophers, undoubtedly due to both its technical sophistication and its impressive philosophical breadth. According to British philosopher Timothy Williamson, Fine’s groundbreaking arguments are often advanced “with characteristic brilliance and rigor.” Fine’s impulse towards stepping outside the narrow confines of contemporary analytic philosophy and towards grappling with a multitude of complicated questions have arguably contributed to the philosopher’s far-reaching appeal and recognition. Rather than simply engaging in arguments aimed at dispelling misconceptions and stereotypes in thought and language, Fine’s work recalls the original purpose of analytic philosophy: that of constructing theories and proposing concepts that enable us to make sense of our own experience. Were one to distill the importance of Fine’s estimable body of work, it would surely be found to reside in the philosopher’s opposition towards the current orthodoxy and in his commitment to rescuing sound philosophical common sense from “the adoption of a theory-driven methodology, one that favours considerations of a broadly theoretical sort over strong and seemingly compelling intuitions” (Modality and Tense, , p. ). This characteristic of his work is perhaps best exemplified by Fine’s position with respect to the Quinean tradition in metaphysics and semantics. Indeed, Fine self-reflexively contextualizes his own modal actualist thinking as a reaction (or “animosity,” in Fine’s own phrasing in his introduction to Modality and Tense [, p. ]) towards W. V. O. Quine’s conception of

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modality. Fine’s modal actualist theories were also defined in opposition to David Lewis’s so-called modal realism, with its “lavish [ontological] extravagance” (p. ), according to which whatever is merely possible is on an ontological par with whatever is actual. This book contains nineteen original essays which critically assess the work of Kit Fine. These papers dwell on both perennial and more recent developments in Kit Fine’s philosophical work, and are organized under a number of specific headings, which cluster around eight topics that fall under the main areas to which Kit Fine contributed; namely, Metaphysics, Modality, Logic, and Language. Specifically, the eight topics are the following: Reality, Arbitrary Objects, Identity, Kinds, Essence, Logic, Semantic Relationism, and Truthmaking, a diversity of topics which bears witness to the scope and richness of Kit Fine’s work. Moreover, the systematic character of Fine’s philosophy provides the ground for an organic growth of, and an interconnection between, those various topics, which are likely to be of interest to largely overlapping groups of individual philosophers, as long as a topic from one area naturally leads to, or shows an affinity for, a topic from another area. This is why we expect the volume to be of interest both to professional philosophers and to graduate and undergraduate students who are interested in metaphysics, language, and philosophical logic. It is our hope that these readers will benefit from the general level of scholarship in the book and from the more particular debates over Kit Fine’s novel theories on meaning and representation, arbitrary objects, essence, ontological realism, metaphysics of modality, and constitution of things. More generally, it is our hope that a thorough discussion of the work of a very innovative and profound author such as Kit Fine can contribute to a better understanding of what is at stake within contemporary analytic philosophy. Concretely, under the rubric Metaphysics (Part I), the contributors to this volume address the following issues: Reality (Fred Kroon and Jonathan McKeown-Green, “Ontology: What’s the (Real) Question?”; Philip Percival, “Beyond Reality?”; Joseph Almog, “One Absolutely Infinite Universe to Rule Them All: Reverse Reflection, Reverse Metaphysics”); Arbitrary Objects (Alasdair Urquhart, “Fine on Arbitrary Objects”; Gabriel Sandu, “Indefinites, Skolem Functions, and Arbitrary Objects”); and Identity (Kathrin Koslicki, “Essence and Identity”; Kit Fine, “Indeterminate Identity, Personal Identity, and Fission”; Graeme Forbes, “Fine’s New Semantics of Vagueness”); Under Modality (Part II), the authors of the essays discuss Kit Fine’s contribution to the metaphysical issues of Kinds (Steven T. Kuhn, “Necessary, Transcendental, and Universal Truth”; Gideon Rosen, “What is Normative Necessity?”); and of Essence (Bob Hale, “The Problem of de re Modality”; Penelope Mackie, “Can Metaphysical Modality Be Based on Essence?”; Fabrice Correia, “More on the Reduction of Necessity to Essence”; Jessica Wilson, “Essence and Dependence”; Scott Shalkowski, “Essence and Nominalism”). The groundbreaking contribution of Kit Fine to the model-theoretic approach to modal logic is summarized and examined by Robert Goldblatt in his “Fine’s Theorem on First-Order Complete Modal Logics.” The section devoted to Fine’s very recent and novel views on the nature of Language (Part III), as well as meaning and truthmaking, includes expository and

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critical essays on Semantic Relationism (Gary Ostertag, “Fine on Frege’s Puzzle”; Paolo Bonardi, “Coordination, Understanding, and Semantic Requirements”) and on Truthmaking (Friederike Moltmann, “Variable Objects and Truth-Making”). Each chapter in this volume, with the natural exception of Kit Fine’s own essay, is followed by his response and provides dialectical exchange that is not only a tribute to the contribution of Fine to contemporary philosophy but also a critical overview of his philosophical work. An up-to-date list of Kit Fine’s philosophical works concludes this volume, which we hope will soon become a resource for all those interested in contemporary approaches to metaphysics, modality, logic, and language. In the remaining part of this Introduction I shall briefly present and comment upon the main upshot of each essay and connect it to Kit Fine’s own response to some of the points that have been made. In “Ontology: What’s the (Real) Question?” Fred Kroon and the late Jonathan McKeown-Green address issues of ontological interconnectedness that stem from Kit Fine’s work in the area. One important case in this regard concerns Fine’s views about what sorts of entities we should, as philosophers, commit ourselves to. In “The Question of Ontology” Fine challenges existing accounts of the philosophical task of ontology, rejecting a Quinean concern with what there is in favor of a focus on what entities are real. But what is it to be real? Adverting to the metaphysical view he formulated in “The Question of Realism,” Fine thinks that the notion of (being the case in) reality is primitive and that the reality of entities should be understood in terms of this primitive notion of reality. Kroon and McKeownGreen’s chapter critiques Fine’s interconnected set of ideas about the task of ontology. It attempts to defend the use of quantificational constructions in capturing ontological commitments, and questions the usefulness to ontology of a primitive concept of reality. Philip Percival’s focus is on the divide between reality and that which is beyond reality, what he calls an “all-encompassing” view of reality. This is the divide between everything and nothing: reality encompasses everything, and beyond it there is nothing at all. Opposed to the all-encompassing view is what he calls a “restriction” view of reality: reality is coincident with some kind of restriction on, or partition of, what there is; it is not the case then that what resides beyond reality is nothing. Consequently, Percival has two main aims: first to classify restriction views of reality, and then to assess a species of the restriction view that pertains to time and modality. Joseph Almog deals in his chapter with de re universe skepticism, which is a form of skepticism about the universe as an objectual unity. Its clearest origin lies in Kant’s doubts about objectual totalities that conform neither to intuition nor to individual concepts. Subsequently, this form of skepticism was taken further by the founders of bottom-up set theory and their conception of sets as limited in size (as, for example, with Zermelo, Von Neumann, Bernays, and, most prominently, Gödel). We witness a reduction of absolute infinity theory and global de re universe theory to localist and assemblage-ist set theory based on the iterative set-of operation. Set theory becomes the official universal language of combination-theory and synthesisof-unities-theory envisaged by Kant. Almog finds this neo-Kantian idealist and concept-driven account in the essentialist ideas of Kit Fine. Key to it is a certain

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conceptual-essentialist constructionism. The chapter contrasts this Kantian view with a top-down globalist de re universe, the ens originarium and absolutely infinite unitary object at that. The expressive incompleteness of the language of set theory (via reflection) is dissected. The epistemological question that fanned Kant’s skepticism viz. how we grasp/think-of this ur-object, is also addressed. The answer is found to rest on the mundanity (in both senses) of our contact with (if not grasp of) the universe. What was classed as impossible for thinking by concept-essence-driven Kantians is found to be such that: without contact-with-it no thinking is possible at all. Alasdair Urquhart’s “Fine on Arbitrary Objects” surveys Kit Fine’s theory of arbitrary objects. It includes a historical survey of earlier writers on arbitrary objects, including Newton, Euler, and Czuber. Then it looks at objections to arbitrary objects, concluding with a sketch of the construction of models containing arbitrary objects. In his chapter, Gabriel Sandu deals with Kit Fine’s notion of arbitrary objects and the relation of dependency between them. In the chapter, Sandu introduces a syntactic notion of dependency between variables which defines a notion of functional dependence on teams (sets of assignments). The idea comes from logics of dependence and independence and the so-called team semantics (Hodges, Väänänen, among others). Sandu argues that this metaphysically much lighter framework can account for some of the natural language examples (anaphora) that Fine mentioned as possible applications of arbitrary objects. At the end of his chapter Sandu touches upon the multi-dependencies between arbitrary objects vs. multidependencies in the framework of team semantics. Kathrin Koslicki evaluates six competing accounts that essentialists might involve in order to meet the Quinean challenge of providing necessary and sufficient conditions for the crossworld identity of individuals: () qualitative character; () matter; () origins; () haecceities or primitive non-qualitative thisness properties; () world-indexed properties; and () individual forms. The first three candidates, she argues, fail to provide conditions that are both necessary and sufficient for the crossworld identity of individuals; the fourth and fifth criteria are open to the charge that they do not succeed in meeting the Quinean demand in an explanatorily adequate fashion. On balance, then, individual forms deserve to be taken very seriously as a possible response to the Quinean challenge, especially by neo-Aristotelians who are already committed to a hylomorphic conception of composite concrete particular objects. Theorists who also accept a non-modal conception of essence, that is, a conception according to which essence is not reducible to modality, face, in addition, the further difficult task explaining how an object’s de re modal profile in some way follows from facts about its essence. Haecceities and world-indexed properties, as Koslicki indicates, are unlikely to be of much help with respect to this second challenge, while the forms of hylomorphic compounds are in fact well suited for this purpose. Kit Fine proposes a new theory of vagueness, radically different from his earlier supervaluationist account, and considers the application of the theory to the question of vague identity. In his response to Fine’s paper, Graeme Forbes is mainly concerned with Fine’s new semantical account of vagueness, which he calls “compatibilism,” while also

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including a brief discussion of Fine’s earlier theory, supervaluationism. Forbes explains compatibilism at some length and discusses its motivation. He ends with a comparison between compatibilism’s treatment of three different versions of the Sorites paradox and their treatment within fuzzy logic of the sort he favors. In his chapter, Steven T. Kuhn addresses the vexing metaphysical problem of how is it that we judge it to be absurd that Socrates might be human and not exist, yet true that he is necessarily human and true that he might not exist. Following Fine’s groundbreaking work in the field, Kuhn says that the answer to that question has profound implications for our understanding of the concepts of existence, identity, and modality. For it requires, according to Fine, that we distinguish between worldly sentences, whose truth values depend on circumstances, and unworldly ones, which are true or false independently of circumstances. Unworldly sentences like Socrates is human express transcendental propositions. Although these are not, strictly speaking, true in every (or indeed in any) possible world, we accept them as necessary in an extended sense. Unless the context gives us special reason, however, we are reluctant to extend the concepts of necessity and possibility further to include worldlyunworldly hybrids like Socrates is human and does not exist. Kuhn argues that this understanding of the relation between necessary and transcendental truth is backwards, and perhaps contrary to what Fine himself has advocated in some of his works. What is taken for (unextended) necessity in the puzzle analysis, Kuhn calls universal truth and he suggests that universal and transcendental truths are both necessary. To further clarify this view he presents a simple formal system with distinct operators for necessary, transcendental, and universal truth. It turns out that the logic for universal truth coincides with something that Arthur Prior had once labeled System A. With the benefit of now-familiar techniques, it is shown that Prior’s conjectured axiomatization for this system is correct. Finally, the formal system is enriched by the addition of an operator for actually true. This raises philosophical questions that sharpen our understanding of the worldly–unworldly distinction. The extended system is axiomatized and shown to correspond to the system SA of Crossley and Humberstone in much the same way that the system without actuality corresponds to S. The new logical systems lose the simplicity of having connectives that apply uniformly to all sentences, but has simpler axioms and greater fidelity to the notions that are to be formalized. Gideon Rosen’s chapter “What is Normative Necessity?” explores Fine’s suggestion that the ethical facts supervene on the natural facts, not as a matter of metaphysical necessity, but rather as a matter of normative necessity. The first part develops an explicit argument against the metaphysical supervenience of the ethical, the main premises of which are ethical non-naturalism and Fine’s essentialist analysis of metaphysical necessity. The second part defends an analysis of normative necessity according to which P is normatively necessary if and only if P would have been the case no matter how the nonnormative facts had been. The last part argues that the basic principles of ethics as the non-naturalist conceives them are indeed normatively necessary in this sense. Bob Hale tackles the issue of how, if at all, one can make sense of the problem of de re modality. Most who have discussed this problem have assumed that modality de dicto is relatively unproblematic. It is, rather, the interpretation of sentences

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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? If, as Hale believes, we should reject a broadly linguistic conception of the source of necessity, that disposes of one major source of the idea that de re modality is especially problematic. Does any serious problem remain? Is there any further, independently compelling, reason to doubt that de re modalities are intelligible? According to Fine, Quine has two arguments against the intelligibility of de re modality: a “logical” and a “metaphysical” one. That the “logical” argument is central to Quine’s attack is surely indisputable. But Hale’s claim that it is his basic argument is, in effect, denied by Kit Fine. Hale can (and does) agree with Fine that there are some significant differences between the two arguments. The most important question, for Hale’s purposes, is whether Fine is correct in thinking that the two arguments have force independently of one another. In his hugely influential paper “Essence and Modality” (), Kit Fine argued that the then orthodox view that essence can be understood in terms of metaphysical modality is fundamentally flawed. He proposed, in its place, the view that all metaphysical modality has its source in the essences or natures of things, where the notion of a thing’s essence or nature can be understood in terms of a broadly Aristotelian notion of real definition. This theory appears to require that the relevant conception of real definition can itself be isolated without appeal to metaphysical modality. In her chapter, “Can Metaphysical Modality Be Based on Essence?,” Penelope Mackie argues that this requirement cannot be met. She then briefly considers the implications of her argument for the relation between essence and metaphysical modality. In previous work, Fabrice Correia has developed Kit Fine’s view that metaphysical modality should be understood in terms of essence, making use of his suggestion that the essence of the logical concepts is given by rules of inference rather than by propositions. Correia here strengthens the case for this “rule-based account” by criticizing alternative accounts and by suggesting replies to some objections to the account. Jessica Wilson argues that Kit Fine’s essence-based account of ontological dependence is subject to various counterexamples. She first discusses Fine’s distinctive “schema-based” approach to metaphysical theorizing, which aims to identify general principles accommodating any intelligible application of the notion(s) at issue. She then raises concerns about the general principles Fine takes to schematically characterize the notions of essence and dependence, principles which enter into his account of ontological dependence. According to Wilson, the problem, roughly speaking, is that Fine supposes that an object’s essence makes reference just to what it ontologically depends on, but various cases suggest that an object’s essence can also make reference to what ontologically depends on it. As such, Fine’s essencebased account of ontological dependence is subject to the same objection he raises against modal accounts of essence and dependence—that is, of being insufficiently ecumenical. After sketching Kit Fine’s deflationary approach to modality, Scott Shalkowski argues for a similar approach to essentialism. Shalkowski argues that nominalizing

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strategies regarding truth, predication, and similarity permit us to give priority to object-language formulations of essentialist claims, thus making essentialism safe for nominalists. Shalkowski does not present these points as a decisive argument against radical nominalism (or radical forms of anti-realism, in general). But the problematic character of the concept of quasi-truth, the need to go meta-linguistic, the apparent inadequacy of the resulting reductions, and the difficulty to provide straightforward justifications of those reductions should give considerable pause to anyone with an appreciation for the kind of simplicity and systematicity that good methodology is able to offer. Robert Goldblatt gives a general overview of Kit Fine’s seminal contributions to the formal development of modal logic. In his chapter, “Fine’s Theorem on FirstOrder Complete Modal Logics,” he presents the technical details of Fine’s influential Canonicity Theorem, which states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This chapter reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterization of when a logic is canonically valid, providing a precise point of distinction with the property of first-order completeness. The critical point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction. Gary Ostertag, in his “Fine on Frege’s Puzzle,” addresses issues pertaining to Fine’s Semantic Relationism program, which gives a new and original framework for understanding the notion of meaning. Thus, in Ostertag’s interpretation, the fact that () “Cicero = Tully” is informative, whereas () “Cicero = Cicero” is not, presents a familiar challenge to the view that the semantic contribution of a name is exhausted by its referent. According to Fine, the problem for this pure form of referentialism is that in (), but not in (), the singular term occurrences are coordinated—they represent their common referent “as the same.” The notion of coordination, or representing as the same, needs unpacking and Ostertag argues that Fine’s account of this notion in Semantic Relationism is inadequate. We thus need an alternative way of understanding it, one on which coordination facts do not enter into the content of what is said or asserted. To borrow from Wittgenstein, coordination lies not with what they say, but with what my words show. To demystify the notion of showing—which will be met with skepticism by some—Ostertag indicates how it can be understood in terms of Grice’s notion of conventional implicature. Paolo Bonardi, in “Coordination, Understanding, and Semantic Requirements,” aims at answering the following question: What is coordination between proper names? Fine proposes two characterizations of coordination (or representing as the same): an intuitive test; and a technical definition. In regard to the intuitive characterization, Bonardi maintains that coordination has been grounded in a notion of understanding distinct from the notion of linguistic competence. Whereas, according to Bonardi, we need a (proper) characterization of understanding in order to elucidate Fine’s coordination, it is unclear how to provide one. Three prima facie appealing proposals to characterize are examined and then dismissed as intrinsically implausible or as incompatible with Fine’s relational semantics. Not even his technical characterization of coordination, in terms of the notion of semantic requirement, will enable us to escape the impasse and so, ultimately, the problem of determining what exactly coordination between names is remains open.

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In her chapter, Friederike Moltmann argues that terms like the number of people that can fit into the bus and the book John needs to write stand for variable objects, objects that have manifestations as particular, concrete entities in different actual or counterfactual circumstances that are satisfiers of, for example, some need. The notion of a variable object is a development of Kit Fine’s notion of a variable embodiment and involves the notion of exact truthmaking or satisfaction, again a key concept in Fine’s philosophy. *

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Since it is the first edited volume on the philosophy of this seminal, original, and profound contemporary philosopher, this whole book on Kit Fine’s contributions to metaphysics, philosophical logic, and philosophy of language will likely have an impact on new generations of young analytic philosophers attracted to the topics, style, and techniques of Kit Fine’s philosophy and will engender a multitude of critical responses. We do hope that this time-slice through Kit Fine’s work and the critical exegeses will encourage the younger generation of philosophers to pursue the topics and the theories that Kit Fine has developed over the last fifty years. After all, as readers can easily appreciate by themselves, Fine’s work is distinguished by its great technical sophistication, philosophical breadth, and independence from current orthodoxy. A blend of philosophically sound common sense combined with a virtuosity of philosophical argumentation and construction, meant to back up the former, seems to me to lie at the heart of Kit Fine’s lasting contributions to the current trends in analytic philosophy. This volume is intended for both professional philosophers and graduate and undergraduate students interested in metaphysics, language, and philosophical logic. It is our hope that readers will benefit from the general level of scholarship in the book and from the more particular debates over Kit Fine’s novel theories on meaning and representation, arbitrary objects, essence, ontological realism, metaphysics of modality, and constitution of things. More generally, it is my hope that a thorough discussion of the work of a very innovative and profound author such as Kit Fine can contribute to the better understanding of what is at stake within contemporary analytic philosophy.

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PART I

Metaphysics

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 Ontology What’s the (Real) Question? Fred Kroon and Jonathan McKeown-Green

Introduction One way to philosophize is to ontologize: about physical objects, moral properties, properties, possibilia, numbers, sets, and much else. For Kit Fine, ontologizing differs from what happens in ordinary life when I affirm that there are chairs and from what happens at school when I explain that there is a prime number less than three. We agree, of course, that ontologists typically pay little attention to chairs, except as examples of purported human artifacts or physical objects, and that they normally focus on the ontological status of numbers generally, rather than of primes specifically; but aside from this, we discern no interesting difference between what philosophers do and what happens in ordinary life and school. We also remain unconvinced by his reasons for rejecting a quantificational account of ontological theses. Although we acknowledge the power of some of these considerations, we nonetheless think that quantificational structures suffice for capturing ontological claims at home, at school, and in philosophy. Fine’s most extensive treatment is to be found “The Question of Ontology” (Fine ) and related considerations are advanced in “The Question of Realism” (Fine ). In the first three, expository, sections of this chapter, we present Fine’s ideas in the context of broader views that he elaborates in “What is Metaphysics?” (Fine b). Then we offer our responses.

 What is Metaphysics? In “What is Metaphysics?” (WM, for short), Fine declares that “Metaphysics is concerned, first and foremost, with the nature of reality” (WM, ). But he adds that this alone doesn’t mark metaphysics off from other subjects. For example, “Physics deals with the nature of physical reality, epistemology with the nature of knowledge, and aesthetics with the nature of beauty.” So what kind of investigation gets to be metaphysics? For one thing, it is distinguished from special sciences like physics by the a priori character of its methods. For another, it is distinguished from a priori disciplines like mathematics and other branches of philosophy by the Fred Kroon and Jonathan McKeown-Green, Ontology: What’s the (Real) Question? In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Fred Kroon and Jonathan McKeown-Green. DOI: 10.1093/oso/9780199652624.003.0002

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       - generality of its concerns: it “deals with the most general traits of reality” (which for Fine include value and mind; WM, ). But metaphysics can also be described, at least in part, in terms of certain tasks that seem peculiar to it: in particular, metaphysics might serve to supply a foundation. Fine thinks this can happen in at least two ways. One (which, as he points out, has “received considerable attention of late”) is supplying a foundation for the whole of reality. According to this conception of the tasks of metaphysics, some facts are more fundamental or “real” than others; and metaphysics, on this conception, attempts to characterize the most fundamental facts which are the “ground” for the other facts or from which they somehow derive. (WM, –)¹

There is also the conception of metaphysics as supplying a foundation, “not for reality as such, but for the nature of reality” (), the sort of foundation that might, in response to a question like “Why is water by its nature H₂O?,” invoke the metaphysical claim that any substance with a given composition is by its nature of that composition. Fine in fact thinks that if the concepts at play in such explanations are sufficiently general they will also have the right degree of modal and epistemic “transparency,” thereby allowing such explanations to terminate—satisfyingly—in general a priori truths of metaphysics.² Taken out of context, however, this (partial) account of the enterprise of metaphysics gives a misleading picture of Fine’s metaphysical views. What it doesn’t show is how different Fine’s conception of foundational metaphysical reflection is from that of most other philosophers. It doesn’t, for example, make it clear that Fine’s version of a prioristic “armchair” metaphysics is very different from the kind defended by David Lewis () and Frank Jackson (). Even the notion of a substance’s nature is for Fine a metaphysical rather than a modal notion (not even a modal notion explained in terms of metaphysically possible worlds rather than logically possible worlds). And while Fine happily allows an appeal to conceptual analysis as part of the methodology of metaphysics, such an appeal bears directly on the investigation of the nature of things, rather than, say, on the way that semantics, or conceptual frameworks, might provide an entry into that investigation (as in Jackson’s work). This might appear to align him with someone like Michael Devitt (), another philosopher eager to disassociate metaphysics from semantics. But for Devitt reality consists of entities that exist in a robust, mind-independent sense. That can’t be Fine’s view, however, since “What is Metaphysics?” cites mind as among the most general traits of reality, and that is incompatible with Devitt’s focus on mind-independence.

¹ Apart from Fine, the main proponent of such a view is Jonathan Schaffer (see Schaffer ), although Schaffer finds elements of such an Aristotelian conception of metaphysics in, for example, the use David Armstrong makes of the notion of “the ontological free lunch” and Lewis’s appeal to perfectly natural properties (Schaffer , ). ² Fine characterizes this second conception of the task of metaphysics as follows: “Metaphysics should be concerned with the nature of reality; it should operate at a high level of generality; its method of enquiry should be a priori and its means of expression transparent; and it should be capable of providing a foundation for all other enquiry into the nature of reality” (WM, ).

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More important for our project, the account sketched above of the enterprise of metaphysics presents a misleading picture of Fine’s ontological views. What he says about reality in “What is Metaphysics?” suggests that the ontological commitments that go with his kind of realism are generous, including the posits of science, mathematics, common sense (water, say), and much more. That it also includes familiar philosophical posits, for example, is confirmed by Fine’s early work on firstorder modal theories and on nonexistence (Fine , , ). He thinks philosophy and logic are well placed to comment on the nature of propositions, facts, possible worlds, and nonexistent entities and, indirectly, to show why we are right to suppose that there are such things. It is true that his comments on the way facts may be grounded in other facts, bottoming out in the fundamental facts, suggest that many of the putative facts about reality may not be fundamental facts, but there is nothing in “What is Metaphysics?” (or indeed his other works) to suggest that nonfundamental facts are anything other than bona fide facts about reality. (Admittedly, he is prepared to call these facts less “real” than fundamental facts, but in the context it is hard to read much into this.³) There is nothing, in short, to compromise the thought that Fine’s talk of reality is talk by someone who has robust metaphysically realist views. But this picture of Fine’s realism is seriously incomplete. Fine’s view is more nuanced: not only is there a distinction to be drawn between a realist and antirealist interpretation (in one sense of that distinction) of talk purporting to be about reality, but there is a further sense in which some true claims about reality may not, even on such a realist interpretation, reflect what is real in the deepest sense: what is true of reality in itself. It is the latter notion, Fine thinks, that best captures and unifies what philosophers are (or should be) concerned with when they speculate about what it takes for a proposition to be genuinely descriptive of the real. To see these refinements, we must turn to other papers.

 The Question of Realism “The Question of Realism?” (Fine ; QR, for short) adds a new dimension to the question of how to characterize realism and, indirectly, of how to understand the project of “What is Metaphysics?” Fine begins by noting that although anti-realism in philosophy has a long and illustrious history, it encounters a problem. Take a familiar kind of anti-realist about numbers who claims that there are no numbers. Fine points out that in our non-philosophical moments most of us, including such an anti-realist, are inclined to say, “There are prime numbers between  and ,” even though this claim implies that there are numbers (QR, ). Similarly, the anti-realist about morality maintains that there are no moral facts, while also thinking that killing babies for fun is wrong, even though this second claim implies that it is a moral fact that killing babies for fun is wrong.

³ See the above quote from WM. While Fine here implies that any fact grounded in other facts is ipso facto less “real” than these other facts, the overall thrust of this passage seems to be that even the less “real” is still part of reality.

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       - Fine comments that if we take the conflict in such cases to be genuine, we obtain an “eliminative” or “skeptical” conception of anti-realism, one that disputes what we ordinarily believe. Anti-realism on this conception urges a reassessment of our ordinary commitments.⁴ Fine’s response to such a form of anti-realism is firm: . . . in this age of post-Moorean modesty, many of us are inclined to doubt that philosophy is in possession of arguments that might genuinely serve to undermine what we ordinarily believe. . . . [and in so far] as the pretensions of philosophy to provide a world-view rest upon its claim to be in possession of the epistemological high ground, those pretensions had better be given up. (QR, )

Fine’s “post-Moorean modesty” explains why he unashamedly continues to accept “what we ordinarily believe,” suggesting an account of reality on which it includes the ordinary objects encountered in perception, say, but also the objects we encounter in mathematics and science, as well as facts involving these objects (and no doubt much more, including moral and aesthetic facts). Even if he doesn’t explicitly say so, when Fine talks about the metaphysical tasks associated with the investigation of reality in “What is Metaphysics?,” he is best read as talking about metaphysics in relation to a notion of reality on this Moorean, common-sense understanding, not an understanding that remains in thrall to skeptical anti-realism. But Fine also wonders whether there is another form of anti-realism that does not put it into conflict with received opinion in this way. Such an anti-realism, he thinks, “requires . . . a metaphysical conception of reality, one that enables us to distinguish, within the sphere of what is the case, between what is really the case and what is only apparently the case” (QR, ). The project of QR is to develop such a metaphysical conception of reality. In fact, Fine develops two such conceptions, answering to two different ways of understanding the above contrast, and yielding two different ways in which propositions may fail to “correspond” to the facts. According to the first, what is real is what is “factual,” and the corresponding type of anti-realist about a given domain denies that there are any facts “out there” that make the propositions of this domain true or false. According to the second, what is metaphysically real is what is “irreducible” or “fundamental,” and the corresponding type of anti-realist about a given domain claims that the facts involving this domain are all reducible to facts of some other sort. Non-factualism, the first of the above kinds of anti-realism, is familiar, but it cannot, on its own, capture the range of positions in which Fine is interested. Consider a refusal to accept moral facts into one’s ontology despite a willingness to assent to “killing babies for fun is wrong.” Admittedly, this might well be explained in terms of an account of moral discourse as expressivist and hence as non-factual. But such an imputation of non-factuality cannot, without a great deal more argument, be the general basis for saying that a class of assertible propositions fails to reflect reality. There are philosophers who are prepared to question the reality of numbers, for

⁴ It is worth noting that this revisionary aspect does not seem essential to anti-realism: somebody who was skeptical or nonbelieving about ley lines, yeti, or non-actual possible worlds is not necessarily in the business of questioning standard worldviews; if anything, he or she is in the business of ensuring that certain views don’t become more widely accepted. This point is discussed briefly in §.

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example, but without denying that ordinary number talk is assertoric or factual (Hartry Field is a prominent example). The same goes for certain ways of dismissing the reality of moral facts. Indeed, the project of fictionalism is founded on the idea of discourses that are assertoric and worth retaining, but useful rather than true.⁵ To find room for such forms of metaphysical anti-realism we need to look to Fine’s second conception of what is real. This second metaphysical conception of the real—the real as fundamental—is, in Fine’s view, even more central to metaphysics. It is not, he thinks, a relational notion, definable in terms of the notion of reduction, say. It is simply the conception of Reality as it is in itself. The example he gives is of the true proposition that two nations are at war, where “we may deny that this is how things really, or fundamentally, are because the entities in question, the nations, and the relationship between them, are no part of Reality as it is in itself ” (QR, ). More generally: One might think of the world and of the propositions by which the world is described as each having its own intrinsic structure; and a proposition will then describe how things are in themselves when its structure corresponds to the structure of the world. (QR, )

Armed with this notion of the fundamentally real as a primitive, Fine thinks that we can define the notion of reduction, using the notion of ground. First, propositions Q, R, . . . are said to ground proposition P just if its being the case that P consists in nothing more than its being the case that Q, R, . . . .⁶ Second, with “real” understood as “fundamentally real,” the true proposition P can then be said to reduce to the propositions Q, R, . . . iff (i) P is not real, and (ii) P is grounded in Q, R, . . . , where each of the latter is either real or grounded in what is real (QR, ). But which propositions are real in this sense? For a start, any true basic (i.e., ungrounded) factual proposition will be real, since “any true factual proposition is real or grounded in what is real” (QR, ) (indeed, Fine takes being real or grounded in what is real to be both necessary and sufficient for being factual).⁷ Fine also thinks that there is a presumption that any given basic proposition is real, and any given non-basic proposition unreal. But, importantly, he admits exceptions to the second presumption. Thus he mentions the case of water, under the assumption that Aristotle is right about the nature of water: any body of water is both indefinitely divisible and water through-and-through. In that case, it is best to see propositions about the location of a body of water as being real, even if they are grounded in other (real) propositions about the location of water (e.g., that the left half of the body of

⁵ Of course, non-factualists are rarely non-factualist about every domain of discourse. Domains where they tend to be factualist include ordinary discourse about both observable objects and the non-observable objects of science. A contrasting kind of pervasive non-factualism is defended by Huw Price (see, for example, Price , ). ⁶ Fine argues in QR that the notion of ground can also be used to settle whether to be a factualist about a discourse. In the end, he says, “the question of whether or not to be a factualist is . . . the question of whether or not to adopt a representational account of what grounds our practice” of dealing with the “facts” in a certain area (QR, ). For recent work on the notion of ground and its logic, see Fine (b) and Clark and Liggins (). ⁷ The reason: “if a proposition is factual, then it must be rendered true by the real world, and [so] if it is not itself real it must be grounded in the real” (QR, ).

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       - water in front of me is here, on the left, and the right half there, on the right). Such propositions will count as both irreducible and grounded (ad infinitum, as it were). One significant consequence of Fine’s remarks about this kind of exception is that it highlights the fact that the fundamentally real, as that notion is used in QR, is a primitive notion. In particular, it is not the notion of what Fine elsewhere calls reality at its “most fundamental,” with the relational notion of more fundamental than understood in terms of the relation of grounding. Recall Fine’s reference in “What is Metaphysics?” to a conception of metaphysics that seeks to determine “the most fundamental facts which are the ‘ground’ for the other facts or from which they somehow derive” (). The notion of the real as comprising “the most fundamental facts” is not Fine’s conception of the real in QR, although he certainly acknowledges connections between the two notions. Instead, the use of the qualifier “fundamental” in QR’s phrase “reality as fundamental” is perhaps best seen as reminding us that this notion of reality is the fundamental one for Fine, that “our interest in other categories of reality [like the factual] derives from their connection with this more fundamental category” (QR, ).⁸ The overall picture we get, then, is one of a unified realist metaphysics that provides a structural overlay to what Fine has to say about reality in works like “What is Metaphysics?” Reality in that work appears to involve whatever there is, with whatever properties it has and whatever relationships it stands in to whatever (else) there is. If we take “true” to mark out correspondence to the facts in this reality, then the true propositions are the (true) factual propositions. But not all factual propositions about reality in this sense are descriptive of reality in a more metaphysically fundamental sense—Reality as it is in itself. To enquire into what is metaphysically real in this sense is to be embarked on a metaphysical task, not a task open to standard techniques of investigation such as those offered by science, say. It requires us to look to the propositions of the given domain, assuming them to be factual, and attempt to ascertain from the overall structure of their grounds how the division into what is and is not real is best effected. (QR, )

 The Question of Ontology More surprises are in store when we turn from “The Question of Realism” to “The Question of Ontology” (Fine ; QO for short). We might have expected that Fine’s metaphysical conceptions of reality, as the factual and the fundamental, would directly inform his conception of ontology. Not so—and, on closer inspection, it becomes difficult to see how such a project could work. We can construe Real, as it is deployed in QR, as either a property of propositions or an operator on propositions. Assuming Fine’s second account of the metaphysically real, “p is real” then either means something like “p describes fundamental reality” or “it is descriptive of fundamental reality that p.” But in asking ontological questions we are surely not ⁸ This is how we think Fine should be understood, although his divergent uses of “fundamental” and his various uses of “real” mean that piecing together the whole picture is not always an easy task.

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asking questions about the status of propositions. We are asking questions about what there is in fundamental reality (whether it contains numbers, for example). And far from this being at bottom a debate about the status of certain propositions, modern ontology as initiated by Quine began with an attack on the thought that propositions formed any part of reality (Quine ). Hence, a theory of the nature of ontology that requires there to be propositions seems unattractively biased from the outset. Like most contemporary philosophers, Fine is impatient with Quine’s doubts about propositions. But his impatience with Quine on the matter of ontology runs much deeper. He thinks Quine misconstrued the whole enterprise of ontology, a mistake that has virtually become orthodoxy. On the Quinean construal, when we ask whether numbers exist we are asking whether there are numbers, and this latter question is in turn regimented into a question involving quantifiers. Let “9” be the existential quantifier, taken as (relatively) unrestricted. The question whether numbers exist then becomes the question whether 9x(x is a number). The commonly accepted view, inherited from Quine, is that ontological questions are thus quantificational questions.⁹ But the philosophical question of ontology, Fine insists, is not the question of what there is. The question of what there is is the question that the skeptical anti-realists, discussed in QR, foolishly address. Recall Fine’s insistence that we all agree that there are chairs and numbers. The existence question is different. He identifies a number of arguments for this conclusion: what we might call the cognitive argument, the disciplinary argument, and the argument from autonomy. The cognitive argument claims that it is an obvious truth that 9x(x is a number) (for it is obvious that 9x(x is a prime number greater than ); and this latter claim immediately implies that 9x(x is a number)). Despite this, we can still sensibly ask, Do numbers exist? So the ontological question—a non-trivial question—cannot be the quantificational question—a trivial question. The disciplinary argument, by contrast, claims that ontological questions are philosophical questions, arising from within philosophy and to be answered on the basis of philosophical enquiry, while the (strictly quantificational) question of whether there are prime numbers, for example, and hence numbers, is a mathematical question, to be answered on the basis of mathematical, not philosophical, considerations. So the ontological question—a philosophical question— cannot be the quantificational question—a question open to modes of investigation from other disciplines. Finally, the argument from autonomy holds that ontology is appropriately autonomous. We may agree with the mathematician that there are prime numbers between  and , say, and hence that there are numbers, but as ontologically minded philosophers we should still be able to raise the question whether there really are numbers. So the ontological question—a question that philosophers can legitimately raise—cannot be the quantificational question—a question that should often be regarded as closed. But what, then, distinguishes genuine ontological existence claims from quantificational claims—how, in Fine’s words, might we create a distance between these

⁹ Quine famously thinks they are first-order quantificational questions, rejecting second-order versions. We think second-order versions and versions admitting plural quantifiers (§) have certain advantages.

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       - two forms of commitment? Not, Fine thinks, by following Quine and counseling ontological commitment to posits that are indispensable to our best overall scientific theory of the world. Fine thinks that indispensability of this kind is neither necessary nor sufficient for an ontological construal of our commitments.¹⁰ Nor can we create distance by downplaying the significance of ordinary commitments—suggesting, for example, that they are in some way figurative or to be taken in a make-believe spirit. As Fine complains, in claiming that there is a prime number between  and  or that there is a chair over there, I would appear to have as good a case of a strict and literal truth as one could hope to have. If these are not strict and literal truths, then one is left with no idea either of what a strict and literal truth is or of what the strict and literal content of these claims might be. (QO, )¹¹

Nor, finally, can we create a distance between the two forms of commitment in terms of the strength of their contents, with the help of a distinction between a thin (lightweight, non-ontological) and a thick (heavy-weight, ontological) sense of the existential quantifier.¹² For how is such a distinction to be drawn? Presumably a quantifier in the thick sense is just a restriction of the quantifier in the thin sense, but that seems to change the focus of ontological claims, so that they concern, not the kind of ontological status things possess, but the kind of thing they are. In Fine’s view, the best way of drawing the distinction falls out of a certain natural feature of existence statements. Fine asks us to consider the difference between the claims “Integers exist” and “Natural numbers exist,” noting that the first seems stronger than the second, even though construing them in the standard manner as existentially quantified claims is to take the first as weaker than the second. But if the usual quantifiers don’t help us, how then does “Integers exist” succeed in expressing the right kind of ontological commitment to integers? (Call this Fine’s quandary.) Fine’s response: 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. (QO, )

But of course for that to be our claim “x exists” can’t just mean “9y(x = y),” for this would turn “every integer exists” into the logically trivial “8x(x is an integer ⊃ 9y(x = y)).” What it seems is needed to express genuine ontological commitment is a special ontologically committing predicate of existence, for which Fine prefers the term “real.” “Real” designates a property of objects and should not, therefore, be confused with Fine’s use of the same word in QR, where it is associated with propositions. Armed with this predicate, we can say that realism about integers ¹⁰ Indeed, Fine thinks that “Quine’s approach to ontology appears to be based on a double error. He asks the wrong question, by asking a scientific rather than a philosophical question, and he answers the question he asks in the wrong way, by appealing to philosophical considerations in addition to ordinary scientific considerations” (QO, ). ¹¹ A similar point is made by Schaffer (, ) in the context of his general argument for permissivism about existence. ¹² For discussion of such quantifiers, see Chalmers ().

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proclaims that, for each integer x, x is real, while realism about natural numbers proclaims that, for each natural number x, x is real. So long as all the various objects are there in at least the quantificational sense—are there to be quantified over—it will follow that the first claim is indeed stronger than the second, since the natural numbers are properly included in the integers. And we already know from Fine’s anti-skeptical attitude towards our ordinary commitments that he thinks such objects are indeed there to be quantified over. (In general, Fine thinks that the intended import of the various ontological realist/anti-realist positions about numbers, integers, physical objects, and so on, rests upon adopting a realist position in the usual non-ontological, quantificational, sense.) But what is it to be real, in this special ontological sense? Is it yet another primitive notion? No; at this point Fine relates the predicate “real,” which applies to objects, to a cognate operator “Real” on sentences, where “Real( . . . )” (“R( . . . ),” for short) means something like “It is constitutive of reality that . . . .” Given this operator, we can say what it is for an object to be real in the QO sense. An object x is real if, for some way the object might be, it is constitutive of reality that it is that way. (That is, x is real = df 9ϕ R[ϕx].¹³) Fine obviously thinks that the debate between realists and anti-realists should include a debate about ontology. So we might have expected that the deployment of “real” discussed in QR would comport with its ontological counterpart discussed in QO. In particular, we might have expected Fine to contend that an object x is real if, for some way the object might be, it is constitutive of reality in itself that it is that way, where the latter notion is as understood in QR. We think that this is indeed the way Fine should be understood, although there is some unclarity in the way Fine articulates his position. Fine has this to say: There have been a number of attempts to clarify the idea of realism in the recent literature; and a critical examination of some of them is to be found in my paper “The Question of Realism” . . . One that has recently found some favor in connection with ontology is to identify what is real with what is fundamental; and one might likewise identify what is in reality the case with what is fundamentally the case. (QO, )

But Fine dissociates himself in QO from this attempted clarification: I myself do not see any way to define the concept of reality in essentially different terms; the metaphysical circle of ideas to which it belongs is one from which there appears to be no escape. (QO, )

Although it looks as if Fine is here explicitly rejecting his earlier account of reality as fundamental, we doubt that this is the case. “Fundamental” in these passages means “not grounded in anything else,” and we have suggested that this is not its meaning when Fine talks of “reality as fundamental” in QR. Recall that in QR Fine sees a strong connection, not an identity, between the real in itself and the real as

¹³ Note that the combination of a special predicate with the usual quantifiers allows us to recapture the contrast between thin (light-weight) and thick (heavy-weight) quantifiers: the heavy-weight ontological quantifiers are just the light-weight quantifiers restricted to things that are real.

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       - ungrounded. QR simply supplies more grist for that mill. In QO, as in QR, Fine takes the notion of the real as it applies to propositions to be primitive.¹⁴ For want of a better term, we will call QO’s account of what distinguishes the subject-matter of ontology partitive—it holds out the prospect that what there is in reality is only part of what there is (in the colloquial, non-mereological sense of the word), and takes this division to be the concern of ontology. In the remainder of this chapter, we argue against such a partitive account. We think that Fine has not adequately motivated this bifurcation, and fails to show that ontological import cannot be expressed with standard quantifiers. We begin (in the next section) by taking issue with his view that ontologizing is distinct from the enterprise that people are ordinarily engaged in when they say that there are chairs and the one that mathematicians are engaged in when they say that there is a prime greater than . Section  continues this critique by suggesting that when Fine casts doubt on the translatability of ontological claims into a quantificational idiom, he not only appeals to an impoverished corpus of examples but also fails to confront what initially looks to be a compelling alternative strategy for salvaging a quantificational construal of ontologizing, one that invokes plural quantifiers. Section  suggests that this alternative strategy nonetheless fails as it stands, and that its failure highlights an important respect in which we think Fine is right concerning the implications for ontology of the examples he considers. With that in mind, we propose an alternative to Fine’s account of these examples, one that combines quantification with a weak notion of pretense in what we call an as-if-ist account. In §, we return to Fine’s account, and argue that it is far from clear how Fine’s deployment in QO of the predicate “- is real,” and the operator “It is constitutive of reality that . . . ” and their cognates is supposed to elucidate the business of ontology, and far from clear, therefore, how Fine’s approach to the question of ontology can do better than the kind of broadly Quinean approach we have defended.

 The Ontological versus the Rest: Ontology for All! We have seen that part of Fine’s case for denying that philosophers’ ontological concerns can be formulated quantificationally arises out of his conviction that there is a distinction between ontological “existence” claims and quantificational “there is” claims. He seems to trace this distinction in turn to one between pure inquiry about ontology, on the one hand, and less lofty talk and thought about what there is, on the other: when people not engaging in professional ontologizing talk about whether there are chairs, numbers, or prime numbers, they have a different, perhaps more “ordinary,” quarry. This distinction is clearly at work when he rules, in QR, that skeptical anti-realists who deny fairly ordinary claims about chairs, or mere mathematical claims about numbers, are on shaky ground. Surely I am just right, he thinks, when I say that there is a chair beneath me, or that there is a prime number greater than ; but of course I am not concerned with ontology when I say this. As he ¹⁴ Part of the problem in interpreting Fine’s views is that these various papers, QR, QO, and WM, may not have been written in the order in which they were published (WM, for example, dates back to the early s, according to Fine).

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comments in QO, “We talk that way—indeed, correctly talk that way—but [an enlightened anti-realist can still assert that] there is no realm of numbers out there to which our talk corresponds” (QO, ). The distinction also surfaces elsewhere in QO: Fine supposes that fairly ordinary claims about chairs and primes can be regimented quantificationally, but he holds that ontological claims cannot. We are suspicious of this distinction between the ontological and the more mundane. As we saw in §, Fine presents three arguments for denying that quantification tracks ontological commitment and these rely on this distinction between the ontological and the more mundane. Take the cognitive argument first. Fine points out that the answers to ontological questions should not be trivial, but says that given the evident fact that there is a prime number greater than , it trivially follows that there is a number (an x such that x is a number); and, similarly, given the evident fact that I am sitting on a chair, it trivially follows that there is a chair (an x such that x is a chair). (QO, )

True, if I am, as a matter of fact, sitting on a chair, then it follows trivially from that fact that there are chairs. An ontologist, however, should be concerned with the question whether I am indeed sitting on a chair. How, in a context where the reality of chairs is a live issue, can Fine declare this to be evident? He may be right to say that the question of whether there are chairs or tables is “an everyday matter that is to be settled on the basis of common observation” (QO, ), but it is surely not to be settled for good on that basis, any more than questions about ghosts or yeti are. The contrast Fine has in mind appears in his disciplinary argument: “ontological questions are philosophical—they arise from within philosophy, rather than from within science or everyday life, and they are to be answered on the basis of philosophical enquiry” (QO, ). But why think this? Philosophers might be more interested than scientists are in whether properties exist, whereas scientists might be more interested than philosophers are in whether the Higgs Boson exists, and crypto-zoologists might be more interested than almost everybody else is in whether yeti exist; but it is not at all obvious that these are different sorts of question, so that the former is somehow more ontological. Both philosophers and theoretical physicists might take just the same kind of interest in whether space exists, even if their methodological preferences for how to investigate that question diverge. One can also imagine any of the above questions surfacing in lay discussions. Maybe these various questions should be approached via different investigative procedures, some more empirical than others. Maybe some ontological questions require philosophical scrutiny, while others do not, but (so far, at least) this seems to be the most that Fine can claim.¹⁵ With his argument from autonomy, Fine explicitly seeks to distinguish ontologizing from various things that mathematicians, scientists, or regular folk do when they

¹⁵ Interestingly, our conclusion problematizes the relationship that Fine presumably seeks between metaphysics and ontology. In “What is Metaphysics?,” he wants (quite reasonably) to carve out a niche for metaphysics that is distinct from that of, say, physics, and that is distinctively philosophical. This is, at least partially, a matter of working out what sensible projects are being served by what passes for metaphysics. We have just argued that, in QO, he fails to show that ontology is distinctively philosophical. A fortiori, he fails to show that it is fully contained within metaphysics.

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       - say that there are prime numbers, chairs made from electrons, or just chairs. However, it is not clear that there are any claims about what there is that are merely mathematical, merely scientific, or merely ordinary, and that cannot be challenged through the reflections of ontologists. True, it may be entirely reasonable for me to say “there are chairs in the spare room,” or to hold a belief that I would express with that sentence, while simultaneously doubting that there are any chairs. (I might, for instance, be attracted to phenomenalism and take it to be analytic that chairs are physical.) It may be equally reasonable for me to tell the class that there is a prime number between  and , while doubting that there are numbers. But none of this requires that there be some ordinary sense of this talk that is importantly different from professional ontology. It may merely reflect variations in standards of communicative precision and other pragmatic niceties.¹⁶ After all, my companion might respond, “You don’t really believe that there are chairs in the spare room, do you?,” to which I might respond that I am indeed unsure about this, but that drawing attention to my ontological scruples would be conversationally wrong in this context. Meanwhile, my student with nominalist sympathies might complain that although he gets my drift when I talk of a prime number between  and , I should phrase matters more carefully; if there are no numbers at all, my claim is false. In short, I might carelessly or disingenuously report that there are chairs in the spare room or that there is a prime between  and , believing that my doubts or pet theories are irrelevant, obfuscatory, pedantic, or embarrassing. After all, most people do believe that there are chairs and prime numbers. Minority views, such as the oldfashioned, pre-Moorean, skeptical anti-realism that Fine dismisses are sometimes casualties of acceptable imprecision. Such loose talk is ubiquitous and doesn’t always sink ships. The candidate asserts that nobody here is denying the importance of stable government, conveniently ignoring the possibility that somebody is. The genial host tells the men that all of their wives are welcome, implicitly assuming that any unmarried men with female partners will take them to be included. Any researcher hoping to pinpoint a speaker’s ontological commitments must search beyond the words uttered to other contextual factors, but the commitments that emerge will be as ordinary as they are ontological. In this connection, it is worth noting that there seems to be nothing special about the words “exist” or “real” that would link them with serious ontology, or about the phrase “there are” that would render it more homely. I might say “there are no unicorns,” “unicorns are not real,” or “unicorns don’t exist” and mean precisely the same thing whichever I say. It may well be possible to understand a metaphysically exercised mathematician when she says that, although there is a number less than every prime, numbers do not exist. Nevertheless, we could understand her to be making the same claim if she said that although a number less than every prime exists, there are in fact no numbers, or that although there is a number less than every prime, numbers have no being and are not really there to be quantified over. Similarly, you may understand me when I say that although Hermione is not real, ¹⁶ Or it may reflect engagement in make-believe or “as if ” talk. It will emerge in § that we are more sympathetic to this possibility than Fine is, but our arguments against estranging the ordinary from the ontological do not require this commitment from us.

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she nonetheless exists, since a complete account of the world in all its richness includes her. Allowing for the unusualness of the subject-matter, however, you might understand me just as well if I made the same point by saying that although Hermione does not exist, she is nonetheless real. To be sure, it is likely that the following constructions are pragmatically out-of-bounds: “There’s a number smaller than every prime, although there are no numbers” and “A number smaller than every prime exists, but numbers don’t exist.” But that can be explained: if, as seems very likely, more than one sort of attribution is in play in each of these sentences, it would be sensible to distinguish them by using different locutions; but this does not require that the same locution is always matched up, as a matter of usage, with the same sort of attribution. To summarize: Despite Fine’s arguments in QO, we so far see no distinction between ontology and ordinary talk and thought about what there is, and hence no special relationship between quantification and one or the other of these. Instead, we have noted that people often speak (and perhaps think) loosely and also that expressions like “there is,” “there exist,” and “is real” are routinely used for a variety of purposes.

 Gods, Integers, and Plural Quantifiers As we saw in §, Fine has another objection to the quantificational formulation—one that does not turn on distinctions among usages or between more or less ontological projects. What we termed Fine’s quandary suggests the quantificational formulation cannot capture the obvious difference in strength between “Integers exist” and “Natural numbers exist.” His solution is to interpret these as universally quantified claims that use a special predicate of existence or reality. But in our view, Fine overstates his case. One way in which we think he does so will be clear from the previous section. It is not clear to us that existential and quantificational locutions differ in the way Fine says they do. If a professor announces a series of lectures that will prove, in the first week, that there are natural numbers, and in the second week that there are integers, her students are not likely to think that it is a repeat lecture, even though on a standard quantificational reading the first claim entails the second. Fine may, of course, argue that this is because of pragmatic rather than logical considerations, but note that the same point applies to existential locutions. Suppose, for example, that I say that gods exist. In that case I am certainly not committing myself to the existence of, or to the reasonableness of talk about, all the gods that have been posited, or of any particular pantheon, or of any particular god. I think I can express much the same sentiment by saying instead that there is a god, or that there are (some) gods or that there is at least one god or that gods are real. All of these formulations seem to be ways of making the same claim in ordinary language. Now replace gods with integers. If I say that integers exist or that there are integers, I need not be committing myself to all the integers that have ever been posited, or to all of the standard integers. After all, I might also think that there are only finitely many numbers, and so I might simply be saying, using existential quantification, that there’s an integer, or that there are some integers. Of course, this is not very likely. An

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       - explicit commitment to the existence of one or more integers, unlike a commitment to the existence of one or more gods, is a difficult commitment to motivate. So Fine is probably right to deny that ontological claims about integers are, in practice, existential quantifications over integers. But we should think more about what, in practice, somebody who said “Integers exist,” or “there are integers,” would thereby be committed to. Sometimes, that utterance would only commit the speaker to the existence of negative numbers. After all, a speaker who already knew that there were (at least some) natural numbers would arguably not be saying anything about those when asserting, “There are integers! I learned that last week.” What is meant here, as a matter of pragmatics, can be expressed by existential quantification: “9x(x is an integer)” is here to be interpreted as a claim about negative, rather than all, integers. Of course, if I was rightly interpreted as being committed to the existence of negative numbers, I might well be prepared to acknowledge the existence of all the negative numbers. Even so, it is unlikely that I can be saddled with this view, simply because I use the words “integers exist.” If I am indeed committed to the existence of all the negative integers, this is at least partly because of facts about the context in which I uttered those words. (To see this even more starkly, note that a student of higher mathematics might, in a different context, say, “integers exist” in order to make a different claim. Such a student might be impressed with the way that you can take the set of natural numbers and build the set of integers out of it using an equivalence relation. When this person says “integers exist,” she is opposing, among others, those who believe in natural numbers without believing in the constructs—the equivalence classes—that, on some set-theoretic accounts, are the members of the set of integers, negative and nonnegative. Again, context will be needed to tell us that this is what’s being claimed.) In short, we worry that Fine has made unwarranted assumptions about what people are committed to when they use particular locutions. Still, we can imagine Fine saying that in some contexts it is clear that no quantificational formulation suffices to express what we mean, and that while pragmatic considerations might be needed to show us which these are, that does not mean the claims aren’t ontologically distinguished in the way he thinks they are. We disagree, but do think that in many cases, arguably including those that interest Fine, quantificational formulations on their own don’t suffice to express what speakers mean. Before we say why, and what lessons to draw from this, we should mention one other way in which Fine appears to overstate his case: he neglects to consider a familiar alternative to the classical firstorder way of formalizing claims of existence. Why shouldn’t we use plural quantifiers to formalize statements like “Integers exist” and “Natural numbers exist” in cases where our interest is in all integers and all natural numbers? Once plural quantifiers are made available, we could say, (EPQ) Ys exist = There are one or more xs such that everything that is one of the xs is a Y, and everything that is a Y is one of the xs. (Formally, 9xx 8z(z≺xx $ Yz), where “-≺ . . . ” symbolizes “- is one of the . . . .”)¹⁷

¹⁷ Here we use the symbolism for plural quantifiers used in Linnebo’s PFO (Linnebo , ).

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Satisfyingly, this ensures that the claim “Integers exist” amounts to the claim that all integers exist, while “Natural numbers exist” amounts to the claim that all natural numbers exist. (It is easy to capture the difference between “Integers exist” and our earlier “Gods exist”: simply omit the second conjunct in the corresponding formulation, so that “Gods exist” = “There are one or more xs such that everything that is one of the xs is a god.”) Plural quantifiers are arguably what we need to formalize the role of plural definite descriptions in claims such as “The people in Auckland like the outdoors” (Brogaard ). The above formulations simply extend such a theory to the case of plural existence claims, on the model of Russell’s account of existential statements involving ordinary definite descriptions. (There is some independent motivation for this, since “The integers exist” sounds rather more precise than “Integers exist,” which, as we argued above, stands in need of contextual disambiguation.) Now consider “[The] integers don’t exist.” Given plural quantifiers, the most obvious reading of this statement is as the straight negation (¬EPQ) of (EPQ), that is, “It is not the case that there are one or more xs such that everything that is one of these xs is an integer and everything that is an integer is one of these.” Again, this is a satisfying result. Of course, the statement is true even if there are some integers but none less than -, say, but context will make it clear that this is unlikely to be an option for the speaker; in any case, she could always clarify her statement to show that she means all integers by adding “In fact, no integers exist.” (Note that the same demand to clarify whether we mean all or some will arise on Fine’s formulation of “Integers don’t exist,” i.e., “¬8x(x is an integer ⊃ Real(x))”.) So on the face of it the case for a quantificational reading of “Integers [don’t] exist” looks far from hopeless, assuming we are permitted to use plural quantifiers to regiment such locutions. In the end, however, we think that matters are rather more complicated than this. To see why, we now turn to another class of statements. These show both why we think Fine may well be right about his preferred logical form for statements like “Integers [don’t] exist” (despite the apparent attractions of an account like (EPQ)) but also why the sense in which he is right may have little bearing on the more important issue of whether quantifiers (whether first-order, higher order, or plural) suffice for expressing ontological commitment.

 Quantification, Pretense, and as-if-ism: An Alternative to the Partitive Account Consider a very different kind of case. Suppose you ask someone whether “those people exist / are real” (looking at a painting featuring a group of people in front of a building). And suppose she answers: () Those people do indeed exist or, as the case may be, () Those people do not exist Notice the use of the demonstrative, “those.” The speaker might use language that is even more overtly perceptual to answer the question. She might say: “If you look

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       - carefully, you can see that the one on the left looks quite inebriated. That’s not artistic license. He really was like that.” Whatever we think of the case of integers, this case cannot be handled in terms of plural quantifiers alone. The reason is implicit in the use of the demonstrative. There is a clear sense in which the speaker is assuming that there are people able to be designated with a plural perceptual demonstrative (“those people”) in order to affirm that “they” exist or to deny that “they” exist. This sense can be brought to the fore with the use of certain coordinate clauses. Thus consider the statement: “Those people—the ones depicted as standing in front of the building—do not exist” where the non-restrictive clarificatory clause yields an affirmative statement identifying “those people.” If we use plural quantifiers alone to analyze this statement, the statement turns into an explicit combination of an affirmative quantificational statement (formalizing “The people being demonstrated are the people depicted in the picture as standing in front of the building”) with a negative quantificational statement inconsistent with it (formalizing “Those people don’t exist”). Note that we avoid inconsistency by adapting instead Fine’s solution to his quantificational quandary. Fine’s partitive strategy yields something like: (a)

8x(x≺those people ⊃ ¬Real(x))

(uttered at a context where there is an attempt to demonstrate certain people by way of their being depicted as standing in front of the building) as the appropriate way of understanding the statement “Those people do not exist.” It should be clear, however, that such an account is no more satisfactory in the present case than an account that simply invokes plural quantifiers. There surely is little temptation in this kind of case to think that the painting literally depicts a number of people, with the only question being whether these people are real. Nonetheless, we think that something like this is indeed the correct logical form of the statement. In our view, the speaker’s sentence should be taken at face value as having the logical form (b)

8x(x≺those people ⊃ ¬Exist(x)),

where “those people” is a complex plural demonstrative and “Exist( . . . )” is a trivial predicate of existence that matches the existential quantifier (that is, Necessarily, Exists(x) iff 9y(x=y)). We avoid the problems facing a literalist understanding of what is asserted with such a statement by invoking the idea of pretense or makebelieve. The speaker pretends that she is able to single out certain demonstratively salient people, ones that therefore exist (from the point of view of the pretense), continuing this pretense when she clarifies whom she has in mind with the further clause “the ones depicted as standing in front of the building.” She then declares (from inside the pretense) that these people do not exist, aiming thereby to say something about how things stand existentially apart from her pretense. (Such a reading is pragmatically justified by the observation that the speaker cannot be intending to ascribe nonexistence from the point of view of the pretense, since the entities quantified over do exist from the point of view of the pretense, and this renders salient the question of how things stand with the reference of the complex

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demonstrative “those people” once we set aside the pretense.¹⁸) What she succeeds in asserting is at a first approximation something like: (P) Outside of the pretense that there are people picked out by the plural demonstrative “those people,” there do not exist people who are picked out by the plural demonstrative. (Or, more simply, ¬9xx(xx are people demonstrated via being depicted in the painting).)¹⁹ This account of relevant claims of (non)existence is very different from Fine’s partitive account. It holds that the question of reality concerns what is the case outside the operation of pretense or make-believe that the world is a certain way. There is no special apparatus here for signaling ontological import of the kind allegedly in play on a partitive account. Ordinary quantifiers (in this case, plural quantifiers) suffice. Could a pretense account do equally well for all the cases that concern Fine? Consider again the anti-realist who wants to deny the reality of chairs or integers, but continues talking as if there are chairs and integers. Earlier we suggested that such an anti-realist could simply be talking loosely. She doesn’t explicitly deny that there are chairs or integers, because it would be conversationally wrong to do so. But how then should we construe her attitude towards statements she sees it would be wrong to deny? Could this too be a case of pretense? (If so, anti-realist denials of existence could again be construed as affirming how things are apart from the pretense.) Perhaps in some cases. (If Hartry Field tells us he merely make-believes that mathematics is true, who will be the first to tell him that he misconstrues his attitude of belief for make-belief?) But we are also sure that on the most familiar way of construing the notion of pretense it is at least contentious to say that philosophers inclined to anti-realism are merely pretending when they claim that there is a prime number between  and  or that there is a chair in the room. In particular, cases of this sort are not accompanied by the usual imaginative phenomenology attending pretense, and to that extent Fine surely has a point when he says that in such cases a speaker “would appear to have as good a case of a strict and literal truth as one could hope to have” (QO, ). Assuming, then, that pretense is accompanied by a distinctive introspectible phenomenology, it is doubtful that pretense offers a good general explanation of any failure of ontological import that might attach to these statements when proffered by anti-realists.²⁰ However, we should not overlook the apparent tension

¹⁸ A question rendered even more salient if we take ourselves to be asking whether “those people” really exist, since this wording suggests that, while “those people” do of course exist at the relevant context of pretense, there is a question whether our words succeed in singling out any person when the context is real, not pretense. ¹⁹ The question of how the speaker manages to assert something like this won’t concern us here. For two rather different pretense accounts of such locutions, see Walton () and Kroon (). These turn on different ways of understanding the meaning of “exists,” both entirely consistent with a quantificational way of understanding what is asserted by means of statements of (non)existence. ²⁰ This condition on pretense may be too strong. What Walton calls prop-oriented make-believe, for example, is pretense that is oriented towards the external world, not focused on some fictional world, and it typically lacks the thick phenomenological features of content-oriented make-believe (Walton ). See

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       - between saying, first, “There is a prime number between  and ” (or even “There definitely is a prime number between  and ”; anti-realists are allowed to be as dogmatic as anyone else) and, in the next breath, “But numbers don’t exist.” We noted in § that there are contexts in which such a sequence of utterances is intelligible, but there needs to be a story about what is being attributed by each utterance. Let’s agree that pretense is unavailable here. Should we then accept Fine’s partitive account as a likely explanation for the coupling of these claims? No, for we think there are more promising alternatives. For a start, people sometimes claim that Xs do not exist, or are not real, because the debate about Xs occurs against a background that takes them to have a certain nature, or to have certain essential features. The background thesis might be that numbers are Platonistic entities, or that chairs and tables are substances in a radically mindindependent sense. In claiming that Xs do not exist a person may be claiming that nothing with the features assumed by the background exists, or is real. This is consistent with believing that there are Xs (or that Xs exist), since one might be happy to agree that thought and talk about Xs does latch on to items that exist, though such items would lack the features assumed by the background. The assertion that numbers, chairs, and so on do not exist, where this is uttered on the background assumption that they have a certain nature, can even be converted into an equivalent absolute claim of nonexistence, one that explicitly states that the purported entities have the features mentioned in the background assumption. Thus one can say that numbers construed as Platonistic entities don’t exist / are not real, or that chairs construed as mind-independent substances don’t exist / are not real, and so on. Here the corresponding quantificational claims do not seem distinct from the claims of nonexistence: there simply are no numbers construed as Platonistic entities if numbers so construed do not exist. This is anti-realism applied to someone else’s conception of a certain kind of entity, and not the kind of anti-realism that Fine takes to be up for discussion. Suppose, then, that you yourself have come to believe that numbers, should there be any, are by their nature Platonistic entities, but that you also deny that there are any Platonistic entities, just as you deny that there are any yetis. Once you are aware of this, you surely cannot rationally assert or endorse the claim that “There (definitely) is a prime number between  and ” if you take this to require genuine belief.²¹ Yet, as Fine insists, it seems that you can assert or endorse the claim. You can think that (standard) mathematics is simply too good and important a theory to give up. How can this be? The possibility of conversations held in the context of background assumptions reminds us of options other than pretense. As noted in §, we sometimes go with the flow—asserting claims that we do not strictly believe, especially if our disbelief is heterodox and our utterance allows us to express some proposition also Brock (), which presents a general argument against the so-called “phenomenological objection” to pretense versions of fictionalism. ²¹ You might at first have thought that there are no numbers construed as Platonistic entities, still believing that there are numbers but disbelieving the Platonistic nature of numbers. When you next recognize the Platonistic nature of numbers, it is surely natural to express your newly found belief as the belief that there are no numbers, not merely as the belief that numbers do not exist.

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that we do believe, or some attitude that we have adopted. If this is what is going on here, there may be no general tactic for recovering the nearby proposition that is believed absolutely, or the attitude that is genuinely adopted. If an anti-realist says “There is a prime number between  and ” in order to convey some information that she takes to be true, or to express an attitude that she takes to be apt, we may need to explore the details of her (nominalist, instrumentalist, expressivist, or other) view as well as the context of her utterance to learn what she thinks. But how then should we describe her tolerant attitude to what she doesn’t strictly believe in such cases? While there may be nothing going on that counts as pretense (assuming, as before, that pretense involves a phenomenologically thick imaginative stance), that doesn’t mean that she takes the statements as an expression of the “strict and literal truth.” Given her honest belief that there are no numbers or chairs, then, in continuing to engage in number- or chair-talk (whatever her purposes may be in doing so), she is not taking her statements to express the “strict and literal truth”— she is only doing as if they express the “strict and literal truth” for the purposes at hand. (If, on the other hand, she does believe that there are numbers or chairs, she not only does as if there are numbers or chairs—very easy, in this case—but her doing as if is backed by actual belief.) In short, it may be that we can do no better than say that these are cases in which the anti-realist does as if the statement is true and does as if she believes it, without really being committed to the truth of the statement.²² Now consider explicit claims denying or affirming existence. We think that there is much to be said for Fine’s claim that such statements sometimes, or even typically, involve us in quantification over the entities whose existence we are affirming or denying. As before, there are locutions that make such a view look almost inescapable. Having announced that ()

There are denumerably many integers < 

I might make my philosophical scruples known by adding that ()

None of those integers exists.

Here “None of the Xs” quantifies over Xs, but deliberately quantifying in this way is easy if it occurs from the perspective of the speaker’s doing as if there are Xs to quantify over. As in the case of speakers’ exploiting a pretense, such a claim can be profitably construed as an attempt to describe how things really are (setting aside the way they are represented to be in the speaker’s doing as if the world is a certain way), using a simple predicate of existence to do so. () can be represented as (0 )

9xx(8z(z≺xx $ (Integer(z) & z < )) & 8z(z≺xx ⊃ ¬Exist(z))),

where (0 ) is used to assert that

²² See also Yablo’s figuralist account of mathematical talk: the idea that mathematical language shares a lot with metaphor (e.g., Yablo ). This is a pretense account of sorts, although one that again deemphasizes the phenomenological aspects of our standard notion of pretense. (In more recent work, Yablo puts a notion of presupposition in place of the role played by pretense; see, for example, Yablo .) For doubts about the distinction between “doing as if ” a claim is true and being genuinely committed to its truth, see Horwich (). For a response, see Daly ().

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       - (A) [Apart from my doing as if there exist all the integers < ] it is not the case that there exist any of these integers. (That is, ¬9xx(8z(z≺xx ! (Integer(z) & z < ))).) By contrast, saying that All of the integers <  exist

()

is to utter a sentence that can be represented as (0 )

9xx(8z(z≺xx $ (Integer(z) & z < ))) & 8z(z≺xx ⊃ Exist(z))),

where this is used to assert that (A) [Apart from my doing as if there exist all the integers < ] there exist all the integers < . (That is, 9xx(8z(z≺xx $ (Integer(z) & z < ))).) This mirrors the manner in which in our earlier case the speaker was able to describe how things really are with her statement that “Those people do (not) exist.” Here, then, we have a general alternative to the partitive account, what we might call an as-if-ist account of existence-affirming or -denying locutions according to which the primary work of identifying ontological commitment is assigned to quantifiers. (To the extent that it doesn’t insist on any special phenomenology, we can take such an as-if-ist account to be more general than a pretense account, and hence to include such an account.)

 Really? We have defended an augmented quantificational thesis. What, then, of Fine’s special predicate “real” and its associated operator R? Would it do equally well, or better, at signaling genuine ontological commitment? At the very least, we think this has not been shown. The problem is not that we have no grasp of a concept of the real, or of how things really stand. The problem, rather, is one of identifying and articulating the philosophically appropriate such concept. Some of the most familiar concepts of the real, it turns out, are bereft of the kind of ontological or metaphysical significance Fine is looking for, despite offering initial hope. Thus consider a typical statement of nonexistence that is formulated in terms of a “really” operator, and where at first glance the operator serves to accord the sentence ontological significance: ()

Barchester Towers doesn’t really exist / Barchester Towers isn’t real.

But seeing such an ontological role for “really” or “real” in this kind of sentence is to misunderstand its function. For sentences of this type are naturally juxtaposed with sentences that declare what the entity really is like (in order to elucidate features that show why it doesn’t really exist): () Barchester Towers is really just a fictional place / In reality, Barchester Towers is just a fictional place Similarly, we can say “Phlogiston doesn’t really exist; it is really just a failed posit of eighteenth-century chemistry,” “The little green man at the bottom of the garden

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doesn’t really exist; he is really just a figment of your imagination”; and so on. Only the most literally minded will think that “really” and “real” have any serious ontological or metaphysical role to play in such statements. To understand Fine’s project, then, we should move away from such commonplace occurrences of words like “real” and “really.” We should instead consider Fine’s story about the way that the concepts designated by these words in his own account function in our thought. It is significant, however, that in QO Fine offers no analysis of these concepts. As we saw earlier, he treats them as primitive: I myself do not see any way to define the concept of reality in essentially different terms; the metaphysical circle of ideas to which it belongs is one from which there appears to be no escape. (QO, )

Fine is unperturbed by the circle, declaring that we seem “to have a good working grasp of the notion [of reality]. We know in principle how to settle claims about the constitution of reality even if we have difficulties in settling them in practice” (ibid.) We doubt this. True, one can apparently insist intelligibly that the objects we normally regard as physical, for example, are in fact nothing more than mental or phenomenal objects, or are complexes rather than simples, and to that extent are unreal. It also seems as though one knows what an argument that confirms or disconfirms this contention would look like, even if it is unclear whether such an argument could ever be available to us. Such an argument would almost certainly tell us something important about the way things are. We suspect, however, that once such arguments are in, and even supposing that they confirm some sort of idealism or phenomenalism, many folk and many philosophers will feel uncomfortable ruling on which of the following is the upshot: (a) that there are physical objects, including chairs, but all the facts about them are grounded solely in facts about the mental or phenomenal; (b) that there are no physical objects and hence no chairs (since any chair is physical), though mental or phenomenal facts make it seem to us that there are these objects; or (c) that there are no physical objects, but there are chairs—mental or phenomenal ones. Maybe some of us have deeply engrained intuitions about which way to go, but some folk and some philosophers may suspect that the choice among these options is merely a matter of bookkeeping and that couching the original question in terms of what is real misleads us into assuming that something more significant is at stake. If this worry about the debate is legitimate, any working grasp of the notion of reality seems not to get us very far, in at least some central cases. Fine’s own illustration of our alleged intuitive grasp of the concept of reality raises even more problems than the one we have just outlined. Democritus thought that there was nothing more to the world than atoms in the void. I take this to be an intelligible position, whether correct or not. I also assume that his thinking that there is nothing more to the world than atoms in the void can be taken to be shorthand for

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       - there being nothing more to the world than this atom having this trajectory, that atom having that trajectory, . . . or something of this sort. I assume further that this position is not incompatible with his believing in chairs and the like. To be sure, the existence of chairs creates a prima facie difficulty for the view but as long as the existence of chairs can be seen to consist in nothing more than atoms in the void, the difficulty will have been avoided. I assume finally that had he been prepared to admit that there was nothing more to the world than atoms and macroscopic objects, then he would not have been prepared to admit that there was nothing more to the world than atoms. But someone who is willing to go along with me so far will thereby have endorsed a metaphysical conception of reality. (QO, )

We object, first, that people who understand what it means to claim that there is, in reality, nothing more than atoms in the void might nonetheless wonder whether anything ontological turns on the question of whether we want to include, as denizens of reality, some or all of the ordinary things that one seems to get when one sticks atoms together. That is, if I claim that chairs are real, because there are atoms arranged chairwise, and you claim that chairs are not real, because there are only atoms arranged chairwise, people might still be unclear whether there is any metaphysically revealing difference between our stories. Such puzzlement undermines the claim that we know what it takes for somebody to endorse a particular metaphysical conception of reality.²³ We object, second, that some people will surely deny that the atomism of Democritus, as elaborated by Fine, “is not incompatible with his believing in chairs and the like.” How, those people might ask, can somebody believe in things that one assumes are not real? Even if somebody can, many will find it counterintuitive. This too is a reason for doubting that the folk generally share Fine’s sort of intuitive grasp of the concept of reality. Even if most of us shared an intuitive grasp of the concept of reality, that grasp would probably not, contra Fine, suffice to inform philosophers’ investigations of what counts as real in any significant way. Maybe the folk do regard it as obvious that certain things are real and maybe they have a systematic way of sorting the real from the unreal, but this does not stop philosophers from disagreeing with one another about what is real. The reality of chairs and other would-be composite physical entities is not taken for granted by all philosophers in our post-Democritian, mereologically savvy, intellectual environment. It is likewise not obvious that there are purported entities whose unreality is taken for granted by all philosophers. Perhaps the most promising candidates are the would-be referents of empty designators that occur in ephemeral discourses like the following: GUILDENSTERN: We’d better not take the path through that thicket, or the Branagh of Hurk will leap out and ask us an empty question. GLADYS:

The What of What will do what?

GUILDENSTERN: that’s all.

Sorry. I was making stuff up. It looks dodgy in there,

²³ This kind of doubt lies behind the theory of quantifier variance that has been advanced by Eli Hirsch. See, for example, Hirsch (, ).

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But some of those who affirm that Hermione and other mainstream entities of fiction are real might be driven by their ontological assumptions to countenance the Branagh of Hurk’s reality too.²⁴ It is far from clear what would settle, once and for all, the matter of which things exist and which things do not. Suppose we knew everything there is to know about the structure of the universe, the distribution of matter and energy throughout it, any spiritual, supernatural, or nonphysical, mental aspects of it and the roles they play, the relationship between normative, semantic, and natural facts and what is grounded in what. Suppose that we even knew all of this from every possible external and every possible imbedded perspective, so that the phenomenology of consciousness was no mystery. Still, without an analysis of reality, of a kind that currently eludes us, we could have disputes about what, in reality, is the case. We could disagree about whether numbers, propositions, and centers-of-gravity are mere instrumental conveniences and, if so, whether this disqualifies them. We could argue about the ontological credentials of fictional events, events generally, things that are mere parts or composites of other things, free will, properties, and possibilia. Even a complete description of the universe in a comprehensible first-order language would hardly shut us up, or down. No matter what was in its intended domain of quantification, we could speculate about whether there was a perfect translation into a first-order language with a different intended domain. So there is good reason to doubt not only Fine’s allegation of a robust intuition about what is at stake when we pose questions about what is real, but also the claim that there are any paradigms of realness or non-realness that might anchor or steer our ontological inquiry and from which lessons could be drawn for the debates that ontologists really engage in. Philosophers must try to demystify claims about what is real and there is too much mystery left if the concept of reality is taken to be primitive.

 Conclusion We have considered Fine’s case for his partitive account, focusing on his reasons for rejecting quantification as the principal vehicle for encoding ontological commitment, and showing why, despite sharing some of his central concerns, we are not persuaded. The problems we have discussed would not matter if Fine’s concerns were different from those of run-of-the-mill ontologists, but he is clear that they are not. Thus when criticizing Quine’s approach to ontology, he remarks that even after it has been judged that numbers are indispensable for the purposes of science, “it will still be in order for the anti-realist to insist that numbers (and perhaps theoretical entities in general) do not really exist—that we talk that way, and even correctly talk that way, despite the fact that there is no realm of objects ‘out there’ to which our talk corresponds” (QO, ). We find Fine’s construal of such a claim counterintuitive.

²⁴ Especially because it would also have been natural for Guildenstern to say “The Branagh of Hurk doesn’t exist,” and one might be led by one’s semantics of negative existentials to conclude that the Branagh of Hurk does after all exist (although not as a person).

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       - Given that ordinary number talk assumes that numbers are mind-independent,²⁵ we think that if we talk correctly when we say that there are numbers it has to be true that there is a realm of objects “out there” to which our talk of numbers corresponds. Numbers in that case form part of reality, even if not part of fundamental reality, and that seems reality enough for most of us. In short, we reject Fine’s estrangement of ontology from ordinary talk and thought about what there is, and urge that the centrality of quantification should be maintained. But we have also agreed that a quantificational story cannot succeed on its own. We noted that, for various pragmatic reasons, people often speak loosely, rely on shared presuppositions, or, most interestingly, behave as if certain claims are true, all the while not believing them. These are facts that we need to appeal to in order to supplement a quantificational story. The result explains the data that impresses Fine, without, as it seems, the need for Fine’s primitive notion of reality.²⁶

References Brock, S. () “The Phenomenological Objection to Fictionalism,” Philosophy and Phenomenological Research. http://onlinelibrary.wiley.com/doi/./phpr./pdf Brogaard, B. () “Sharvy’s Theory of Definite Descriptions Revisited,” Pacific Philosophical Quarterly (): –. Clark, M., and D. Liggins () “Recent Work on Grounding,” Analysis. doi: ./analys/ ans Daly, C. () “Fictionalism and the Attitudes,” Philosophical Studies : –. Devitt, M. () Realism and Truth (nd ed.), Princeton University Press. Chalmers, D. () “Ontological Anti-realism,” in Chalmers et al. (), –. Chalmers, D., D. Manley, and R. Wasserman (eds.) () Metametaphysics: New Essays on the Foundations of Ontology, Oxford University Press. Field, H. () Science without Numbers, Blackwell. Fine, K. () “First-Order Modal Theories,” Studia Logica : –. Fine, K. () “First-Order Modal Theories I—Sets,” Nous : –. Fine, K. () “First-Order Modal Theories III—Facts,” Synthese : –. Fine, K. () “The Question of Realism,” Philosophers’ Imprint : –; reprinted in Individuals, Essence and Identity: Themes of Analytic Philosophy (eds. A. Bottani, M. Carrara, and P. Giaretta), Kluwer, , –. Fine, K. () “The Question of Ontology,” in Chalmers et al. (), –. Fine, K. (a) “A Guide to Ground,” in F. Correia and B. Schneider (eds.), Metaphysical Grounding: Understanding the Structure of Reality, Cambridge University Press. Fine, K. (b) “What is Metaphysics?,” in Contemporary Aristotelian Metaphysics (ed. T. E. Tahko), Cambridge University Press, –. Hirsch, E. () “Against Revisionary Ontology,” Philosophical Topics : –. Hirsch, E. () “Ontology and Alternative Languages,” in Chalmers et al. (), –. Horwich, P. () “On the Nature and Norms of Theoretical Commitment,” in P. Horwich (ed.), From a Deflationary Point of View, Oxford University Press, –.

²⁵ Ordinary number talk allows, for example, that the number of species of mammal that existed before the evolution of humans is far greater than a thousand. ²⁶ Thanks to Glen Pettigrove and Justine Kingsbury for useful comments on an earlier draft.

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



Jackson, F. () From Metaphysics to Ethics: A Defence of Conceptual Analysis, Oxford University Press. Kroon, F. () “Descriptivism, Pretense, and the Frege-Russell Problems,” Philosophical Review  (): –. Lewis, D. () “David Lewis—Reduction of Mind,” in S. Guttenplan (ed.), A Companion to the Philosophy of Mind, Blackwell, –. Linnebo, Ø. () “Plural Quantification Exposed,” Noûs (): –. Linnebo, Ø. () “Plural Quantification,” The Stanford Encyclopedia of Philosophy (Summer  edition), Edward N. Zalta (ed.). http://plato.stanford.edu/archives/sum/ entries/plural-quant/ Price, H. () Facts and the Function of Truth, Basil Blackwell. Price, H. () “Naturalism without Representationalism,” in D. Macarthur and M. de Caro (eds.), Naturalism in Question, Harvard University Press, –. Quine W. V. O. () “On What There Is,” Review of Metaphysics : –; reprinted in From a Logical Point of View, Harvard University Press, , –. Schaffer, J. () “On What Grounds What,” in Chalmers et al. (), –. Walton, K. () Mimesis as Make-Believe, Harvard University Press. Walton, K. () “Metaphor and Prop Oriented Make-Believe,” European Journal of Philosophy : –. Yablo, S. () “Go Figure: A Path through Fictionalism,” Midwest Studies in Philosophy : –. Yablo, S. () Things: Papers on Objects, Events, and Properties, Oxford University Press.

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 Beyond Reality? Philip Percival

My focus is the divide between reality and that which is beyond reality. On what I call an “all-encompassing” view of reality, this is the divide between everything and nothing: reality encompasses everything, and beyond it there is nothing at all. Opposed to the all-encompassing view is what I call a “restriction” view of reality: reality is coincident with some kind of restriction on, or partition of, what there is; it is not the case that what resides beyond reality is nothing. I have two main aims: to first classify restriction views of reality (§), and then to assess a species of the restriction view that pertains to time and modality (§).

 A Classification of the Restriction View of Reality There are different species of the restriction view of reality. I will structure them around what Kierland and Monton (: ) call the “Reality Principle”: (RP)

Reality consists, and only consists, in things and how things are.

Some might accept the Reality Principle but object that it is misleading: its talk of both things and how things are implies that there is significant contrast when in fact this is not so. An extreme criticism along these lines would be to the effect that “how things are” is superfluous because, trivially, how things are is among the things: the phrase “how things are” is a nominal, and so is either empty (and therefore superfluous) or else non-empty on account of naming one of the things (and therefore superfluous). In response, I agree that “how things are” may be read as a nominal. This fact, and the thrust of the objection, is captured in an utterance such as “There are two things that really annoy me: one is Gordon Brown, the other is how things are.” I take it, however, that the intended reading of the Reality Principle invokes a different use of “how things are.” On this alternative usage, “how things are” is more akin to propositional variable. This use is exemplified by the following exchange: A: B:

What do you mean by “how things are”? I mean grass is green, and Gordon Brown is Prime Minister, and so on.

Philip Percival, Beyond Reality? In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Philip Percival. DOI: 10.1093/oso/9780199652624.003.0003

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  ?



Assuming this alternative reading to be intended, one acquainted with the writings of Arthur Prior and amused by his neologisms might be tempted to rewrite the Reality Principle as: (RP*)

Reality consists, and only consists, of things and thethers.

Nevertheless, it might still be objected that the Reality Principle remains misleading even when “how things are” is given some such non-nominal interpretation. This objection has two forms, depending on whether it is talk of “things,” or talk of “how things are,” that is taken to be superfluous. To facilitate my discussion, I shall take the matter of “things” to be the concern of ontology, and the matter of how things are to be the concern of what I shall call “ideology.” I shall use “how things are” non-nominally. Incorporating ideology into ontology: truthmaker maximalism—It might be thought that from the viewpoint of truthmaker maximalism, the Reality Principle’s talk of “how things are” is superfluous: according to truthmaker maximalism, ontology incorporates ideology because the things determine how things are: the things include e.g. states of affairs, things that embody how things are. In my view truthmaker maximalism is more pain than gain. It does not eliminate the necessity of conceptualizing reality in terms of how things are. An ontology of states of affairs might seem like a reduction of ideology to ontology, but it is not: it is a reduction within the category of ideology. It is the reduction of how things are in general to how things are in particular vis-à-vis what exists. This fact is masked by those truthmaker maximalists who maintain that their theory invokes a primitive relation R such that both (TM) (**)

Necessarily, if

is true then for some x, R(

, x). For all x: necessarily, if R(

, x), then entails

is true.1

The primitive relation R is supposed to hold between true propositions and entities. Clearly, if (TM) is true then ideology is determined by ontology. The need for (**) is suspicious, however. We must ask, how come (**) is true? How does the actual obtaining of a relation between two entities guarantee that necessarily, one is true if the other exists? Surely, it cannot be the actual obtaining of this relation that guarantees this much; rather, it is the entities themselves that guarantee it. R is therefore entirely superfluous. We could have instead: (P)

Necessarily, if

is true then for some x, entails

is true.

If (R) is now defined as holding between a true proposition and an entity iff the existence of the entity entails the truth of the proposition, given (P) we can recover both (TM) and (**). Of course, the objection is that the truthmaking relation cannot be defined in terms of entailment in this way. If this objection is conceded, however, the correct response is not to resort to a primitive (true) proposition–object

¹ Obviously (TM) and (**) are independent, although Rodriguez Perreira () writes as if (TM) entails (**).

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   relation; rather, it is to replace “entails” by a primitive proposition–proposition relation R* to obtain: (P*)

Necessarily, if

is true then for some x, R*(

, ).

i.e. where, ex hypothesi, “R*(

, )” entails “ entails

is true.” While the external reduction of ideology to ontology is an illusion, the internal reduction of propositions to propositions specifically about existence is no advance. For one thing, at least for some species of the doctrine the entities invoked are highly obscure: for example, states of affairs qua values of first-order variables are constructed by means of non-mereological composition. For another, existence of the kind Truthmaker Maximalists require—is at best obscure. (As Williamson () has pointed out, to avoid the absurdity that every truth is necessary, the truthmakers of contingent propositions must be contingent existents. But it is not possible for a thing to fall outside the domain of unrestricted quantification!²) Reality without things? Once it is recognized that the how things are component of the reality principle is non-nominal, and as such not superfluous, one might object that the reality principle is misleading from the other direction. For it can appear that nothing substantial would be omitted were the reality principle to have taken the form: (RP*)

Reality consists, and only consists, in how things are

It would seem that some thing might be omitted from how things are only if some thing makes no contribution whatsoever to how things are. That much seems impossible, however. No thing could be such that how things are is entirely insensitive to how it is. Accordingly, someone who advocates (RP*) might well think that (RP) is true but misleading. Kierland and Monton’s preference for (RP) above the simpler (RP*) is therefore puzzling. My suspicion is that they prefer (RP) because, somewhat paradoxically, they mean to exploit the fact that adding the further qualification “things” encourages a narrower interpretation of “how things are” than would otherwise be natural. The most natural interpretation of “how things are” (and therefore of (RP)) takes “how things are” to be captured by, or, even, to consist in, that which could be judged truly. The most natural interpretation of “how things are” in the context “things and how things are,” however, takes “how things are” to be captured by, or, even, to consist in, that which could be judged truly of some thing or things. This latter interpretation, which I suspect Kierland and Monton intend, is narrower than the former to the extent that that which could be judged truly includes, but is not exhausted by, that which could be judged truly of some thing or things: not all judgement is de re judgement (singular or plural)! On this interpretation, then, (RP) amounts to the thesis: (DR) Reality consists, and only consists, in things and how the properties and relations of those things stand. ² For an argument in addition to Williamson’s: Every proposition that is singular with respect to x strictly implies its particular quantification with respect to x: for the latter weakens it (see Percival ).

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  ?



Later, we will identify interpretations of the Reality Principle that are even narrower than (DR). There are two ways reflection on the Reality Principle might lead one to the view that there are matters beyond reality. First, one might accept that there is nothing beyond things and how things are, but reject the Reality Principle as embodying an inflated view of reality: reality is confined to some element of a partition of things and how things are. Second, one might accept the Reality Principle, but reject an all-encompassing view of things and how things are.

. First route to a restricted view of reality: rejection of the Reality Principle The first line of thought denies the Reality Principle on the grounds that: (a) Reality does not encompass things and how things are in their entirety: things and how things may be partitioned into reality and that which is beyond reality. The partitioning of things and how things are may be a partitioning of ontology (things) or ideology (how things are), or both.

..      At least with respect to ontology, there is a case for the view that rejection of the Reality Principle is common sense. Isn’t it common sense to hold that the things include the fictional (fictional characters, and how these characters are), and that what is fictional is not part of reality? According to common sense, the fictional is contrasted with reality, not included within it! And of course there is an ontological tradition that follows suit. The most prominent strand in this tradition is Meinongian. It divides the things in the domain of unrestricted first-order quantification into the existent and the nonexistent, and typically equates the existent–nonexistent distinction with the real– unreal distinction.³ A less prominent strand in the tradition identifies nonexistence with exclusion from the domain of unrestricted first-order quantification. For example, Nathan Salmon () holds that notwithstanding the fact that Socrates is an object of singular thought, Socrates escapes unrestricted quantification: Socrates is a thing such that no thing is identical to it. In response, Nathan Salmon’s viewpoint strikes me as absurd. If Socrates is an object of singular thought then Socrates must belong to the domain of unrestricted quantification. Every proposition that makes some claim regarding Socrates may be unrestrictedly quantified (particularly, and with respect to Socrates) so as to generate a proposition that makes that very same claim not of Socrates, but of some object or other. The latter proposition is weaker because it involves removing some information—namely, the identity of the object quantification is effected with respect to. It is therefore entailed by the former: every proposition entails its weakenings. Accordingly, if any proposition is true, so ³ Cf. Routley ; Parsons ; Fine , ; Routley’s position is subtle: in at least one publication it is not the view that there are nonexistent things, since the locution “there are” is viewed as restricted quantification, i.e. the restriction being to things that exist.

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   too is . But of course, as Salmon is keen to point out, some proposition is true. To my mind, Salmon’s alternative is a desperate attempt to recover a presentist viewpoint in the face of Quinean scruples about existence and recognition of Kripke’s observations about names and singular thoughts. Salmon learnt from Kripke’s knee that we are able to have singular thoughts about Socrates and other past objects, and he learnt—or, as I would prefer, takes himself to have learnt—from Frege/Russell/ Quine that existence is membership of the domain of unrestricted first-order quantification (better, that “exists” is a predicate such that any member of the domain of unrestricted quantification satisfies this predicate automatically). Having conceded this much, then, the only way of preserving the presentist credo that Socrates does not exist is to adopt the absurdity that Socrates enters singular thought but escapes membership of the domain of unrestricted quantification. This reasoning shows that Salmon’s strand of the doctrine of nonexistent/unreal things is untenable, and that any development of this doctrine should be Meinongian: no values of first-order variables fall outside the domain of unrestricted quantification, and the real–unreal (existent–nonexistent) distinction must be made within that domain, i.e. by means of a predicate “real” or “exists” that some but not all things are taken to satisfy. Any partition of ontology by means of a predicate “real” or “exists” would surely lead to a partition of ideology: for if e.g. Socrates is unreal, how could reality extend to how Socrates is? Call this a Meinongian route to a restricted view of reality. Standard formulations of this route either take “real/exists” to be a primitive, or else define “real/exists” in such a way that it is trivial that the entities deemed to be unreal/ nonexistent do not satisfy the predicate “real/exists.” I have no sympathy for van Inwagen’s () extreme anti-Meinongian line to the effect that there is no such sense of “exists.” The term “exists” is sufficiently vague/ unclear for explication of it (in Carnap’s sense) to be legitimate, and a Meinongian definition may well be a permissible explication.⁴ However, anti-Meinongian explications, i.e. such as the definition “x exists” = “x is identical to something,” are equally legitimate. But I am skeptical of the alternative line that “exists” or “real” express clear primitive notions with respect to which inquiry into what is real/exists in this sense—or even an opinion as to the matter—is profitable. This is not to say that I am paid-up member of the Quinean fan club, however. Like Frege, contemporary Quineans take it for granted that if “x exists” is defined, using unrestricted quantification, as “something is identical to x,” then it is trivial that everything in the domain of unrestricted quantification satisfies this predicate. I do not take it for granted, since I think the question as to whether the domain of quantification is included within the domain of predication remains open. It is possible—epistemically, and for all any of us know metaphysically—that the domain of quantification is broader than the domain of predication.⁵

⁴ Fine’s () neo-Meinongian analysis of “exists” presupposes a partitioning of how things are, and therefore cannot be employed to motivate a partioning of how things are (see below). ⁵ See Percival .

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  ?



..     Perhaps one might read “how things are” non-factively. After all, in the Tractatus (.; Wittgenstein ) the “general form of a proposition is: This is how things stand.” On this reading, one might respond to the question “What does ‘how things are’ mean?” by saying: “Here are two examples of how things are: ‘first, Obama is President, and second, Obama is not President.’ ” The Reality Principle is absurd, however, on a non-factive interpretation of “how things are.” On the intended interpretation, how things are is factive: how things are is contrasted with how things are not. I might begin the task of delineating reality by saying “Obama is President,” but not by saying “Obama is not President.” A relevant partitioning of how things stand, then, is a partition that either is, or at any rate runs parallel to, a partitioning of the true thinkables (assuming that how things are does not extend beyond the thinkables). Recently, Kit Fine () has effected such a partition by extending the ethos of Meinongian ontology to the sphere of the ideological. According to the Finean ideology that results, analogously to the Meinongian real–unreal partition among things, which is effected by a predicate, an “in reality”–“beyond reality” partition amongst how things are (factive) that is effected by means of an operator. This operator is “it is constitutive of reality that . . . ” It partitions how things are by partitioning the truths into the truths that embed in the operator truthfully and the truths that do not embed in the operator truthfully. On the all-encompassing view of reality, every truth will embed truthfully into the operator. On the restriction view, however, some will not. For example, Fine’s operator might be invoked in pursuit of a metaphysical anti-realism about morals. The metaphysical anti-realist about morals might concede that how things are includes how things are morally—e.g. that Hitler was wicked—or, even, how things are with respect to moral properties—and yet exclude such matters from reality on the grounds that among the substantial moral truths, no such truth p is such that it is constitutive of reality that p. So, on this view, for example, Hitler was wicked but it is not constitutive of reality that Hitler was wicked. That is, on this view while the matter of whether Osama bin Laden is wicked calls for judgement (as to truth or falsity!), the judgement called for does not engage reality. An even more radical alternative might result from reflection on Aristotle’s suggestion that to speak truly is to say of what is that it is, or to say of what is not that it is not. This might tempt one to partition how things are into what is and what is not, i.e. with reality being confined to the former. On this viewpoint, negative truth extends beyond reality! For example, how things are includes both Obama is President and Hilary Clinton is not President, but only the former embeds truthfully in “it is constitutive of reality that.” This route to a restricted view of reality is problematic, since it is often hard to determine whether a truth pertains to what is, or to what is not. For example, is the property being bald positive or negative? It is instructive to consider the relation between Finean ideology and Meinongian ontology. Finean ideology does not entail Meinongian ontology: one might suppose that everything exists, and that among the truths of the form “F’s exist,” every such truth p is such that, it is constitutive of reality that p, and yet e.g. be a Finean anti-

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   realist about morals. Equally, Meinongian ontology might repudiate Finean ideology by rejecting as unintelligible the reality operator. On the other hand, Finean ideology offers a reduction of Meingonian ontology, i.e. just in case whenever the Meingonian holds that “F’s do not exist” the Finean holds that this much is to be explained in terms of the ideological partioning of how things are. Initially, Fine’s own conception of the relation between ontology and ideology was left somewhat unclear, but more recently he has offered an explanation of this kind. First, “F’s are real” is held to be of the universal form “All F’s are G.” Second, “x is real” is explicitly defined as “for some φ, it is constitute of reality that φx.” Third, “F’s exist” is defined (somewhat tentatively, in the narrow sense) as “F’s are real.”⁶ Finean ideology is two letters distant from Meinongian ontology: it requires a move from a predicate “real” of things to an operator “really” (in effect) on truths. As such, it is two letters away from Quinean contempt: just as for Quine the question “what exists?” has an immediate and trivial answer—“everything”—so too for Quine* does the question “how are things really?” have an immediate and trivial answer— “how things are.” Quinean* contempt for Finean ontology lacks the basis of Quinean contempt for Meinongian ontology. The latter defines existence as identity to something, and then presumes—not unnaturally!—that everything satisfies this predicate. I cannot think of any analogous argument the Quinean* might pursue other than a trivial deflationary complaint: “really” is defined by “really p” =df. “p.”⁷ Of course, the Quinean’s insistence that when the Meinongian asks “What exists?” the answer is “everything” involves no commitment to an inflated ontology: it is consistent with rejecting as false many Meinongian assertions “Some things are F’s.” Similarly, the Quinean* insistence that when the Finean asks “how are things really?” the correct answer is “how things are” involves no commitment to an inflated realism: it is consistent with rejecting as false many “realist” claims that how things are includes e.g. facts about who is wicked. A deflationary view of “really” need not lead to a deflationary view of how things are. Dismissing the objection that the meaning of “it is constitutive of reality that” is such that it is trivial that every truth embeds truthfully within it prompts a complaint from the other direction. If the operator “it is constitutive of reality that” is primitive, the question arises as to how opinion is to be formed regarding its application. How, in the first instance, is one to decide between the all-encompassing and restriction views of reality? And, if one decides in favor of the restriction view, which species of it should one adopt? Fine suggests that the keys to such matters are the equally primitive notions of factuality, fundamentality, and, most crucially, ground (these notions likewise being expressed, respectively, by sentential operators, “it is factual ⁶ The simplest way of modifying Quinean ontology so as to incorporate Finean ideology is to define “F’s are real” by “it is constitutive of reality that there are F’s” and to identify being real with existing. But this route is precluded by Fine’s doctrine that such quantificational truths are not constitutive of reality (amongst other considerations he advances). He admits that “exists” is ambiguous, and some of its uses are broader than the narrow one he defines. ⁷ Perhaps this is unfair: doesn’t “it is constitutive of reality” have much more force than “really”? But “constitutive” might surely be replaced by “part.” There seems little difference between “it is part of reality” and “it is really the case that.” And what is the difference between the latter and “really”?!

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  ?



that,” “it is fundamental that,” and “because”). How things are in some respect is not constitutive of reality if how things are in that respect is not factual; nor (with perhaps a few exceptions) is it constitutive of reality if it is not fundamental. Opinion is to be formed regarding whether how things are in some respect factual and fundamental by consideration of what grounds what. Unlike classical truthmaking, grounding is not standardly cross-categorical. That which is classically truthmade is a proposition, whereas that which classically truthmakes is ordinarily something that is not a proposition. In contrast, grounding occurs within the category of propositions: for example, if the mental is grounded in the physical, then Bill has a headache because BLAH, where BLAH articulates how things are in some physical respect. I am prepared to run with Fine’s notion of “ground.” I am very skeptical of his idea that what is grounded in the factual—the factual but non-fundamental—is not constitutive of reality. My skepticism in this respect will emerge when we consider rationales for adopting a restricted view of reality that endorses the Reality Principle.

. Second route to restricted reality: beyond things and how things are A second route to a restricted view of reality accepts the Reality Principle but rejects the all-encompassing view of reality on the grounds that the matter of things and how things are is itself not all-encompassing. According to this line of thought, the Reality Principle is correct, but (b) Things and how things are is not all-encompassing: in addition to things and how things are is that which resides beyond things and how things are. That is, one might hold that reality comprises things and how they are but that there are matters that lie beyond this much. What could motivate this restricted view of things and how things are? Again, this question falls into two parts, depending on whether the restriction is on things (ontology) or how things are (ideology).

..   How might ontology extend beyond the things there are? Ontology might extend beyond the first-order realm of things. In addition to the values of first-order variables we have the values of second-order variables (as illustrated by the inference from {Brown is wicked, Blair is wicked} to “there is something Brown and Blair both are”).⁸

..     In a trivial sense, how things are is not all-encompassing. For example, on one interpretation of “how things are,” how things are contrasts with how things are believed to be by Osama Bin Laden. But this is immaterial: our interest is confined to notions of “how things are” which might plausibly be taken to coincide with reality

⁸ Note: Has anyone who has extended ontology in this way taken the extension to be into the unreal? There is a problem about even asking the question, since “unreal” is a first-order predicate.

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   and which are restricted. “How things are believed to be by Osama Bin Laden” is ambiguous between the contents of Bin Laden’s beliefs and the fact of his believing that content. On the latter reading, reality includes how things are believed to be by Bin Laden; on the former reading, how things are believed to be by Osama Bin Laden partitions into (a part of) how things are and for all we have yet been told, nothing at all (since it is just a lot of falsehoods). Neither reading gives a defensible version of the Reality Principle. Some contrasts with how things are, then, are of no interest to us: they do not point to matters beyond reality. But what of the distinction between truth and falsity that is raised by consideration of the trivial contrast between how things are and how things are believed to be by Bin Laden? On one reading, this is parallel to (at least a restriction of) the distinction between how things are and how things are not. Might how things are not be a matter beyond reality? Might it not be that beyond reality lies falsehood? This idea might seem absurd, but I take it to be the view of Prior and, I think, McDowell (). In Prior’s view (and McDowell’s) reality is constituted by the facts and the facts are the true propositions. They deny that reality thus conceived is all-encompassing, however: in addition to the true propositions there are the false propositions! (Their view does not reduce to the ontological species of the first route: the true–false distinction amongst propositions is not tantamount to a real–unreal distinction amongst things; on their view propositions/thinkables are not things.) If propositional quantification is allowed, we might articulate this position as: for some p, it is not the case that it is constitutive of reality that p, and for all p, it is constitutive of reality that p iff p. Of course, the idea that reality consists in the true propositions is contrasted with a correspondence theory of truth i.e. according to which reality “corresponds to” the true propositions (that are themselves nevertheless part of reality). It should be noted that while the idea that reality consists in the true propositions might be allied with the view that reality is restricted, it is consistent with the view that reality is all-encompassing: to obtain the latter view, one need only hold that false propositions are nothing! If Bigelow is to be relied upon, it would seem that this is the Stoic view: for the Stoics, only true propositions “exist” (Bigelow : ). While the Prior/McDowell view is interesting—indeed, quite attractive—its conception of that which is beyond reality downplays its interest: trivially, on their view that which is beyond reality is not nothing, but it is not a matter for true judgement. On an alternative interpretation of “how things are not,” however, how things are not is a matter for true judgement. This interpretation leads to a restricted view of things and how things are that is usefully contrasted with a species of the view that reality is restricted that rejects the reality principle. Whereas one species of the first route to the view that reality is restricted partitioned how things are into what is, and what is not, one species of the second route takes how things are to contrast with how things are not. On this conception, only positive truths report how things are (and hence reality); in so far as they concern how things are not, negative truths are concerned with matters beyond reality. The problems that beset this alternative are similar to the problems that were seen to beset the view that the reality principle is false because reality is confined to how things are positively. The intended contrast between how things are and how things are not is only clear to the extent that the contrast between

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  ?



positive truths and negative truths is clear. At the very least, however, work needs to be done here: is instantiation of the property of being bald a matter of how things are, or of how things are not? It is the contrast between things and how things are and the past concerning things and how things were and how things might have been that is my main concern. Some species of presentism contrast things and how things are with the past about things and how things, and yet endorse the reality principle. Similarly, some species of actualism contrast things and how things are with the possibilities of things and how things are, and yet endorse the reality principle. Relatedly, in so far as one contrasts e.g. the past about things and how things were with things and how things are, one will of course contrast it with things and their instantiation of properties and relations. A fortiori, one will also contrast it with what Lewis terms being: things and their fundamental properties and relations. It should be noted that although the restriction view that things and how things are i.e. according to which things and how things are is not all-encompassing can be combined with the Reality Principle, they need not be. If they are so combined, they lead to a restricted view of reality. But a restriction view of things and how things are can be combined with the no-limitation view of reality once the Reality Principle is rejected.

. Matters beyond reality: Do they supervene on reality? Are they grounded in it? The restriction view of reality has been seen to split according to whether the restriction is within the sphere of how things are, or confined to it. It also splits along another axis, according to the account given of how supra-reality “relates” to reality. Where supra-reality is confined to the sphere of true judgement, two “relations” are especially germane: (i) supervenience and (ii) grounding. The most conservative view holds that supra-reality supervenes on reality and is grounded in reality. I take it that this is the view of Kit Fine (at least as far as the factual is concerned). Compare Fine’s insistence that the nonexistent supervenes on the existent. Obviously, since everything that is grounded in what is fundamental supervenes on the fundamental, the conservative view of what is factual results if (a) whatever is fundamental and factual is included in reality, and (b) something factual that is not fundamental is not included in reality. But Fine believes (a) and (b). (Indeed, he is inclined to believe that reality does not extend beyond what is fundamental.)⁹ The most radical view holds that supra-reality is not supervenient upon (and hence not grounded in) reality. An intermediary view between these extremes holds that supra-reality supervenes upon but is not grounded in reality. How should the species of presentism and actualism just canvassed be located within these alternatives?

⁹ Of course these ideological claims, to the effect that truths that are not constitutive of reality supervene on those which are, run entirely parallel to his very early ontological claims, to the effect that the nonexistents supervene on the existents.

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   Whether the past can supervene on reality as conceived by restricted presentism is moot. Regarding the present things, the supervening of the past about those things on how things are is facilitated by the admission of such properties as having been sun-burnt etc. But what of aspects of the past that appear not to be correlated with past-tensed properties of present things, i.e. because it concerns things that do not presently exist?

 Certain Restriction Views of Reality Pertaining to Time and Modality . The relevant species of presentism and actualism The species of presentism and actualism I wish to focus on both endorse the Reality Principle, but they then claim that what so doing amounts to is as follows: (PR)

(i) The things are exactly the present things; (ii) How present things are coincides with how the present things presently are; (iii) How present things presently are coincides with the matter of which properties and relations present things presently instantiate. (AR) (i) The things are exactly the actual things; (ii) How the actual things are coincides with how the actual things actually are; (iii) How actual things actually are coincides with the matter of which properties and relations actual things actually instantiate. They then hold, further, that the matter of how things are (thus conceived) is not allencompassing: it does not exhaust matters of judgement. For: (PB) (MB)

Beyond things and how things are (thus conceived) resides the past about things and how things were. Beyond things and how things are (thus conceived) resides the possibility of things and how things might have been.

My intention is that (MB) is a modal analogue of (PB). That is, it is to be understood by analogy with (PB). We obtain austere versions of presentism and actualism thus understood if we sharpen their conceptions of present/actual things and how things (presently/are) to present/actual being in Lewis’s sense, i.e. so that, in effect, they construe reality as present/actual things and their present/actual instantiation of fundamental properties. To get these austere species of presentism and actualism we must replace PR(iii) and (AR)(iii) by their austere versions: (PR)(iii*)

(AR)(iii*)

How present things presently are coincides with the matter of which non-tensed properties and relations present things presently instantiate. How actual things actually are coincides with the matter of which non-modal properties and relations actual things actually instantiate.

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  ?



. Sanson and Caplan’s view I take Sanson and Caplan’s advocacy of what they call “brute past presentism” to involve what I have called the intermediary version of the restriction view of reality. Their “brute past presentism” involves the same view of reality as the non-austere presentism I have just characterized. In rejecting the view that “facts about how things once were . . . just are special facts about how things are now (for example, facts about the present instantiation of primitive ‘tensed’ properties)” (: ), they seem to maintain that how things were is not included in how things are. I take them to reject the “austere” version of presentism I have just characterized, however: in their view, present instantiation by various things of various past-tensed properties is constitutive of reality. Sanson and Caplan’s “brute past presentism” adds to the version of presentism in which I have characterized the view that the explanation of truth about the past must advert not to reality, but to the past about things and how things were. Though beyond reality, the past about things and about what relations then were instantiated by the things that were then is fundamental to the notion of truth about the past: it is in virtue of it that truths about the past are true (i.e. and not in virtue of things and how things are, and therefore not in virtue of reality). Inevitably, marks of the past about things and how things were are to be found in the present: if in the past, Bill instantiated being sunburnt, and Bill is among the present things, then presently Bill instantiates the past-tense property having been sunburnt. (Such marks of the past should not be confused with present evidence of the past: arguably, no mark of the past in this sense is present evidence for the past about things and about what properties then were instantiated by things.¹⁰ Evidence of the past is confined to present things and which non past-tensed properties present things presently instantiate. A Dummettian realist species of the view would hold that possibly, the past about certain things and how these things were has left no present evidence for it whatsoever.) In Sanson and Caplan’s view, however, marks of the past in the present should not be confused with the past itself, which is more fundamental. In their view, the present instantiation by things of pasttensed properties is grounded in the past about things and the relations things instantiated. An austere species of brute past presentism emerges if one adds to austere presentism the view that present instantiation by various things of various pasttensed properties is grounded in the past about things and about how things were. Sanson and Caplan’s view opposes Bigelow (: ), for whom the past is grounded in (the present) reality as conceived by Sanson and Caplan and seemingly by Bigelow himself. Sanson and Caplan have little sympathy for the modal analogue of brute past presentism, however. They hold that the possibilities about things and about how things might have been are grounded in things and their modal

¹⁰ Williamson’s conception of evidence (Williamson ) is going to make this difficult to stick!

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   properties. In their view time and modality are disanalogous in this respect: whereas brute past presentism is at the very least credible, its modal analogue is not.¹¹

. Brute possibility actualism On this view, similarly, the possibility about things and what things might then have instantiated is fundamental to the notion of truth about the possible: it is in virtue of it that truths about the possible are true (i.e. and not in virtue of things and how things are (as conceived by the actualist), and therefore not in virtue of reality). Inevitably, things and how things are contains marks of the possible. For example, if it is possible for Bill to be sunburnt and Bill is actual, then Bill actually instantiates the property possibly sun-burnt. It is not in virtue of his actual instantiation of this property, however, that possibly, Bill is sunburnt. On the contrary, the proposition “possibly Bill is sun-burnt” is true in virtue of the possibility whereby Bill instantiates the property being sun-burnt.

. Two contrasts with brute past presentism: Prioreanism and eternalism To better understand brute past presentism it is helpful to contrast it with what it is not. Brute past presentism is the doctrine that present truth is grounded in the past about things and how things were. As such, it is neither Prioreanism nor eternalism. Brute past presentism is not Prioreanism. Prioreanism holds that the past is an aspect of the present in the following sense: it identifies present reality with all true propositions i.e. including propositions of the form it was the case that (), and therefore takes the relation between reality and past-tense propositions to be the same as the relation between reality and present-tense propositions. In contrast, brute past presentism denies that such truths as it was the case that there are dinosaurs are constitutive of reality (or directly reflected in how things are)—indeed, such truths as it was the case that George Bush is President—belong similarly to (or reflect) supra-reality. In the latter case there may well be a mark in reality of this aspect of supra-reality—namely, George Bush’s instantiation of the property having been President (and perhaps in the former case too, although here the object of the instantiation might have to be a spatial region, or “the whole world” (Bigelow). But this mark is distinct from (and grounded in) certain matters that are supra-reality: the past about things and how things are. ¹¹ Sanson and Caplan suggest “what, then, is the difference between the modal and temporal cases? There is a natural and plausible conception of modality reality that treats the actual as fundamental and understands modality in terms of the actual modal properties of actual things or, perhaps, the actual powers and dispositions of actual things. There is no natural and plausible analogy when it comes to the past, because—to fall into a metaphor—the past, having happened, is able to assert its independence from the present in a way that the merely possible is not able to assert its independence from the actual” (: ). This could be understood as begging the question—it assumes that the explanation of the truth of past-tense propositions that adverts to things and their past-tense properties is neither natural nor plausible—but there are other defects. Primarily, it means that brute past presentism and its analogue, brute future presentism, are not on a par. The future hasn’t happened yet, and so, for all Sanson and Caplan have to say, cannot assert its independence from the present. Their explanation of the asymmetry between time and modality commits them to an asymmetry between the past and the future: an explanation of the truth of “Arnold was pale” must advert beyond reality, whereas a natural and plausible explanation of the truth of “there will be a sea-battle” tomorrow is confined to reality (and the future-tense properties of things).

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  ?



Brute past presentism is also contrasted with eternalism (about the past). The eternalist agrees that an explanation of the present truth of “Arnold was pale” points to Arnold’s past instantiation of being pale. But from the eternalist perspective the ground is closer to the instantiation of being pale than it is to the past instantiation of being pale. The eternalist takes “past instantiation” to be itself explicable in terms of something more fundamental (token-reflexive truth conditions or whatever)— namely tenseless instantiation. So to speak, from an eternalist viewpoint the ground of the truth of “Arnold was pale” is Arnold instantiating being pale (only not now), i.e. the point being that the “location” (point in time; time) at which Arnold instantiates being pale is something that exists (and so Arnold’s instantiating being pale then is something that is).¹²

References Bigelow, J. () “Presentism and Properties.” Philosophical Perspectives : –. Fine, K. () “The Problem of Non-Existence I-Internalism.” Topoi : –. Fine, K. () “The Question of Ontology.” In Metametaphysics: New Essays on the Foundations of Ontology, ed. D. Chalmers, D. Manley, and R. Wassermann. Oxford: Oxford University Press, –. Kierland, B. and Monton, B. () “Presentism and the Objection from BeingSupervenience.” Australasian Journal of Philosophy (): –. McDowell, J. () Mind and World. Cambridge, MA: Harvard University Press. Parsons, T. () Non-Existent Objects. New Haven, CT: Yale University Press. Percival, P. () “Predicate Abstraction, the Limits of Quantification, and the Modality of Existence.” Philosophical Studies (): –. Perreira, R. () “Truthmakers.” Philosophy Compass (): –. Routley, R. () “Exploring Meinong’s Jungle.” Notre Dame Journal of Formal Logic : . Salmon, N. () “Existence.” In J. E. Tomberlin (ed.), Philosophical Perspectives I: Metaphysics. Atascadero CA: Ridgeview Press. Sanson, D. and Caplan, B. () “The Way Things Were.” Philosophy and Phenomenological Research (): –. van Inwagen, P. () “Being, Existence, and Ontological Commitment.” In David John Chalmers, David Manley, and Ryan Wasserman (eds.), Metametaphysics: New Essays on the Foundations of Ontology. Oxford: Oxford University Press. Williamson, Timothy () “Knowledge as Evidence.” Mind (): –. Williamson, Timothy () “Truthmakers and the Converse Barcan Formula.” Dialectica (–): –. Wittgenstein, L. () Tractatus-Logico-Philosophicus. London: Kegan Paul.

¹² Prior to the publication of the volume, Philip Percival informed the editor that he would be unable to complete the paper or otherwise prepare it for publication and that he ‘disowned it completely’. However, he kindly agreed to let the paper be lightly edited so that it could still serve as a basis for Kit Fine’s response.

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 One Absolutely Infinite Universe to Rule Them All Reverse Reflection, Reverse Metaphysics Joseph Almog

Three questions are to guide us. They are about the three segments in the title: the nature of the universe proper, reflection (up-to and down-from) the universe and the method with which to describe the metaphysics (as opposed to constructive justification epistemology) of the universe.¹

¹ It is with much pleasure that I dedicate this chapter to Kit. I knew him when I was still a student and he was in Edinburgh and later as a colleague. He was always thrillingly generous with his ideas, a true ideaman. I still recall trekking in the Grisons in Switzerland and Kit expounding enthusiastically his qua objects. I did not think there were any such but his thrust was irresistible. As I write below on a seemingly disparate series of topics, I know Kit will see they are all tips of one big iceberg having struggled with them all in his own way. Such is the theme underlying the whole chapter—should we pursue the—essence(concept) of sets or rather their nature (in the sense of the distinction I develop in Almog  and elsewhere)? This topic of essence vs. nature is a domain where we both drew at one point certain related critiques of a certain influential modal account of Kripke’s but opted for different positive accounts. Another important common issue coming up a bit later (in the Appendix to this chapter) is Kit’s claims about the relation between the essences of the individual B. Obama and the singleton set {BO} or the question of relating (in Cantor’s thinking) concepts, inbegriff, pluralities, sets, ordinals, and cardinals. More generally, the neo-Kantian idea of “constructional metaphysics” that accompanies us from start to finish is one Kit brilliantly contributed to now for  years. The setting for the dispute between constructive and realistic metaphysics—the idea of an absolutely infinite universe—evokes mathematical topics, such as the status of the universe of all sets and ordinals. This has a way of making such an essay too technical for the metaphysician and too metaphysical for the technically minded. I tried to keep technical ideas to a minimum, especially while dissecting reflection principles. I also owe thanks to joint teaching on related matters to John Carriero, David Kaplan, Barbara Herman, Tyler Burge, and recently Olli Koistinen who goaded me to think about the pre-set theoretic origin of the idea of absolute infinity (while I was writing a monograph on Spinoza, a dashing user of the notion). Special thanks are due to many years of teaching together the philosophy of set theory with Tony Martin (who disagrees probably with every word I say here except his own, viz. that Gödel is a conceptualist metaphysician; see Martin ) and in the same vein to comments on the epistemology of large cardinal/ determinacy principles by John Steel. I also owe debts to conversations with Mandel Cabrera, Sarah Coolidge, Antonio Capuano, and Noa Goldring and to very helpful correspondence with Hugh Woodin and Kai Hauser. Thanks also to Mircea’s kindness as editor. None of these interlocutors should be taken to share the chapter’s stance. This standard qualifier should be read here literally. Joseph Almog, One Absolutely Infinite Universe to Rule Them All: Reverse Reflection, Reverse Metaphysics In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press (2020). © Joseph Almog. DOI: 10.1093/oso/9780199652624.003.0004

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       



 Three Questions about the Universe The first question is about the nature of the universe proper, though don’t get impatient if by the next sentence we are saddled by a multitude of questions about a multitude of universe-s, none of which are genuinely uni-verses, the total single object you’d expect. The first question is of metaphysics and on the face of it not of mathematics. The question is this: is this global ur-object, what Cantor called “the unity of (the) all,” indeed what was just said about it—an object, the ur-object, the urone-with-unity? In modern analytic philosophy, at least after Kant, “it” is often denied its “itness,” denied objecthood, real being, ensemble-ness and essential-unity, and/or any “essence” or “concept” by which we may think of it. It is left as a mere “unfinished (-able)” and “unsynthesized” manifold. So there, our first universe-question: unitary object or not? This first query seemed of metaphysics and not of mathematics; the next seems of mathematics and not of metaphysics. When it comes to mathematics and its Cantorian (absolute) infinities of sets and ordinals (and you will see below why I speak of them separately, like early, but not later, Cantor), it is often said that we have a mathematical answer to the question whether the universe of (just!) these absolutely infinitely mathematical objects is itself an object or “part of” such an object. The answer is “no,” it is not, and could not be an object by a theorem due to Cantor. This is not quite right. We do have a mathematical answer, if we demand that the universe of all mathematical objects be a set (Cantor’s menge); then the answer is no. We also have a mathematical answer, if we demand that the universe of all these mathematical objects be a proper class (an absolutely infinite Vilheit (multiplicity)): the answer then is yes, that is what we, following Cantor’s letter to Dedekind of  standardly take it—really them—to make. But for all the mathematical riches of sets and (proper) classes, I do not see that a mathematical answer has been given yet to the original question—is there an object that can claim, “all these absolutely infinitely many mathematical objects are my-own objects,” in the way that there is a certain object, a certain human body B (say, of Barack Obama) that can claim that all these organs are its-own organs? The question is further complicated by the commonly used phrase “the universe of sets.” This last suggests the alleged item—object or multiplicity—is of-the-sets, as if the Obama body B was of-the-organs, rather than the organs being of-B. If we allowed that the sets are of-the-universe in the way the organs are of B, thinking of them now in the genitive as the universe’s mathematical objects, we would soon start wondering whether the answer to the question regarding the all-ness of these mathematical objects is itself mathematical (model theoretic) and not metaphysical.² ² The question can be raised even if the mathematical objects, say the sets, were prior to the universe. Unless we think the only way for a multitude to unify into an object is the “set-of ” operation, we could wonder whether the multitude of all the sets unify into an object, even if it—that object—is not a set. It is not part of the mathematics of set theory that every object is a set even if often, because of its foundational/ reductionist aspirations to serve as a universal language, it so offers to represent the very idea of “object.” The question becomes very live indeed when the universe is taken as the mother object prior to its objects, in the way I am assuming B is prior to its organs.

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   So there goes our second question—are the universe’s mathematical objects—or the universe of all mathematical objects— unifiable in an object or not?³ Call the first question the absolute universe question, with “absolute” indicating no adjectival restriction such as “mathematical,” “physical,” etc. but just the universe tout court. Call the second question, the mathematical universe question. We are in a position to make two points about these two questions. (i) the two questions may well be—naturally if not logically—connected, e.g. answer the absolute question negatively and it is very likely you will so answer about the mathematical objects; answer the absolute question positively and it is likely—such is the thrust of the present chapter—that you will answer the mathematical universe question positively too. (ii) the answers to the two questions may lead us to pose a third question: Do we have here “splitting universes,” a universe-dualism of mathematical vs. non mathematical universe-s or is there—as the lexical meaning in the vernacular suggests—one and only one universe, the universe and at that, it is a unitary object and, at that, the ur-unitary object so that any other (mathematical) unitary object is such because it is one of (now in the genitive) the universe’s generated objects? This third and last question may be called the uniqueness of the universe question.

 Intermezzo: Universe Skepticism vs. Universe Realism Behind our trio of questions lies a rather simple clash of Welt-anschaungs (properly so-called in this case!): We start with the received view—universe-skepticism—about this global object—the Welt/universe—marked by both epistemological inaccessibility and metaphysical/ontological incoherence, the former breeding the latter. As I address our three questions, I examine the (i) metaphysical/ontological skepticism, (ii) the epistemic skepticism, and (iii) not least, the (“Kantian”) metathesis, that the study of ontology is to be conducted only within the bounds of constructive epistemology. In all, this leads us to reverse positions on all three fronts. The “object” that has become the paradigm in neo-Kantian analytic philosophy of non-real objectuality (it is a mere “ideal”), the object that cannot exist as a unity and the object that is unknowable turns on our reversed development to be rather (i) the prime object (with all local objects getting their object-hood from it), (ii) its existence (“the absolutely ³ “Unifiable” suggests to many an act of mind, a suggestion often followed, even if in an idealized form, by the pursuers of the idea of an iterative multitude of the sets (ordinals), sometimes described also as a “generative” process. This reliance on putting together the together-object (the “ensemble”) is surely influential since Zermelo, but in fact earlier, both Cantor and Dedekind, were responding to (neo-) Kantian ideas of (“transcendental”) mental operations combining ingredients unto new unities. The answer as to what process (agency?) makes up the unity of any ingredient-unities (sets and ordinals included) is not to be presupposed here by the use of the verb. It is indeed one of the key disputes between what I call in a moment the neo-Kantian “conceptualist” tradition and the realist tradition. This last has the generation (of any unity, not just sets and ordinals) take place (and a place it takes) away from minds and in and by the universe-object itself.

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       

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infinite universe exists”) is the prime existence axiom, and (iii) not least, it is the prime object we think of and know of, in all our local episodes of thinking/knowing local/limited objects. Indeed, if it were not for thinking of it, the existential prime, we would not think and know any of its local/limited products. In this last claim, our inevitable (necessary!) cognitive access to the universe, I am driven by Descartes’s apercu about the presence of the universal absolutely infinitary object in any local cognition. I introduce now this Descartes’s apercu so that it stays with us both as a motif for the chapter but also as a point of light at the end of the tunnel, as we indeed tunnel for a while though the doctrines (and motives) of the influential view of universe-skepticism: . . . when gazing at the sea, we are said to behold it, though our sight does not cover it all nor measures its immensity; if indeed we view it from a distance in such a way as to take in the whole with a single glance, we see it only confusedly, as we have a confused image of a chiliagon, when taking in all its sides at the same time; but if from near at hand we fix our glance on one portion of the sea, this act of vision can be clear and distinct, just as the image of a chiliagon may be, if it takes in only one or two of the figure’s sides. By similar reasoning I admit along with all theologians that God cannot be comprehended by the human mind, and also that he cannot be distinctly known by those who try mentally to grasp Him at once in His entirety, and view Him, as it were, from a distance.⁴

Universe-realism submits a symbiotic ontological-and-cognitive primacy of the global object thesis: it is the prime existent making all others and in every cognition of any such local other it is cognized. And this primacy affects how we are to approach the existence and cognition of mathematical objects such as sets and ordinals.

 Universe-Skepticism: The Effects of Kant’s Symbiotic Idealism/Realism Before I go on addressing the three universe-bound questions, let us take a short pause and notice an oddity. I speak here rather liberally of “the universe” and I just raised three questions about its fundamental nature. But the three questions may well seem three questions too many in modern analytic metaphysics, at least as we practice it since Kant’s seminal work in the s. They may well be regarded as three transgressions of what many call (I just repeat a phrase I don’t really understand) “the limits of intelligibility.” For the source critic, Kant, the three may well be involvements with a downright ens non gratum, the universe.⁵ ⁴ Descartes . ⁵ As for Kant’s own stance, as often, the apostles are more zealous than the prophet. Kant himself, for all his powerful critique of both the notion of “universe” and of “Ens Realissimum” (“God”), has engaged both ideas in an insightful way in , wherein he provides no less than a basis for the existence of a necessary being generating all others and itself infinite (see Kant ). In , Kant knew how to speak intelligibly of a—the—being that is maximally real. He did not specify it by “perfections,” a notion he harshly dismissed already then. Indeed he did not specify it—at all—by predications/determinations. He rather started his reasoning with a certain mundane fact—that something is possible (in my view, this also meant to Kant then: something is thinkable). The existence of this possibility—even if we counted it as a purely

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   Of course, those familiar with work in transfinite ordinal and set theory will find endless allusions to “the universe of sets (ordinals)” either (i) by “building” it directly (axiomatic theory-free) as Cantor did for his ordinals or as we do now building V as the union of all initial segments V-alphas or (ii) indirectly, as axiomatic-theory involving, viz. as a (“the”) model of a (rich extension) of the ZFC axiomatic set-theory. So, with so many nominal uses of “the universe,” you might think I am overdescribing the effects of the universe-critique. But as we shall see, there is no objectuniverse behind all these mere manners of speaking. There may be a proper class or an inbegriff (Cantor’s term I will systematically use below and which I roughly translate as concepted multiplicity, with “begriff ” translating as concept). So when all is said and done, the modern actual (absolute) infinity theories have no official de jure object, the universe, which is absolutely infinite. There are absolutely infinitely many objects, yes. But not one of them is the one as it were, the uni-verse, the prime unity whose sets and ordinals they are.

logical possibility (one which for Kant involved thinkable concepts only)—seemed to Kant to require a material ground. When he gave examples later in his paper, he spoke of breaking up various complex concepts and reaching elemental ingredients—his example is space—which is not conceptually definable anymore (in those days Kant spoke of space period, without the later famous theory of forms of intuition). Here then was the existence of space, period. Period because no concept or definition can substitute for that brute existential fact of an “object.” Kant thought this ur-being is material yet necessary in its existence. And this existence was necessary not because of its attributes (concepts, perfections) but because of its place in the chain of beings (and possibilities). The supposition that this ur-being was possibly not-in-existence took away the very ground from all possibility and thus of our initial premise. In my own words, logical possibility required a material being whose existence was necessary; no span of possibilities without an urmaterial necessary being. In a second stage of the argument—that need not concern us here—Kant goes on to argue that this being is eternal and furthermore intelligent and with a “free” will. For Kant, this made the being God. By  and the Critique of Pure Reason, Kant seems to take his distance from such discourse. In his critique, he accuses the notion of supreme perfect being of being a mere assemblage of concepts (“perfections”) and the metaphysician to leap from the conceptual assemblage—this is the fallacy he calls subreption—to “it” being a universal maximal object. For this critique of a leap from concept to object, I have nothing but solidarity. But our path to the universal object-universe is not conceptual. Kant also goes directly, without mention of an assemblage of concepts, at the notion of a universal object, an object that he views as “the sum of all possibilities.” As of this, I will just say: exactly, our universe as object is not a sum or a combination or a construction bottom-up. It is then left for Kant to consider a third and last barrier, one resting on his epistemology—if there were such a universal object, it would be unknowable. If an object is both unavailable in “intuition” nor can it be descriptively-conceptually characterized uniquely, it cannot be thought of singularly. Kant asserts this is the case with the universe or any ens that is the maximally real ens (Kant : –): “The concept of the supreme being satisfies all questions a priori which can be raised regarding the inner determinations of a thing, and is therefore an ideal that is quite unique, in that the concept, while universal, also at the same time designates an individual as being among the things that are possible. But it does not give satisfaction concerning the question of its own existence—though this is the real purpose of our enquiries—and if anyone admitted the existence of a necessary being but wanted to know which among all [existing] things is to be identified with that being, we could not answer: ‘This, not that, is the necessary being.’ ” It is exactly this last point of Kant—rooted in his epistemology of objects—that we challenge below. We reverse his epistemology binds metaphysics methodology and then urge that—the universe as object is (i) the prime existent and (ii) ipso facto (and inevitably) we think of it (in any of our thoughts) and know it. Indeed if we did not think and know it, we would not think and know any local singular item.

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       

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Kant’s influence on exiling the universe from metaphysics has been deep and lasting. For example compare it to another famous critique on limits of intelligibility, Quine’s on modal notions. We are all aware that while Quine’s critique has been answered and in effect pushed sideways by the modern responses of Kripke and others, Kant’s critique has been incorporated into the very core of modern foundational studies of formal infinity-metaphysics, including those of modal metaphysics. My focus in this chapter is not modal metaphysics but since both Kit Fine and myself have indulged in it, I will just say that for all the rampant talk of “possible worlds,” this has been merely a manner of taking a set theoretic model theory for a modal language (indeed at that level not modal yet but just a certain universal operator called “Box”) and interpreting its central set of indices as involving possible—this is the key modal adjective—“worlds.” But in truth, all those who so speak are engaged in what I call below limitative local conceptualism, not direct de-universe realism, viz. instead of investigating head on the one and only (absolutely infinite) real universe directly, we review a spread of surrogate conceptual constructions, aufbaus, whether they are driven by sheer combinatorics (as in David Lewis’s famous work) or combinatorial constructions restricted by some local parameters (as in Kripke’s famous critique).⁶ Either way, the worlds are combinatorial constructions. I don’t mean this only in the overt case of reduction projects in which worlds are exchanged for (sets of) propositions etc. I rather mean this even when it is asserted unabashedly that the worlds are here to stay. For still, in effect they are all combinatorial/model-theoretic constructions and, ironically, especially so where the constructor—I refer here again to David Lewis—proudly calls himself a realist. As I said, possible worlds are not our subject here but they are anything but a counterexample to my sense of modernity’s flight from a metaphysics of the real universe. If anything the “possible worlds” constructions are grist to the realist’s mill—they have been a manner of deflating the essence or nature (I do not separate these two at this point) of this one real universe, and re-present it within the bounds of controlled model-theoretic constructions. The “vertical” deep structure of this one real universe is gone and is replaced by a “horizontal” spectrum of formalcombinatorial-conceptual constructions.

 Set Theory—A Hotbed of Realism about an Absolutely Infinite Universe? Modal metaphysics (in truth: a model theory of a Box operator on a certain algebra) is not our focus. The focus of this chapter is the treatment of absolutely infinity in the setup of transfinite ordinals and set theory, as initiated roughly in between  and  by Cantor, Dedekind, and others we shall have the occasion to mention. And though these studies are often regarded as the culmination of a realistic attitude ⁶ By historical local parameters, I mean that we allow in specifications of worlds indexing of the sort: this is a “world,” viz. a combination, involving Nixon, not his twin brother; and if it involves QEII, it also involves her actual origin Q, not the origin Q* of her sister Margaret; and so on. We return to Kripke in a long footnote below.

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   towards infinity and an infinite universe, I believe the situation is much more complex. I explain. The original launching pad for the present chapter was an attempt to understand the provenance of the modern universal characteristic, the modern universallanguage and language of universality, the language of sets—where do such “things”—sets—and their “universe”—the universe of all sets—come from? The “where” is not metaphorical but meant in the quite literal sense of a quest for the whereabouts of sets—if they exist at all, mustn’t they be somewhere? In standard expositions, transfinite set (ordinal) theory is taken to reflect the most exuberant form of ontological Platonism and realism (two often fused themes).⁷ But to me what was striking was how in the succession of founding fathers from Cantor himself—both of the early  Grundlagen and all the way to his letter to Dedekind in —and through the writings of Dedekind, Von Neumann, Bernays, Zermelo, Gödel, and Ackermann, there dominates a retreat from a brute realism of an objectuniverse metaphysics. Variations and all, the aforementioned thinkers to a man follow (and consciously so) what I would describe as a neo-Kantian theme—metaphysics within the bounds of justification-epistemology—according to which the universe (of sets) itself is—and the adjectives are now quoted—“unknowable,” and thus inside this methodology, “un-thinkable,” and thus, again inside this methodology, deprived of the ontological status of “real being(ens)” and of making up an “essential unity.” There is here a fusion of idealism and realism that is initially surprising, an internal practice-realism but practiced within an overall-idealism. Cantor says famously in his Grundlagen “I am thoroughly a realist but no less an idealist.” This form of idealismcum-realism may well serve as the leitmotif of the metaphysics within the bounds of justification epistemology. It preserves the symbiosis of existence and thinkability introduced above in the wake of Descartes’s sea passage as the mark of universerealism but, crucially, inverts the order of determination: it is the thinkability of an object that determines its coherent existence. I turn now to an essential “technical” manifestation of this symbiotic idealism/realism.⁸ ⁷ This modern fusion of idealism (Platonism being a form of idealism) and realism struck me, as I was teaching the texts of the key founders of set theory, as being enhanced by the already mentioned symbiotic idealism/realism in the great thinkers about infinity in the modern tradition, starting with Leibniz and all the way to Gödel. We return to this symbiotic fusion in our discussion of separate points of Dedekind, Cantor, Zermelo, and Gödel, as we advance through our discussion of (i) bottom-up reflection and (ii) metaphysics done only within the bounds of our justificatory concepts. It is striking how to a man all the great figures, in different centuries (all the way from Leibniz to Woodin) are symbiotic idealist/realists. I know “in my bones” that this is true but have no good explanation of this pattern. Something similar applies closer to home in our own domain of modal metaphysics, when one examines the works of Kripke, Fine, and others. See below fn.  (and Appendix) on this last pair of thinkers. I lay some of the responsibility for this symbiotic idealism/realism on Kant’s deep influence. But this cannot be the whole story. ⁸ The quoted adjectives are from Gödel’s remarks on the reflection principle and indefinability of V to which we turn momentarily. A presentation of Gödel’s conceptualism that I believe captures overall Gödel’s philosophy of the universe of sets by-way-of a concept-of-set is given by the crystalline Tony Martin (). What Martin calls realism is what I term “symbiotic idealism/realism” of the great infinity figures. If you are realist about designed-for-minds ideal-entities you are for me not a realist tout court, you are a realist within your overall idealism. The above assertion of Cantor, “I am a thorough realist but no less an idealist,” applies to Gödel with bells on.

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       

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. Reflection: the executive arm of symbiotic idealism/realism Our second focal point continues the dissection of the nature of the universe by way of a fundamental (heuristic) principle of set theory, called the reflection principle.⁹ It is said to be about the mathematical universe V of all sets, though, as we shall see, it has its source in a metaphysical-within-the-bounds-of-epistemology principle due to Kant about the universe-as-object. Reflection has many variant technical formulations—weaker and stronger—but the guiding idea is clearly put informally by Gödel:¹⁰

In this vein, before we plunge in earnest into the metaphysics and epistemology of the universe, I adjoin an explanatory word about the standard terminology of contemporary “epistemology.” I am here using the terminology (inherited from Kant) of an a priori–a posteriori distinction. A related idea, much relied upon in contemporary discussions in the philosophy of mathematics, following Gödel, is to segregate “internal (intrinsic) justification” from “extrinsic justification” and then launch into the “routine” of classifying which propositions are justifiable which way. In relation to this terminology, I should say two things. First, in the pre-critical, pre-Hume-Kant, age, in the universal nature-philosophy of writers like Descartes and Spinoza, there is no a priori–a posteriori distinction or related, an internal/external justification of truths or beliefs. This is so in spite of the common-lore “imagery” of “Cartesianism” as obsessing with “internal justification.” Justification and more generally “epistemology”—indeed the very title—is a brainchild of late eighteenth-century thinking and the bringing in of normative and legal notions. Descartes and Spinoza were not legally minded. Kant was. I cannot expand on this here but see Almog . Second, I should like it noted that at this point in this chapter I use these epistemological notions as a mouthpiece (“expositor”). Expressed in my own language, I do believe a duo of theses denying livelihood to these epistemological dualisms. First, I do not believe any truths—including the simplest ZFC axioms—are known a priori or worse are “intrinsically justified” (one could put this badly by saying that all evidence is extrinsic or a posteriori; I do not want to so put it. The very terms of the distinction should be questioned). Second and stronger yet, I believe that the very categorization (as separate from whether any example falls under a priori or intrinsic justification) rests on a mistake. Perhaps after we have drilled our way through the tunnel of deuniverse realism and into the sunlight of the “new plateau,” it will be clearer why I think the very Kantian distinction is internal to the justification-conceptualism framework and with no intuitive trans-framework basis. I return to this point in and about modern epistemological categorizations towards the end of our chapter. By the way, I take Quine’s deeper point in his famous “Two Dogmas of Empiricism” to be not just an attack on the analytic–synthetic distinction but just this stronger point, viz. the ill-founded-ness of the justification-theory dualism of a priori–a posteriori (as applied to both pieces of knowledge and to truths). ⁹ Reflection is a model-theoretic transfer principle. There are many variations thereof used in current discussions both in directly mathematical work as in philosophical analyses. I try to minimize technicalities in the discussion that follows, not least because the various model-theoretic forms used intra-mathematically are not the issue so much as the underlying metaphysical (in my estimation—Kantian) motivations. For more technical background, see in the intra-ZF case, the Levy–Montague formulation. See also Bernays’s contributions for stronger forms. Strong generalizations are introduced in Reinhardt and discussed most interestingly in a (see the footnotes below). The motivating early segments of Reinhardt’s paper (including the remarks on visualization/imagination and means of “construction(synthesis)”) and the more general discussion of axiomatics in the concluding pp. – are distinct in their clarity and diagnostic power. A most interesting attempt to explain philosophically the unifying force of the reflection principle is given in Corazza’s discussion of V to V maps preserving elementary-structure as ground zero-principles. For the role of the principle (broached below) in recent large cardinal theory see Steel’s very clear exposition () and the opening of Martin and Steel . ¹⁰ I assemble here a variety of Gödel’s remarks. We return in detail to his quotes below. Interestingly, Cantor says in Grundlagen (the very locus I see below as (sometimes!) supportive of de-universe realism) things that sound very much like Gödel’s: “Das Absolute kann nur anerkannt, aber nie erkannt, auch nicht annähernd erkannt werden” (: ). Like remarks about uncharacterizability—and thus to a neo-Kantian, unthinkability—of the universe are made by Zermelo in his elegant  paper on boundary

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   All the principles for setting up the axioms of set theory should be reducible to Ackermann’s principle: The Absolute is unknowable . . . in the last analysis, every axiom of infinity should be derivable from the (extremely plausible) principle that V is indefinable, where definability is taken in more and more generalized and idealized sense . . . Consider a property P(V; x), which involves V. If, as we believe, V is extremely large, then x must appear in an early segment of V and cannot have any relation to much later segments of V. Hence, within P(V; x), V can be replaced by some set in every context. Hence, there is a property Q(x), not involving V, which is equivalent to P(V; x). According to medieval ideas, properties containing V or the world would not be in the essence of any set or monad.

To Gödel, the insegregability by familiar properties (expressed in (generalizations of) first-order language) of V-proper from initial segments V (alpha) meant—and we should take notice of the sequence of leaps undertaken here—that () the universe V is indefinable using accepted means of expression; thus (and now in an absolute vein), () V is “indefinable” tout court, thus () “unknowable,” thus () “unthinkable,” and finally, at the directly metaphysical level, thus () V is not a “real being” and () not an “essential unity.”¹¹ What is more, as can be gleaned from the last stretch of the quote, every real being (unity, set)—which it is assumed must be thinkable and knowable and thus, to begin with, “characterizable”—had better have no reference in its characterization (in its “essence” says Gödel) to this creature of darkness, the universe—for if anything is true of the universe, it is already true of a set in it, a limited-coherent-unity part thereof. As mentioned, as formulated technically inside set theory, this (cluster of) reflection principle(s) is (are) a model-theoretic principle(s) transferring truths—but of course only those expressed in the characterizing language—from the universe down to local sets. But, let me wonder out loud: Conceptually speaking, is the reflection really from a prior universe down to the posterior local sets? There is here model theory and there is the underlying metaphysics (and here guided, indeed induced, by an underlying constructive epistemology). What is the motivation or raison d’être of this type of reflection principle? It turns out that “raison d’être” is no show of linguistic floweriness at this point— the procedure encoded by “reflection” is literally the reason (and by a conceptual actof-Reason) for the legitimized being of the “new” unitary being, the “new” set, which is first thusly-justified and, only in turn, second, thusly-conceptually-and-by-reason made to-be. I would like to make three basic observations about this justification-cum-existence model-theoretic “mechanism.” numbers discussed immediately below, though his “reasons” did not allude, as Cantor and Gödel did, to an alleged absolutely infinitary object that may well be—and be the prime being—even though we cant speak of it inside mathematics. I return to the Cantor–Gödel ambivalence below and to alleged Gödel–Zermelo differences in a moment. ¹¹ The adjectives in quotes are all terms used by Gödel. See Hao Wang’s commentaries (). See also our quote below from Zermelo’s  key paper. Of course Gödel selects the kind of properties, often called “structural” or “intrinsic” or “essential” that can express the characterization. V has absolutely infinitely many sets (ordinals)—let alone containing all sets/ordinals—and that would segregate it from initial segments but this is not a structural property. One is to recall—and we will in a moment below—that mathematical formulations of variations of the reflection principle are model-theoretic embedding principles that are relative to the expressive power of the language whose model theory is discussed.

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

. Reflection is (just) a model-theoretic principle The fact that model theoretically a certain entity, domain or cardinal is not discernible from another such is in itself short of denying existence to the “higher” entity (domain, cardinal); in turn it falls short of showing that we have no way to think of— and in time, to know—the pertinent item. Model theory, for all its riches, is just that—model theory. It brings out what in the pertinent and mostly schematic formal language can be said (defined). In this respect, the reflection principle is of a kind with Lowenheim–Skolem theorems or for that matter the compactness theorem.¹² For example, the latter assures us that for a first-order language if a sentence has arbitrarily large finite models, it has infinite models. Should this have convinced Cantor in —if only he had by that time such a model theory of a prescribed language—that the infinitary ordinal omega does not exist and in turn is not an “essential unity,” is unthinkable and unknowable, because uncharacterizable by familiar operations? Cantor adopted a richer informal language, one allowing itself (what we could call today) non-elementary notions. Likewise a tad later with Skolem, the prime mover of such model-theoretic techniques. Skolem drew the correct consequences viz. the language of purely algebraic closure operations can only do so much. Of course, Skolem knew full well the transfer does not work merely downwards (to uncharacterize “bigger” entities), it works just as much upwards, the first-order language is simply being schematic in its resources vis-à-vis complex cardinality notions, once we cross the infinitude line. Indeed, Skolem knew that the finite–infinite line is also very fragile and exploited it to produce models—called nonstandard—in which “entities” (ultraproducts! The very idea used later in large set theory elementary embeddings of “reflecting” the universe) that have infinitely many “predecessors” and are thus not genuinely the natural numbers, even if our first-order concept is happily satisfied. He did not think this is doomsday or the collapse of the natural numbers or of higher cardinalities. He knew full well that if you add uncountably many constants to the language—to our as it were “conceptual resources”—and use compactness, you can transfer things up to the level the constants lead you to. Such “tailored transference” did not reveal reality or unreality—of the cardinals or the sets proper—it revealed the limits of the tailoring-concept, the axiomatics used (and interpreted by the very set theoretic model theory). Instead of drawing end-of-theworld conclusions (in both senses of the phrase), he drew the natural response—the language I use may well be the limits of my concepts, yes; but it is not the limits of the world. In the case of natural numbers, one needs, says Skolem, to take as primitive an

¹² Of course model theoretically we have transfer principles running both “up” and “down.” Those running up are not thought to have wonderous ontological revelatory powers (as noted in our common reaction to compactness in a moment, though of course some would so “justify” the revelation of the existence of infinitesimals). The interpretation of “downward” transfers as ontologically revelatory is driven by a metaphysical prejudice already percolating in the conceptualist thinking. The model theory is an executor on behalf of the prejudice. I am after an articulation of this prejudice below.

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   extra set theoretic-cum-model-theory notion of recursion or finitude. In a word, the limits of my language are the limits of my concept but leave the world out of it.¹³ Of course, one may reach a language that one views as mathematically optimal and yet one still has a result of model-theoretic indiscernibility. This is of intramathematical significance but it still offers no license to draw ontological consequences (let alone have the ontological consequences fall out of the cognitive and epistemic claims of unthinkability and unknowability). To reach (all) such metaphysical conclusions, one needs to view the model theory and the language over which we interpret as the limits of intelligibility. This intelligibility boundary—the limits of my/our language are the limits of my/our (thoughts of the) world—is not itself an intra-mathematical claim but one driving us to restrict the metaphysics to what our conceptual resources can construct and justify by means of certain rule-based combination principles. In a word, we are ruled by the Kantian guideline— metaphysics can be done rationally only within the bounds of justificatory epistemology, a theme we broach in a moment. Model theory may be wheeled in as the executive arm of such a metaphysics-within-the-bounds-of justification-theory but this itself needs philosophical justification.¹⁴

¹³ As is well known, Skolem predicted the supposedly shocking—and drama-involving—unresolvability of the CH by the model theory of first-order ZF(C) (perhaps less known is the fact he anticipated the Henkin tricks of constructing models out of the very language “modeled,” which, with important technical twists, lead eventually to forcing, as recorded in a reminder by Cohen himself; see Cohen ). I mention this to give the sense that some users of model theory—with Skolem, on ne peut plus—tried to minimize the metaphysical drama and read the results for what they were—limits of our deductive theories/model theories for them. No further metaphysical results are inevitable. Or if they are, one wants a philosophical argument, not just intimidating theorems. In like vein, Skolem used in effect an infinitary omega logic in his metamathematical work in the s (anticipating the completeness theorem) to be able to say (and prove in the metalanguage) more. In like manner, around the time Cantor told us infinitary ordinals exist (and can be thought of and known), the s, Dedekind faced a similar “model theoretic” problem of expressive resources. As is familiar (see his very clear  letter to Keferstein summing the reasoning), he set out to characterize the natural numbers N but in a (set theoretic) language not allowing the arithmetic notions, in particular the notion of “finitely many (predecessors).” He in effect discovered the transfer of arithmetic truths to nonstandard models (structures extending N with “alien intruders,” nonstandard integers). To block this (filter out these structures) he let himself augment his range of conceptual methods (what he calls in a Kantian language notions of the “understanding”) to include taking the least structure of those he considered thus far, in effect allowing himself to “quantify” over arbitrary subsets of the domain, as one could in what we later called a full second-order language. Like Cantor, he did not conclude from the initial model-theoretic “uncharacterizability,” any ontological or cognitive (“we cannot think of”) “inaccessibility” conclusions about the natural numbers. ¹⁴ The point made here is not restricted to first-order model theory, which from Skolem – onwards is known to allow a spread of nonstandard models. Even when the basic conceptual resources are enlarged (at the price of the deductive completeness of the underlying theory), model theory is still just model theory viz. a theory of models of an underlying language/theory, and no substitute for metaphysics. Enhanced (say with “finite” or with “recursive’ or with “arbitrary subset” quantification), it may well lead to categorical characterizations (e.g. as in Dedekind’s above-mentioned second-order characterization or with first-order PA, but now with addition or multiplication made “recursive” as in Tannenbaum’s theorem for first-order PA) but we have not inverted yet the neo-Kantian idea of constructing the metaphysics by means accepted-justified by a certain bound-human epistemology. We now give ourselves more such conceptualmeans but the basic direction of constructing new unities by means of our concepts is upheld (for instance, in Zermelo’s  second-order constructions of domains of sets mentioned immediately below). I linger on this (be it first-/second-order) constructional conceptualism throughout the coming paragraphs.

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

. Reflection bottom-up—the executive arm of conceptualism Our second point about the reflection principle dissects the content of reflection. Putting on the back burner the just made point about relativity to the model theory presupposed, let us look at what the reflection principle says. It tells us the universe is not separable from its initial segments by “adequate” or “structural” properties. Now, technically this comes in the form of reflecting down from the “universe” V (or a proper class) to a set. One might surmise the conceptual content is thus a “step” from the universe down to the local unitary object. Not so. The principle works the other way round, as contra-posed versions verify: if none of the already constructed local objects—the sets—bears the structural property P, the universe will have no independent source to “make itself” bear the property P. What is the import of this claim? At the heart of this claim is the neo-Kantian idea of “constructing” bottom-up new unities from available ingredients; what was a concept-given-multitude referred to as a “universe” but truly a model of an underlying theory/concept is now-projected and made into a unitary set. We keep using freely the notion of “universe” in these cases as when we say we reflect the “universe” down to a set, where (i) the whole point is that there is no and could be no such universe and (ii) what we reflect over is best captured in my view by Cantor’s term inbegriff, which as mentioned, I translate as concepted collection (multiplicity). The reflection is technically carried out (e.g. in the large cardinal applications immediately mentioned below) over models of set theory. Such a model is often, again speaking technically, a proper class. But what matters to us is that the “frame” in question is given as a concepted-multiplicity, it is the frame that satisfies some previously justified principles, ones thought-through the pertinent begriff (“concept”). What is key for us is not the model-theoretic tricks (e.g. how we get the embedding j from V to V to fit into a class-free set-language only) but a point of conceptual analysis: the model/domain is given via the concept (the principles), viz. given only via our way of “conceptualizing” the multiplicity of all the sets. The application of such Kantian concepted modes of construction is quite clear in the “simple” axiom of infinity or one “level up” in the axiom of in-accessibles. We apprehend by means of a concept (viz. the previous principles) a certain multiplicity and we now, by an act of mind, what Kripke described well (elsewhere and not for reflection principles but my point is that his description applies to such with bells on) as a lassoing—we lasso out of the concepted-multiplicity a unitary set. The new “essential unity” is afforded only by way of the guiding concept/lasso—the vehicle that makes the multiplicity thinkable for us and thus knowable and affords now a characterization of the new unity. What is key to this Kantian procedure is that we advance from cognitively already given—and critically for Kant, epistemically justified—“data,” and now by means of a concept, really run a lasso around the old inbegriff. The advance is from lower local unities and their multiplicities to combinations thereof and then by “synthesis” to new local unities. The procedure has written into itself—as noticed very insightfully by Zermelo in the quote below—that no original global maximal entity (i) can be built and (ii) can sustain cognitive access from us. Whatever can be said to exist has

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   been conceived (here meaning: constructed by concept) and by justified means; to be is to be conceiv-able. Says Zermelo:¹⁵ The “ultrafinite antinomies of set theory,” which the scientific reactionaries and antimathematicians eagerly and delightedly call on in their campaign against set theory, these specious “contradictions,” arise solely from a confusion between the non-categorical axioms of set theory and the various particular models of them: What in one model appears as an “ultrafinite un- or super-set” is in the next higher domain a perfectly good “set” with a cardinal number and order type of its own, which serves as the foundation stone for the construction of the new domain. The boundless series of Cantor’s ordinal numbers gives rise to an equally boundless series of essentially different models of set theory, in each of which the whole classical theory can be expressed. The two polar opposite tendencies of the thinking mind, the idea of creative progress [schöpferischen Fortschrittes] and that of all-embracing completeness [zusammenfassended Abschlusses], which lie at the root of Kant’s “antinomies,” find their symbolic expression and resolution in the concept of the well-ordered transfinite numberseries, whose unrestricted progress comes to no real conclusion, but only to relative stoppingpoints, the “boundary numbers” that divide the lower from the higher models. And so the “antinomies” of set theory, properly understood, lead not to a restriction and mutilation, but rather to a further, as yet unsurveyable, unfolding and enrichment of mathematical science.

In sum, on our reading, the foregoing rather standard way of using the reflection principle is to use a limiting principle on metaphysical construction. I would like to call this form of use of reflection—not the model-theoretic embedding proper but the justificatory conceptual role—the justification-reflection principle. The principle displays a concepted procedure that justifies the new existence claim. The justification reflection principle asserts only justified-existence. The use of model theory as the executive arm of this constructional justificationconceptualism is natural enough if we view the underlying language (on the relevant model theory) as indicative of the range of justified concepts we have. But whatever differences we have about the range of concepts—and what leap to amended concepts (reflection on reflection etc.) we may allow ourselves at a given stage—the Kantian theme rules justification-reflection—to be is to be conceived by essentialunity licensing blueprints.¹⁶ ¹⁵ Zermelo . ¹⁶ A word about my “lumping together” Zermelo and Gödel as neo-Kantian conceptualists, both using justification-reflection. There are both technical differences between the two, as well as different philosophical emphases. Nonetheless on the main front here under focus, the construction-by-means of bottom-up reflection, I believe they share a basic conceptualist outlook. Zermelo is much farther away than Gödel (or earlier Cantor) from any discourse about the absolute infinite universe V, as if it were an object (in spite of our having to pass in silence over it or if not in total silence then in partial silence viz. up to what our model-theoretic reflection principles over adequate properties allow to say). In any event, Zermelo speaks more in the vein of turning what was a potential infinity (a model) into a local objectual reality, a set, and he does not whisper sotto voce of a vision of an ultimate (“divine”) absolutely infinite universe. Furthermore, he uses second-order language (essentially) to specify a categorical description (up to a given rank) of the cumulative frame, a language Gödel stays away from (though Gödel himself in remarks to Wang and in remarks to Reinhardt encourages a climb to high-order forms of reflection; see Reinhardt a, especially the beautiful discussion on pp. –, to which we return below). What is more, in Gödel , Gödel envisages a sort of generalized deductive completeness theorem for all set theoretic truths from some (I take it first-order) ultimate set theory extended by some ultimate large set (ordinal) principle. Such a deductive completeness was not on

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

. Justification-reflection and large cardinals by way of licensing concepts Our third and last point about reflection principles concerns the introduction of large cardinals. Contemporary lead set theoreticians Martin and Steel say in their justification of the use of reflection principles: Because of the richness and coherence of its consequences, one would like to derive PD itself from more fundamental principles concerning sets in general, principles whose justification is more direct. We know of one proper extension of ZFC which is as well justified as ZFC itself, namely ZFC + “ZFC is consistent.” Extrapolating wildly, we are led to strong reflection principles, also known as large cardinal hypotheses. (One can fill in some intermediate steps.) These principles assert that certain properties of the universe V of all sets are shared by, or “reflect to,” initial segments r—:. of the cumulative hierarchy of sets. (Reflecting reflection, we get ordinals K such that certain properties of V reflect to smaller V’s. This is the form of K B the principles below.) Reflection principles have some motivation analogous to that for the axioms of ZFC themselves, and indeed the axioms of infinity and replacement of ZFC are equivalent to a reflection schema.¹⁷

Zermelo’s mind (who in general dismissed the restriction in the proof theory (and justification) to finite deductions). In this respect, the use of an infinitary omega-logic, as in recent work of Woodin, is an interesting twist on the “logic-internal-to-models-of-set theory,” a twist on the philosophies of both Zermelo and Gödel, a sort of mid-way (of course the key idea in Woodin’s work is the post-forcing reabsolutization of sentences across models of higher (large)-set theory and this was not so vividly on Gödel’s or Zermelo’s pre forcing agendas; see Woodin  to which we also return below). I cannot go into yet other distinctions between Zermelo’s mathematical writing style contrasted with Gödel’s generalized grand universal-logic style. Gödel is more of a philosopher in the sense of priming epistemology (“intrinsic justification”) and principles of evidence; Zermelo is not focused on epistemology. What is more, Gödel seems to have been friendly, by the time of his conversations with Reinhardt, to an intensional theory of proper classes (properties) on top of sets, in a way that Zermelo never was. What matters to the current chapter is that both view models of set theory as given by way of a concept and then seek natural mind-apprehensible closure conditions to project from the inbegriff-multiplicity a unitary object, a set (it is clear that Zermelo, writing in , expected to so conceptually project smaller large cardinals than Gödel, who by the time he advises Reinhardt, is thinking of larger (trans-measurable) cardinals). In general, Zermelo and Gödel (and in the same way, a few generations later, Woodin) strike one as operating in the distinguished lineage of a series of idealists/realists (read the phrase in one breath), a lineage running from Leibniz to Kant, on to Bolzano, on to Cantor and to Dedekind. Cantor as we saw says directly that he is both idealist and a realist and they are (this is my word) symbiotic for him. I read Kit Fine’s method in metaphysics as being in this distinguished lineage. See immediately below the footnote on the essentiality of origin and the Appendix dedicated to Fine’s essentialist analysis of singleton sets. ¹⁷ The question of evidence-types for “new” axioms (or for that matter old ones, like infinity and replacement for instance) is not our question in this chapter and as mentioned I view the whole journey through Gödel’s quest for “intrinsic justification” and “extrinsic justification” epistemology as misguided. E.g. I do not think (even) infinity or replacement are intrinsically justified. They are deeply true of the universe of sets and precisely for this reason of organic truth viz. they are true of sets because true of the universe, I do not think they are intrinsically justified. What makes these axioms—these fundamental truths—true is not a concept (of set), any more than what makes all humans mortal is the concept “human being.” Concepts don’t kill humans and in like way concepts don’t make the universe (absolutely) infinite (thus the maker of truth of infinity and replacement). As mentioned, epistemology is not our task and commenting on the alleged evidential superiority of proving PD from a conceptually justified principle is not our de jure assignment. Furthermore, it might seem odd for an accidental tourist to differ with the judgment of two leading practitioners and with Gödel’s word—which has become the canon in modern studies—in the background. Nonetheless, it seems to me

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   It has become commonplace to connect the use of model-theoretic principles on V to ultimate forms of justification for such large cardinals. The extension of such principles to larger and larger cardinals (more and more similar to the

that on three fronts there is in the “epistemological” assertions here substantial points to disagree with. The disagreement is with both “intrinsic” and “extrinsic” justification, as Gödel shaped the two ideas. First, concerning “intrinsic” justification. PD is proved from a certain large cardinal hypothesis about Woodin cardinals (see Steel ). This large cardinal assumption itself may well be evidenced by reverse connections from determinacy theory rather than the evidence running the other way (much work of Woodin, Martin, Steel, and others has set just such a back-and-forth correspondence between determinacy theory and large cardinal hypotheses). The “justification” (if that is what our quest is for) runs the other way, so to speak from the countable ordinals up to large cardinals. It may then not be a priori or conceptual or intrinsic evidence, as the one presumed via reflection for a proper class of Woodins, but then so much the better. But see immediately below on the kind of evidence PD or AD(L(R) is sustained by. Second—and related—PD itself is evidenced without large cardinals from the organizational depth vis a vis a host of issues in the second number class, where, as has been pointed by Woodin and others, it provides a deep complete theory of that level of ordinals. In my view, PD is known to be true (as is known the Lebesgue measurability of projective sets)—and known for certain—now for decades and the connection to large cardinals is mathematically important but not game changing as to evidence. What is already certain does not need the reputed extra justificatory power due to a proof, though of course deep correspondence with another structured domain is revelatory. My point is that it is not certainty-enhancing. If I hold up—like the philosopher G. E. Moore—my two hands or better yet wash the dishes with them, there can be no better evidence for their existence (though by this time looking for evidence—let alone intrinsic justification is only a philosopher’s odd quest; one does not “internally” justify what one lives-by daily). No concept-based evidence for their existence (from a general concept e.g. of wellformed human being) would make the certainty higher; when one is washing the dishes one is certain and that is that. Of course it may well turn out that the evidence for certain kinds of elementary embeddings of V is not conceptual at all but a “washing the dishes” level evidence of working with them (as indeed this seems the point of view of Woodin). At that point, the back-and-forth connection between PD (and for Woodin, AD in L(R)) and large cardinals would be part of “washing the dishes” and the hypothetical absence of such a derivation would be a worry. A very clear analysis of the working-power of PD in the hereditary countable sets level is given by Woodin (). Third and last, Gödel and the almost universal discussion following his terminology understands by “external justification” attention to “fruitful consequences.” Indeed, the just-cited “justification” of PD is often thought to be of this kind. Of late, various model-theoretic axioms—generalizing the original combinatorial statement at the level of aleph- of “Martin’s axiom”—have been said to be “justified” on such grounds of umpteen rich consequences. This is not our just-made point about the (certain) truth of projective determinacy. When a truth is seen to be the source of the structure of a domain this is not (just) a case of “fruitful consequences.” What is at stake is not just the provision of a complete theory of the pertinent domain (as e.g. Gödel’s original axiom that V=L does). Rather this “fruitful” completeness must be seen to not be coming at the price of negating truths—higher up—about the universe of sets (in this way, V=L in its original form restricted to Gödel’s original L was known to offend even before Scott’s proof). The fundamental truth of PD is certain not because it settles many (all) problems at the level of countable ordinals but because it is seen to be emanating from a more global insight about the universe of sets. In this sense determinacy for projective sets or for those constructible over the reals (taken as primitives) is arguably a manifestation of some general feature of ordinals—beyond the countable level—in structuring the full universe of sets. The issues broached in this footnote are obviously very complex—both at the level of evidence principles and the intra-mathematical level of what is the fundamental source of what. The main point of the note was and remains to cast doubt on conceptual justification (for reflection) grounding (I) the truth of a proper class of Woodins and (II) PD and, in turn, of justifying (II) from (I). I don’t see these complex issues with sufficient clarity but there may be some points about the limits of conceptualist justification that are sufficiently clear.

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       



uncharacterizable “universe” V) is often presented as a realist’s demarche—we are given more, ever more, and what could be more realistic-in-attitude about sets? I beg to differ with this reading of the use of reflection inside large cardinal theory. Of course, to reiterate once more, the model theory is just model theory: if there is an elementary embedding j whose critical point is such and such, there is an elementary embedding j whose critical point is such and such . . . what more can be said?¹⁸ But the metaphysical point is not inherent in the model theory. On my reading, the use of justification-reflection is anti-realist (as I understand the key word “realist”), deeply moved by a realism-averse projective conceptualism. These may seem just more long words of the “ism”-variety but there is a genuine question here about what our approach to describing (vs. constructing/projecting) reality is and what a genuine existence, as opposed to justification, principle is. The ever more large sets (ordinals)—by way of reflection-justification—are not witnesses of a shift in orientation, letting now in concept-free existence principles. Rather the use of stronger and stronger blueprints preserved in the elementary embeddings is a form of allowing oneself more complex conceptual means for assembling (and thus “en-sembling”) a previously concepted-multiplicity. The program remains the neo-Kantian justification program of Gödel to boot—only through a concept can you project from a large multiplicity, a large set; to be is still to be conceiv-able. Perhaps the point could be put in terms of a rhetorical question—how can a negative principle of un-knowability and un-thinkability—the universe-object is unthinkable and unknowable—be, as it is often repeated, a “positive” realistic principle of (and for) a richer and richer ontology? The answer is: the principle is not a realistic ontologically maximizing principle. Quite the contrary. It may maximize surrogates for the real thing—as many as could be encompassed by the justifying conceptual mechanism but no more. To hark back to our ur-body B (of Obama) and its assembly of organs, if I had—per impossibile of course because here we are dealing with on ne peut plus organic objects—a principle justifying larger and larger surrogate simulations of the assemblage of organs (cells) of B by means of artificially assembled “organ collections” but it was inherent to the mimicking that B itself does not exist, I would not view the projection of larger and larger such artificial assemblages of organs (mimicking truths about B) as a form of realism, denying as the procedure does the natural source-existence of B. In like ways, reflection-as-used in justifications for “large” sets denies the existence of a certain maximally real object. It denies this real existence and goes on to construct closer and closer mimicking concepted unities—thus the inherent conceptualism—that imitate the denied black hole object “up to” what we can peel off with a rich conceptual definition (the elementary embedding). In this respect, the modern synthesis of large-large cardinals, e.g. a measurable cardinal or higher up Woodin and supercompact cardinals, is in my view conceptually of a kind—obviously “technically,” what is preserved by the embeddings has

¹⁸ For a summary of the use of those, see Steel .

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   much more structure—with lower-level syntheses, all the way “down” to the first infinite set (or Cantor’s generation of the first infinite ordinal by his second principle) or, one level up, with the projection as unitary object, the cardinal Aleph-omega, the previously mere-model of the Zermelo system Z by the deployment of the axiom F, replacement. We justify as mind-friendly certain closure conditions.¹⁹ Those allow us to justify the assertion of a new objectual existence.²⁰ The case of the simplest infinite set (or Cantor’s ordinal omega) should divulge the key. We could be reading—and we will so be reading with the realist in a moment below—the claim that an infinite local object-a set—exists by reflection on the already given absolutely infinite universe, viz. the infinite set is generated from the prior universe by claiming it (omega) is in the image of the original universe, a restriction of the prior universe to a posterior cut off piece-fragment of the universe. But the classical use of reflection to justify omega is not from the universe down to a fragment but rather the concepted justificatory use working bottom-up. We have and can think-of—and thus we can know—the multiplicity of finite ordinals. We can now apply a blueprint that lassos that multiplicity into a cohesive unity. I believe Cantor— and clearer yet Dedekind (with his idea of Dedekind-infinity and his argument in his theorem  to there being a Dedekind-infinite simple infinite system)—saw with this simple case the key to the general case: reflection is a way of projecting by means of concept, an inbegriff-given system into a subunit (and thus a unity!) thereof.²¹ Elementary embeddings higher up the infinite ranks are conceptually in the same spirit as an application of the conceptualist limitative metaphysics—(i) to be is to be conceived by way of blueprints (written into the embedding j) and (ii) inherent to this conceiving via the j-conditions is the reconfirmation that the universe itself— given our blueprints-language—cannot be conceived. We cannot concept-justify the universe as unity by displaying it as the image of an already given conceptedmultitude. The universe is not and can not be a subunit of a previously given multitude.

. Enters reverse reflection—top-down unfolding reflection So much for the justification-reflection bottom-up principle. But what if in truth—if not in “intrinsic justification”—the direction of generation was so far all upside down? What if when Cantor framed his three construction principles for ordinals (including his third “domain restriction”) and called them “generation principles,” he reversed—for epistemological reasons of justification—the true generation order of the ordinals, from the absolutely infinite universe down? What if the Gödel stricture in the above quote—to give the essence without reference to the universe—should be ¹⁹ In the case of replacement, we are using first-order specifications of the function that projects as a set the image of the function. Use of full second-order means in the allusion to arbitrary functions might be questioned (by some) as to “intelligibility.” ²⁰ Notice that over classical first-order set theory, Montague’s reflection theorem is equivalent to the power of infinity and replacement. Cantor already knew that replacement is a “large cardinal” principle and in the very sense here alluded to—a large cardinal axiom is a manner of using a closure condition to turn a universe-large inbegriff into a unitary set. ²¹ See Dedekind, in Ewald . It is quite possible that Bolzano had this idea in his work on (what we’d call today) the concepted multitude/proper class of all truths.

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       



inverted and the only way to understand the nature of local sets—their true DNA as it were (rather than how we come to justify their construction)—is by relating them to the originator-body, the universe as ur-object? We would not only reverse the order of generation (and with it of understanding, if not of proof and justification); we would not only reverse the order in which the metaphysics determines our knowledge thereof; we would now also be reversing the very direction of the “reflection.” I explain this in a moment but, before I explain, let me note that both Cantor and Gödel seem to have played with this idea of every local unity being a product subunit of an original ur-object that determines structural features of the sub unities in its image. Says Gödel: The principle that every math[ematical? metaphysical?] proposition has a generalization for arbitrary higher cardinality (but not the other way around) expresses one of the most general properties of the structure of the world. Namely: Everything is mirrored in everything. (The symbol and the reference are structur[ally] the same?) God created man to His likeness. The same thing appears at different levels. Here we have an “unfolding.”²²

I will borrow from Gödel the word “unfolding” (“enfaltung”) to introduce a title for our reversed form of reflection from the universe to local subunits, unfolding reflection.²³ In the standard use of justification reflection we say: if the universe has infra-structure property P, so already does a local witness set S; in the converse and reversed version we now say: if a local set S has infra-structural property P, so does the universe. Why would we say this? We must recall that in both cases contraposing the conditionals is revelatory about what is the source of what. With justification-reflection, we see in the contraposed form that: if none of the already available local witness-sets S bear P, it is not true that the universe makes (on its own) P true. The ground of truth for the universe’s being P is the features of the prior local sets. In reverse reflection, the contraposed form tells us: If the universe does not have property P, none of the generated local witness sets bear P. Here, we do not have justification driving us, but the determination in situ— the local sets, restrictive transformations of the prior universe, are in its image. If the universe does not have a given infra-structural property, there is no other “maker” to make unfoldings-thereof bear P. Either the universe makes the unfolded local sets the way they are or nothing does.²⁴ This idea is one of metaphysics, not of model theory, a determination of local unities by the mother unity that made them the unities they are. Let me lay out the

²² Gödel quoted from Leibniz notebooks in Gödel-Nachlass –. ²³ I harp on the productive-generational process connotation of unfolding (as in “unfolding carpet”). Elsewhere in his  remarks on the concept of set, Gödel speaks of an unfolding concept. For us, the current quote and its process productive ring is closer to our intent of the universe itself doing its own unfolding. ²⁴ Of course we focus here, as does the standard use of bottom-up reflection, on infra-structure properties. The reversal of reflection is separate from a theory of exactly which properties should so count, a matter I do not enter into here.

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   type of reversal targeted by attending to a more local case familiar to us from mundane metaphysics.²⁵ We consider the relation between the human being QE II and the species Homo Sapiens. We are focused here on the nature (also called by others, like Saul Kripke, Kit Fine, or Gödel above, the essence). We can see the two inverted forms of reflection as two ways of pursuing what is fundamental to this specific woman, QEII. As before with our question whether the universe is of-the-sets or rather the sets are of-theuniverse, here we are to hold before our mind the question: is the species of-theindividual humans or are they—the humans—of-the-species?²⁶ The essence-path of the conceptualist first. The conceptualist is well aware that the universe and its biological species had a long history before QEII came into it. He is also aware that she is a human being and that this is a fundamental (“structural”) fact about her. How does he go about it? He often abstracts from the nonlocal universe-history, including that of the human species, classifying them all as mere historical facts but not “structure pertaining.” Instead, he reverses the historical/ generative order to fit the epistemological situation of one who is set out to prove— and thus to justify—his classification of “human,” but say not “horse-rider,” as fundamental to her nature. The conceptualist-essentialist gambit is in effect to encode within her concept/essence, the way we think of her and think of her as a unity, encode all the key determinative factors that naively one may think come to us from the generative history. A local concept replaces a global universe-process. In tow, the local entity before us—the Queen—becomes the entry point, whereas the species is as of yet unmentioned. It may come in as a late entry, at that assembled, again by a concept, from the multiplicity of other individuals like the Queen, the humans. On this picture—from single individuals up to their species—we have what I called a bottom-up justification-reflection: if the species has some structural property P, say its members carry DNA D or are rational, we expect to witness it in some local human, say Elizabeth. Contraposed, we say with the conditional: if none of these local humans bears the property P, the species Homo Sapiens cannot bear P. As before, we need to filter unwanted structural properties that pertain to the species as a whole (it is ancient, its members interbreed, it has a close to / female/male distribution, etc.). What is key is that we view the species as a posterior assembly—by a guiding concept—from the multiplicity of individuals, reversing the historical-generational order, the order those individuals come into existence by being generated-by and ²⁵ Obviously there are formal-descriptional ideas induced by the determination and its preservation of original universe structure, the “in its image” transformation “loses” the absolute infinity etc. but retains other structural features. This demands us to craft a language for the universe, the language-of-theuniverse as it were. I prefer to develop at this stage intuitive basic control of the idea—see the next few paragraphs—rather than to launch into premature model-theoretic encodings. ²⁶ The answer is not to say in the standard distinction-making clever way, “well, these are two different uses of ‘of ’.” Perhaps so. The question is what is the difference and to what different pictures of metaphysical generation (existence) they lead us. The two inverted ways correspond to the distinction between the universe-involving idea of the nature of local entity x vs. and the localist and universe-free notion of “essence of x.” I have discussed a cluster of examples of how the two pictures differ in the earlier Almog . In any event, I do not use “nature of” and “essence of” interchangeably. I do relate “concept of” and “essence,” as does Gödel.

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       



being nodes-on that species’ tree. The reflection is bottom-up—how things are with the locally available individuals induces the property of the posterior and at that conceptually constructed species. The order-of-ontology-making, e.g. species-making, follows the order of epistemological construction and assembly. Justification rules existence. The reverse unfolding-reflection does not invert the metaphysical-generative order only to simulate it in an epistemology friendly way, local to global. We let the process of generation determine the local products. Rather than cleanse the structural properties of allusions to the primal global object, it is exactly those relational processes that make for differentiation and identity formation of distinct humans. When in our cognition we think-of Elizabeth, we follow the metaphysical-generative order: we think of she who was made by the species by this specific process and if there was no species to so make-her, we would not be thinking-of-her. This does not mean that we entertain in our feeble minds a long description of the lineage of Elizabeth ever since our mother Lucy. We think de re, de-Elizabeth, of her. But our thinking of-her is in effect only possible because in-existence, she was made by the species in the way that she was made. The reversed form of reflection is now topdown: if Elizabeth has a structural property P (DNA D, is rational, etc.), the species has P. Contraposed: if the species was not bearing P, no local products would bear P. And what is good for such structure-properties is sure good for the sine qua non of it all: existence. If Elizabeth exists, the species exists, viz. if the species had not existed, she would not.

 Justification vs. Unfolding Reflection Let us return from species to ordinals and sets. The contrast between justification reflection and unfolding reflection and the attendant contrast as to what-determineswhat (the global the local or vice versa) has been articulated. Doubtless, there will be many—with what I called “split universes” outlook, an outlook that is integralessential to the conceptualist approach—who will separate the causal-historical generation of species from the ahistorical, indeed acosmic, existence—with no coming into it—of the mathematicals. Indeed if anything, the conceptualist looks forward to reducing the assembling into existence of the historical species to the model of the mathematical inbegriff and its assembling-ensembling a set, whereas our path is reversed here too: we look to understand the universe’s generation of ordinals and sets (mind you, two separate generation processes as we shall see) on the non-conceptual and process-ual model of the species. Doubtless, no energy-transfers involving reproductive exchanges are involved in the universe’s making of the ordinals and sets. Nor is the difference due in my eyes to sets and ordinals, unlike species, being “out of” this universe and as it were in “another one.” No-thing is out of this universe, to be a thing is to be of this universe. Rather, the generation of the ordinals (and, in turn, by them of the sets) is not a case of energetic causation because, unlike the species or the planets or the atoms, the very existence of the originator being, the universe, has written into it at the out-set, that it has built-in the ordinals and in their wake, the sets by self-causation (better, selfgeneration). There was a moment in the life/existence of the unfolding universe without the species, but none without the ordinals and sets.

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   So, on the unfolding universe idea, it is not that the ordinals and sets are “made” elsewhere, in another universe or a pre-universe factory, but it is that the very existence of the universe—and by structure-preservation as it unfolds—makes the existence of any local-image sub-unity. So not only has the absolutely infinite originator universe in-it the absolutely infinitely many ordinals and sets, but any produced local images, the moon, or Barack Obama, have in them too—each has singly—the (absolutely infinitely many!) ordinals and sets.²⁷ This requires a few more words of explanation and we turn to these matters below. But to summarize the foregoing discussion about reflection, we need only to note that for the unfolding universe metaphysician the reflection is from the originator universe downstream: it is because it is an absolutely infinite being/unity, that its ordinals—its images—make an absolute infinite plurality; and in their wake, the sets.

 Reverse Metaphysics: Conceptually Justified Construction or Generative-Universe Processes? The reversals we attended to lead us to our final reversal, the one regarding method— the relation between metaphysics and epistemology. More important even than the modernist legalistic concentration on epistemology and its justifications, we are led back to the more “ancient” relation between metaphysics and our cognition of the world and our understanding of it. Part of the reversal broached here is not just an order-of-investigation reversal between epistemology and metaphysics but a shift from the modernist, post-Kantian, interest in the—as if juridical—matters of proof and justification and back to a pre-Kantian attention to our understanding. The truth about the universe may or may not set us free but it does set us more perceptive; the proof of these fundamental truths may—as is familiar from courtroom dramas—set us justified but partially blind. And so, before we prove and justify this, that, and the other, we may want to understand this, that, and the other. And to understand this, that, and the other, we may have to understand the universe. If in truth, this, that and the other are the universe’s own unfolding into this, that, and the other, to understand those locals, we may have to walk the path the universe took to make those local images-of-itself. This much does not indulge in proof which—on the present turn—cannot precede understanding. And in tow, it does not indulge in the basis of proofs, viz. “axioms,” at least not in the standard justification-epistemology sense of “conceptually transparent” starting points of argumentations and justifications. We can argue and justify later. If axioms there are, for us they are fundamental truths of and about the universe, transparently justified or not. Let us now see how these ideas cohere into an alternative to justification-bound constructed metaphysics, in particular in the

²⁷ Recall Gödel’s enfaltung-prinzip of fn. . This universal imaging from universe to subunits should remind the reader of Kit Fine’s question: In what way does Barack Obama have in him the singleton {Obama} and, in turn, {{Obama}} and . . . ? Also, in what way BO has in him the absolutely infinitely many ordinals , , . . . omega, omega+ . . . .? I have written on this in Almog  and revisited the matter at the end of Almog . I enlarge on this below in the Appendix dedicated to Fine’s views on singletons.

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       



laboratory case of the absolute infinity of sets and ordinals and their relation to the “universe.”

 De-Universe Realism I: The Micro-cosm Case of Omega We can get a bit of a grip on the macro-cosmic case—the universe U—dispute between the conceptualist and realist by looking at the micro-cosmic case of the infinitary object, obj(Omega).²⁸ This focus on the micro-cosmic “small”-scale analog is driven by Cantor’s methods of introducing the new idea of an infinite actual object in his  Grundlagen. Of course, as I mentioned, Cantor himself is symbiotically an idealist/realist and there, in the mathematical-philosophical motivational sections –, Cantor practices both (i) an idealist bottom-up universe-free local constructive conceptualism but also, (ii) in various “metaphysical” remarks, he dabbles in a global top-down realist account—he urges us to consider a form of absolutely infinite object-universe realism, as the underlying part of the iceberg whose tip—but it is only a tip—is the bottom-up generative (“construction”) principles he himself proposes. In those moments, Cantor proceeds as if the absolute infinite object-universe (not to be spoken of officially) makes possible the reverse engineering by way of the bottom-up reconstruction of the ordinals he will offer. When Cantor criticizes in sections – of Grundlagen the conceptualism of Aristotle (arguing in effect for a top-down actual infinity metaphysics) we are focused on the case of obj(omega). There, in connection with the “simple” case of the first infinite ordinal-object, we already have many of the controversy-making ingredients that will recur higher up with the full universe U.²⁹

 The De-Universe Realist II: Obj(Omega) First The realist sees the conceptualist as missing the true metaphysics, jumping in only in the second half of this “the unfolding universe” film. For the conceptualist synthesizes obj(omega) from below by a concept (recall the justification reflection principle or Cantor’s own two from-below generation principles). To the realist this is a late in the day—second half of the film—attempt to re-construct (and justify) obj(omega) by ²⁸ We need to keep our nomenclature clear. The sign V has been used for the “universe” in various ways—(i) as an (ultimate) model of set theory, (ii) as the union of all segments V-alpha built iteratively but still just a proper class (inbegriff) and (iii) as an alleged black-hole originary absolutely infinite object. To speak of the universe as the ur-object (with no connotation of set or ordinal theory yet) I will use U. To indicate I relate to a given ingredient, say omega, as a unitary object I will write obj(Omega); if we relate to the corresponding concepted multiplicity(“frame”), I write inbegriff/frame(omega). ²⁹ Later, in the s and culminating in the elegant  letter to Dedekind, Cantor shifted to a local conceptualist and universe averse view. The modern class (as many, as manifold) vs. set (as unity) shows up in an early form in his  letter. Fine’s elegant defense of Cantor’s abstractive (from concepted sets!) notions of ordinals and cardinals is given in Fine’s work on Cantorian abstraction. This theory of sets, cardinals, and ordinals is in my view from the universe-averse and concept-friendlier period – of Cantor. It may be that in his metaphysics Cantor remained through and through a de-universe-realist. But in his mathematics, concepted set-reductionism took over.

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   projecting it from inbegriff/frame(omega). But where is obj(omega) coming from? asks the realist. And where does inbegriff/frame(omega) come from?³⁰ From the ground-zero object  or  by taking successors and eventually a “limit” of all these successors, says the conceptualist. To this, the realist retorts: you are totally missing the first half of the film that makes possible your second half reconstruction—the reversed engineering, In the first half, in the existential order, obj(omega) is primal. This is the universe realist ground axiom: obj(omega) exists Because this objectual claim is true, a further—but this one is derivative—existential global claim hold, now about an inbegriff: inbegriff/frame(omega) exists Because these two originary claims are true, local finite ordinals such as  and  and  exist and indeed make it true that— + = . Obj(omega) determines inbegriff/frame(omega) at once. And this makes all the finite ordinals exist at once. We have to separate two parts of the “unfolding omega film,” metaphysics first, then reconstructive epistemology. In part I, Obj(omega) is the primal existent making up inbegriff/frame(omega) and the individual finite ordinal numbers. The reversed engineering comes in stage-, encoded in the succession of constructions carried in PA like theories, with a final eventual climb to infinitary type omega is possible at all because, to reiterate, in the metaphysically antecedent part , pre-our reversed engineering history, in the history of the universe’s own engineering, obj (omega) made inbegriff/frame(omega). In the existential order, it is the infinitary object that generates the manifold, rather than manifold being “collected” by an act of mind into the infinitary object.

 The De-Universe Realism III: The Primal Existence of U This two-part film—top-down metaphysics underwrites our bottom-up reconstructive engineering—is generalized higher up when it comes to the universe of sets. In part I, we start with the full object-universe existence fundamental truth (“the axiom of primal universe existence”): U exists In the metaphysical order of determination, this is our ground axiom. The symbiotic idealist/realist minimizes direct existence axioms; he proceeds prudently by justification-reflection; and proceeds from the “small”-local to more and more extended locals, never reaching the genuinely global. Notice that here the notion of ground axiom is metaphysical—a fundamental truth determining all others—without any promise that this truth is epistemologically “self-evident.” It will turn out, ³⁰ I use “frame” as the English cognate of inbegriff, instead of multitude or multiplicity which already have various subsidiary connotations.

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       



however, that we do know this truth and with the highest certainty. But this is a separate question, not one to determine what an axiom is, what in the order of truth determination is primal.³¹ In contrast, the realist starts with Reality—with a true objectual existence. The nature of this prime global object gives us in turn, as local-restricted-images an absolute infinity plurality of locals. Realism—properly so-called—starts with existence, not with concepted essence; and starting with existence, it starts with the prime existence underwriting the existence of all and of the-all. Now, what-who exactly is this global ur-object U? There are trans set theory questions about whether U is one and the same as the so-called material universe.³² But at this point, what is “logically” key is that we start with “U exists.” And far from it being indescribable—the black hole object CantorGödel-Ackermann spoke of (while saying we cannot speak of it)—the universe realist argues that, to the contrary, U is the prime object that, in studying sets/ordinals, is (i) known, (ii) thought-of, and indeed (iii) is in existence, for it is Existence (with capital e, not the derivative predicate applying to existents and meaning: is of Existence). When something local, e.g. the first infinite ordinal omega or the first uncountable ordinal omega  exist, they exist as product-transformations of the absolutely infinite U. Of course they preserve, by their very nature, some of that ur-object structural properties, and in this sense they reflect the object-universe U; however, some structural features they lose e.g. the absolute infinity of U. The local sets (ordinals) cannot and thus do not reflect the object-universe on the very fronts making it the (unique) object-universe, and making them “mere” produced-parts or un-foldings thereof. As existents, the local ordinals(sets) are given their identity-and-nature by determination from U. We reverse the conceptualist’s explanation. Ordinarily, we scan the transitive closure of a local set S, tracing its members (and how those members got generated from below etc.) but we stop at a ground zero that is the empty set or the number  or  or some other local ur-element, say Barack Obama. But in the de-universe realist picture, the determination is from U downwards and the segments differentiating  from  or omega from omega  trace the unfolded workings of U. In this sense, we can gauge the type of preservation of U witnessed in this or that local set. The very finite set of the coins in my pocket—{A,B,C}—preserves less the structure of U than does the set N, the set of natural numbers; the uncountable set R, the set of real numbers, whatever its cardinality in the aleph-sequence, is more like U than the set N is. So the finite sets which are the paradigms the Kantian conceptualist starts with, those very finite sets that many expositions of set theory seek to generalize bottom-up to the infinite, are from the de-universe realist perspective, the most

³¹ See immediately below on the certainty we have of our knowledge of U’s existence. ³² So-called because “material universe” by its very use of the restrictive adjective suggests to many—as in “italian actor”—that there are other types of items of the kind (actor, universe) who are not bearing the adjective (are not italian, are not material). If we read the phrase (especially with the definite noun phrase taken strictly) “The material universe” to mean “the universe, which in particular has material features (objects) . . . ,” I do not see the terminology as begging metaphysical questions.

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   un-characteristic unfoldings of U. E.g. when we x-ray U’s structure and the large landscape of infinite sets, we realize it is an oddity (the exception!) that finite sets cannot be put one one with their proper parts (subsets). An “average” set—read on the model of “the average swede”—in inbegriff/frame(V) is one that does (i) allow such mappings that (ii) make the set and its one-one proper subsets share an aleph. In like manner, the situation common to infinite ordinals of a number class, say omega and omega+, that share an aleph is better at reflecting universe structure, than the exceptional situation involving finite ordinals, wherein distinctness say of  and  sends them to different cardinals. Indeed from the overall universe point of view, the first number class is anything but a basis for generalization; it is very odd indeed, every two ordinals thereof running to different cardinals. In these ways, the infinite sets (and ordinals) are more like U than the finite sets. Of course U as an absolute infinite object does not have an aleph at all. But it induces the absolute infinity of inbegriff/frame(ordinals). This absolutely infinite inbegriff/frame of all ordinals is “more like” U than any given large set, even if that large set reflects (in the technical sense) a substantial range of discursive truths about inbegriff/frame(V). Notice that we have not yet said anything about which among the three categories of infinite-sets, infinite-cardinals, and infinite-ordinals is more “like” U. But the “likeness” just noted of the inbegriff of all ordinals to U is very important.³³

 The De-Universe Realism IV: Thinking and Knowing U Local sets are transforms of U, their nature and identity determined by the path U took to make each. So much concerns the metaphysics—existence-identity-and-nature—of a given local set. But what about thinkability and knowability? Ackerman (Cantor-Gödel) made U be beyond the pale. If each local set is now U-dependent in its metaphysics, will we not end up putting local sets beyond the cognitive pale? On the universe realist picture, any piece of knowledge, be it about the “tiniest” set, like {Barack Obama}, and all the way to N or R and yet higher up all the way to the full inbegriff/frame(V) are manifestations of thought about and knowledge of U. Every sighting of a particular human being, say QEII, is a sighting of—the species homo sapiens; every sighting of bits of water on the Santa Monica Beach or on Vancouver Island is thought about and knowledge of—the Pacific Ocean.³⁴ It may well be that a space station full cover image of the Pacific Ocean gives you more qualitative information about that body but whenever one cognitively interacts with the ocean, whenever it impacts our cognition, we think (and eventually) know of the ³³ We return to this in our ordinals-first development below. ³⁴ The ocean example is due to Descartes in his replies to objections on meditation III where he starts with an actual infinite substance and views all being-s as limitations/restrictions thereof. Descartes observes that in spotting the wave, that local determination, we perceive (if not form an image) of the full ocean and in spotting/cognizing local beings we perceive (and through them) the infinite substance in its local manifestations. Spinoza further develops some of these themes of connecting-cognitively with the infinite global object by way of its local determinations.

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       



Pacific Ocean. Likewise: in thinking and knowing a local set one thinks about and knows-of the full U.³⁵

 De-Universe Realism at Work V: Concept vs. Object; Finite vs. Infinite; Set vs. Ordinal; Cardinal vs. Ordinal I would like to turn to one last suggestion we are led to by the de-universe realism picture. The suggestion interrelates four separate dimensions that have been lurking in the back of our discussion: (a) (b) (c) (d)

The relative priority of concept(s) and object(s). The relative priority of finite and (absolutely) infinite The relative priority of sets and ordinals. The relative priority of ordinals and cardinals

The conceptualist offers here one natural—and rather “standard” by now within set theory—organization of the four dimensions. The de universe realist offers quite a different organization. Here is my reconstruction of the local conceptualist’s thinking. The fundamental language of thought—and thus for him, of Nature—is that of sets as conceptedunities. The binder concept makes a together-object, an ensemble, a unitary-set. Absent a binder concept, we are left with a mere inbegriff/frame of given items. Given this idea of concepted unities based in the finite case, two extensions are now going to seem natural. The first is still working on fixing the “grammar” of the fundamental language of thought, the language of concepted unities. The conceptualist seeks an extension to principles of infinity to proclaim the existence of some infinite local concepted unities. The series of bottom-up strengthened extensions, e.g. the axiom of infinity, ³⁵ Our focus was for a while the absolute universe U. We also spoke of the “material universe”—with Barack Obama and the Milky Way—viz. the cosmic universe (or Nature with capital N). In each case, both for U and this alleged “material” variant, the determination of local structure traces back to preservation of structure inherent in the original object-universe. But how could U effect—induce structure into—the material object BO that seems to lie a universe apart? Are we not back with split universes? We are not. There is one object universe U and it has in it all the absolutely infinitely many sets of inbegriff/frame(V). In this sense, U, the sole universe there is, is itself absolutely infinite—(i) it-proper is absolutely infinite and (ii) it has in it absolutely infinite many objects (ordinals, sets). Indeed Barack Obama himself, a local tiny cosmic object made by U, has in it not just {BO} but the full absolutely infinite inbegriff/frame one may generate from ur-element Barack Obama, inbegriff/frame V(BO). I do not want to dissect the many reasons—some from epistemology, some from metaphysics— philosophers (and some mathematicians, e.g. famously Hilbert in his “On the Infinite”)—have thought the universe—the material one that is—is too “small” to have infinitely many “things” and surely too small on the vast sizes of transfinite ordinal and set theory. In my view, this is a misconceived problem analogizing with the problem of suitcase-packing of physical objects. If you want to think of the universe as a suitcase, it sure has problems of packing a certain amount of matter (though heaven knows we are in the dark about that!). But as a suitcase, it can take all the absolute infinitely many ordinals and sets there are. Indeed when, as our astrophysical theories suggest it was  billion years ago, the universe was very tiny, it already had in it all the absolutely infinitely many ordinals and sets; it-itself, the object U, was thus already absolutely infinite. We return to this matter in the Appendix dedicated to Fine on singletons, wherein we discuss the absolutely infinitely many sets that are in Barack Obama, viz. Inbegriff/frame V(BO).

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   the axiom of replacement, the axiom of the first strong inaccessible, and so on higher up (at least for a while), all take this form—we search for a concept to unify and objectify as a local set something given to us previously as an inbegriff or a frameof-objects. This was one dimension of the extension from finite to infinite sets. Another dimension of extension (in both the finite and infinite cases) is from concepted sets to numbers, in particular ordinal numbers. There are two major strands of conceptualism at this point and their inner differences are very interesting. But on the main front I am focused on, they share the same idea—ordinal numbers are concept-driven assembled sets. The first view is due to the late – Cantor and was developed beautifully by Kit Fine.³⁶ We start with a concepted-set of  elements and to get the ordinal number , we abstract from the specific identity of the elements, keeping the ordering. This well-ordered abstracted/constructed set is the ordinal number . We can take a further step, and further step it is, and drop the ordering, ending up with the cardinal number . In the case of finite sets, we are often told the ordering will not matter anyway. And so the locution “the number of items in the set” can skip over ordinals and we can think of it as the cardinal number . When we transit to infinite base sets, the ordering suddenly does “matter.” We may abstract the ordinal omega+ as one well-ordered set; we then abstract omega+ as another wellordered set. The difference between the two ordinals traces to the concepts governing the abstractions; the concepts are different. We may then bring into play a yet higherlevel concept and abstract-synthesize the cardinal number aleph- shared by both of the just mentioned as well as infinitely many other countable ordinals. Alternatively, we may give the finite ground case yet more power and regard orderings as altogether posterior matters. What is now the key on this second view— it is inspired by Frege and earlier Aristotle—is the concept we operate with—say “red”—and the cardinal number will be synthesized from it as the number of that base-concept. In the general theory,  will be the concept unifying all -membered concepts, not unlike the idea that Barack is to be understood as the old philosophical constructional idea—the object satisfying all the Barack-true concepts. So cardinal numbers are prior and the ordinal number  will now seem as derivative of the cardinal number .³⁷ There has been much discussion of the relative merits of these two positions. I will not enter it. I want to stress their common ground—numbers—ordinals and cardinals—are made into concepted unities, a certain kind of set. In my view, one manifestation of this conceptualization of ordinal numbers is a very interesting discussion Cantor engages in his Grundlagen, at an early time (), when he was still not set or concept-involved at all but rather seemed to start with ordinal numbers. At one point, in responding to his philosophical critics’ skepticism about transfinite numbers, he mentions a potent conceptualist argument of Aristotle from the arithmetic of numbers. ³⁶ Fine . ³⁷ This basic idea of Frege to start with the concept (of all one-one lower-level concepts) is independent of his further idea to take the extension of the concept as the cardinal number.

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       



Put roughly (it is so put in the Cantor original), Aristotle’s complaint is that with transfinite ordinals the operation of addition is not commutative. This should dampen our thought that with infinitary ordinals we are operating with the same notion of addition, and consequently, with the same notion of number (or ordinal) as we did with  and , where + made +. When it comes to omega+ and +omega, we see immediately a difference due to the orderings. Aristotle now makes his second critique: the treatment of a unit such as “+omega” would amount to the disappearance of the finite sizes, “swallowed” as they would be by the infinite component. Pervading the Aristotelian objection is a sense that it is cardinal numbers that present the “natural” notion of number. A conceptualist bent on our first blush use of finite numbers to classify concepts or sets (by size) is naturally led down this path. I believe Frege so read numbers (in his  Grundlagen). His wish to make them “logical objects” is subsequent to the their being concept-based. In any event, the kind of “object” he makes them be is a concepted-object par excellence.³⁸ So quite apart of the details of Frege’s eventual (ill-fated) theory of logical objects (“extensions”), the philosophical idea operative in the Aristotelian objection and the priming of cardinals is that numbers are concept-induced. I am inspired to read the objectual ordinalist—and set free—discussion of Cantor in  as leading us to quite a different picture priming ordinal numbers (and as objects!) prior to any concepted entities, in particular sets. With this ordinalist development, we are presented by the de universe realist a new reorganization of the above (a)–(d) factors: regarding (a), the key to object(s) existence is not concepts but how the originating primal object produces them-ordinals as its images; regarding (b), the absolute infinity (of this ur-object) precedes ordinary infinity which in turn precedes finite unities, the most uncharacteristic unities; regarding (c), ordinals precede sets and indeed we are to understand sets as induced by the absolute infinity inbegriff/frame of the ordinals; finally, as for numbers, ordinals precede cardinals. Let us develop this reorganization of (a)–(d). What picture emerges here? Kronecker is notorious for saying “God created the integers; all the rest is the work of man.” Now, Cantor saw Kronecker as his nemesis but as often in such exacerbated passions, there is an underlying blood relation fanning the flames. I see early Cantor as making us think—within functional (set-theory free) mathematics—that we need a slight generalization of Kronecker’s philosophy that arithmetic is the backbone of the universe. On my reconstruction, Cantor leads us to a gentle twist on Kronecker: God (the universe) made the ordinals; all the rest is the work of man. Now, regarding the first half of this conjunction, on my understanding of Cantor, he tells Kronecker that God could not make just the integers without making the transfinite ordinals; the finite integers are just a tiny fragment of the former.

³⁸ This remains true if we move from Frege’s own brand of logical objects to attenuated modern neoFregean versions bent on avoiding his law V.

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   What is more, if we considered the species—ordinals—in full and not just a tail end—the finite ordinals—we’d see its true laws, as in the non-commutativity in the general case, with commutativity reflecting structural facts about whether the two summands are symmetric—omega and  are not symmetric in their paths to existence (they originate in different number classes). What is truly fundamental about all ordinals comes out by looking at the full inbegriff/frame(ordinals) itself—it is in the image of the universe as the sole absolutely infinite object U. Now, moving to the second conjunct, we find “all the rest” and “the work of man.” As for all the rest, we should separate (i) the rest coming from functional mathematics to which Kronecker was very friendly—rationals, reals, complex numbers, elliptic curves, etc. But also we can mark as part of the rest, Kronecker’s enemy, (ii) (the then nascent) set theory. As for, “the work of man,” I will understand this to mean “the defining work of human concepts.” I believe that on the objectual ordinalist metaphysics we are now considering, the ordinals are meant to provide the metaphysical generative basis for both the ontology of functional mathematics—e.g. the real numbers—but also by a different route, to ground all the purely set theoretic ontology as induced by ordinals. The first part, e.g. provide for the real numbers, involves questions about finding in the ordinals and tools of functional mathematics (e.g. functions, orderings, etc.) the makers of the topology of the real line ordinarily understood by means of the universal language of set theory. I do not here want to enter this complex issue but only say that the ordinalist metaphysician is indeed committed to finding within ordinals and the set theory-free tools of functional mathematics—the means practiced before the advent of set theory, e.g. in Cantor’s work in the s on the sizes of various number multiplicities such as the algebraic numbers, the reals, n-dimensional spaces, etc.—the makers of the ontology of arithmetic (including rational, real, and complex numbers). The real numbers are just more numbers and the ordinal numbers are them-all; and so, we must find ways to engender the real numbers from the ordinals. On the second front—where do sets come from?—the objectual ordinalist is reversing the conceptualist’s diagnosis. Originally, the foundational language of concepted sets was introduced to reason about, in a sort of extended Begriffschrift—a concepted-unity-schrift—about the transfinite arithmetic of transfinite ordinals. The Begriffschrift of set theory was going to give us a medium for epistemology—a medium for Reason’s reasonings— and provide a complete account of the untamed transfinite ordinals. But as often in such an epistemology concept-writing, it is initially introduced to provide a proof theory and mind-friendly control of the untamed metaphysical objects and their structural truths. But soon we reverse engineer things and declare the metaphysics as done from now on only within the bounds of the nascent conceptual-writing epistemology. It is thus that Cantor turns in the s to concepted unities, his limited ensembles, as the conceptschrift with which to reveal the structural truths of the ordinals. The foregoing question of relativity priority of ordinals and sets is not of mathematical interpretability, for we can go back and forth between the language of sets and the language of ordinals. The question is one of metaphysical fundamentality—

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       



what objects the originator object U generated first so as to generate next further objects in its image and in their (the now already induced absolutely infinitely many items) image? At a certain point of his review of recent work on inner models of set theory (with large cardinals), Jensen draws the following distinction, which I quote in full in view of the intriguing character of the distinction:³⁹ Could it be that the duality in modern set theory is nothing but a new manifestation of an ancient conflict between two points of view—I almost want to say two emotional states—which have always existed in mathematics? I call them the arithmetical and geometrical points of view. I also call the first one the Pythagorean point of view, for Pythagoras expressed it in its purest form: Everything consists of numbers. In other words, every mathematical structure can be interpreted in the natural structure of the positive integers. This idea is naturally very attractive; it gives to all of mathematics the intuitive clarity of the natural numbers. I conjecture that, if it could really be carried out, it would still be the dominant point of view today. In reality, however, the geometric point of view has been dominant since the rise of analysis. Thus I also call it the Newtonian point of view. The Newtonian directs his gaze to the real rather than the natural numbers. He is less impressed by their clarity than by their boundless multiplicity. The real numbers constitute a gigantic, unfathomable sea. For every principle that generates real numbers, there must be a number not attainable by that principle. This excludes the possibility of an interpretation of the real numbers within the natural numbers. If one accepts set theory, then there is no doubt that Cantor has refuted Pythagoreanism in the strongest terms by showing that there are more real than natural numbers. But Cantor also introduced the ordinal numbers, which are in every sense the transfinite continuation of the natural numbers. They share much of the intuitive clarity of the natural numbers. Thus Cantor, who refuted the old Pythagoreanism, made possible a new Pythagoreanism in which the ordinal numbers take over the role of the natural numbers. . . . In this sense, Gödel’s axiom of constructability seems to me to embody an entirely coherent Pythagorean picture of the world. And this picture cannot be refuted, for Gödel showed that V = L is consistent if the other axioms are. But this axiom provides—modulo the ordinal numbers—a complete description of all sets and is therefore unacceptable for a Newtonian. For him, there must be a real number not generated by constructible processes. Since one cannot prove in set theory that such a number exists, one must seek new axioms. Thus, the ancient conflict is fought in a new arena.

Perhaps the distinction I am after is not quite the one Jensen points to. After all, the “Newtonian” or geometric notion of “arbitrary” set of integers, I rather trace to a generalized logic of concepts, taken as a Begriffschrift—be it Frege’s schrift or Zermelo’s set based schrift—in which we reason with concepted-unities given in terms of the items that satisfy the governing-synthesizing concept. This is another— logical/conceptual/model-theoretic—route to the arbitrariness of subsets.⁴⁰ And on the other side of the divide, the Pythagorean metaphysics, I am sure Jensen would not ³⁹ Ronald Jensen, “Inner Models and Large Cardinals,” BSL, . I owe thanks to recent interaction with Jensen, Kai Hauser, and Hugh Woodin on the distinction introduced in this penetrating quote. ⁴⁰ Many modern writers (e.g. in the case of “justifying” the to them obvious falsehood of CH) speak of the allegedly obvious “heuristic” truth of set theory looking to maximize the span of subsets (of the integers). This is a conceptual observation often tweaked further to support model-theoretic axioms such as Martin Maximum thought to “maximize”—via forcing extensions—subsets (and refute the CH). The battle “cry” maximize is not clear to me at all but inasmuch as I understand a bit, it emanates from the

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   want to be saddled with a metaphysics of an ur-object universe and, as if this was not enough metaphysics, one of a single unified universe, one universe for physics, mathematics, set theory, one universe to rule them all, with ordinals as the language of that universe’s primitive order, and with induced concepts and their progeny, synthesized-sets, as ways of ours to unfold the universe’s original-and-ordinal-order. This is much more baggage than Pythagoras or Jensen would be happy to carry. So I will use the distinction between the two cultures in my own voice only. The model-theoretic culture set theory had developed and pursued includes current attempts to close off the span of models, to “absolutize back” to a concepted unique universe, by means of extra axioms (richer concept). The model-theoretic path seems to me to vindicate the idea of set theory’s being a mathematics within the bounds of our concepts and thus within the bounds of epistemology, even if the concepts are a tad unfolded and sometimes the epistemology is a tad a posteriori. I read the model theoretization of our path to the universe of sets as the technical manifestation of the conceptualization of the path to the universe of sets. The case of the model-theoretic principle—reflection—as driven by metaphysical conceptualism about sets, what Gödel calls the concept-of set, is a potent example of this conceptualization by the technical way of model theoretization.⁴¹ Gödel’s general hunch that the route taken will be of an unfolding concept was like many of his hunches quite right. The recent Woodin efforts on the CH-looking for an ultimate inner model, ultimate L, indiscernible from the full inbegriff of sets V, do not refute this hunch. We attend to the kind of large cardinals involved and the kind of inner models sought and the kind of absoluteness engineered, which makes the concepted path to the universe more complex in its demands. More complex, but still within the bounds of an unfolding concept of set.⁴² concepted-multiplicity origin of (sub)sets. Conceptualism—and by way of model-theoretic axioms—leads naturally enough to combinatorial maximization of subsets. ⁴¹ As explained earlier, I believe Zermelo’s  proof of categoricity for second-order ZF (up to a rank) is another such methodology of carrying out constructive conceptualism by way of model-theoretic tools. His language is full second order and assumes a primitive maximalist-combinatorial notion of subset but with no proof theory to track it. Of course, Zermelo did not expect (or hope) for a uniqueness result for the “height” of the universe which he regarded as “essentially” evolving. ⁴² I so read the intent behind efforts of Woodin in “Strong axioms of infinity and the quest for V” and related reports. Within this set up, as in Gödel’s earlier ordinal-guided construction of L and in Jensen’s (many) measurables involving inner models, the continuum hypothesis turns out true (and really it is no “turn out”). Now, the question of various “evidence pieces” for and against the CH is very complex and some much beyond my competence. But there are many “conceptual” motifs often repeated about “wanting more” (subsets of integers? countable ordinals?) and other rather (to my mind) dogmatic battle cries for “maximizations” as inherent to the idea/concept of set, all mixed (as in the case of reflection “justifying” a proper class of Woodins) with intra-mathematical model-theoretic principles, wherein a model-theoretic principle such as “Martin maximum” (itself bred by a forcing axiom) is made to encode the conceptual battle cry and said (in the minds of many) to “refute” the CH. As I said, the question of what is being built into large-cardinal covering inner models, in the vein of Jensen, then recently Woodin, is technically complex and I do not have a native’s feel for it. But at the conceptual level, Cantor’s old  idea that the ordinals are the DNA of the (set theoretic) universe and they control all-numbers, including the real numbers and the subsets-of-integers coding such real numbers, is in the vein of the ordinalist-first metaphysics outlined above. Sets do not have their “origin” in concepts and the model-theoretic codings of these combinatorial-conceptual ideas. Rather sets are induced by ordinals. This of course does not lead to this or that specific idea of generation (“definition” from) the ordinals and—technically—there are many

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       



The original universal metaphysics inherent in Cantor’s  remarks about a single unified absolutely infinite universe—“the unity of all” he called it—within which spring the absolutely infinite manifolds of ordinals (and in turn, the ordinalinduced sets) is now gone. It is most definitely not the metaphysics lurking in the wings of works on models of set theory. Absolute-universe theory is an outmoded way because it is an out-modeled way of understanding things; model theory (of an underlying concept/theory) is the modern way. The natural numbers (and perhaps obj(omega)) are perhaps model-and-concept free, an absolute universe preceding our (first or even Dedekind-like higher-order) conceptualization/modelization. But when it comes to the absolutely infinitely many ordinals and the sets, the concept (and spread of models it induces) precede and make-up/construct the phenomenon.

 In Sum Let me remind us that I set out to contrast two views of absolute infinity. There is the modern view, embodied in my view in the practice of modern set theory and its models. I called the outlook local conceptualism. It avoids the object-universe like the plague and swears it can characterize all the pertinent objects, each set, without reference to the universe as object or as an inbegriff, the outlook that turns justification-reflection into a central guiding principle. It is a bottom-up methodology, starting from given “small” objects—locally available sets—and proceeding by graspable combinations, building-up, by stages, the frame of all sets (ordinals). Metaphysically, and this is the point of the current chapter, there is a nonconceptualist alternative way of thinking of absolute infinity and its source—the universe being-so. From time to time some speculative thinkers—e.g. Cantor and Gödel—have transgressed the Kantian boundaries of “intelligibility” and nibbled at the big dark object, the universe. In my view our flight from that dark object is not because it is paradoxical or unknowable or unthinkable or worst, does not exist. If anything exists, it does; if anything is thought of, it is thought of; if anything is known, it is. It is Existence and all thought—and eventually knowledge—is of it and its modus vivendi. I rather think it is a certain dominant view—resting on justification epistemology—of how we think and how we gain knowledge that has declared this dark object out of order (roughly because it is out-of-court, the court of justificatory epistemology). Ironically, it is this out of order exiled item that is, in my view, the source of any order (“ratio”) we have. The reverse metaphysics approach suggests that we lay to rest the justification epistemology and its strictures—as in Kant—as to what it is to think of an object. We

such. But the heuristic that sets are generated from ordinals is making, in the case of the second number class, for a more friendly frame for the truth of the CH. The change of mind of those who worked on the CH—Cantor, Gödel, Woodin—is itself indicative of the complexity of the issues (not just what can be proved but to begin with, what one wants proven). An ordinalist philosophy on which “less” sets is more (more real structure of the universe itself, models of theories aside) while at the same time capturing all familiar large cardinals is an interesting development in the spirit of what I see as Cantor’s ordinals-first  philosophy. It is of course said repeatedly that most of the practicing set theorists “believe” the CH to be false.

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   might rather prime understanding of the real-unconstructed world over proving from concepts/model theory of an aufbau made-up-world. Understanding of the universe(and its processes and objects) would then be seeking “axioms” in a different sense than those looked for by the justificatory conceptual epistemology—we do not start with evidentially transparent propositions justified by our ready-made concepts, with the bottom-up reflection principle epitomizing the procedure of projecting objects from our prior concepted models. We rather take snapshots/photographs of the world of processes and things, and do so not by binding ourselves at the outset by discursive theories/definitions and models encoding them. We can study the world by looking at the world, by recalling that there is much more to looking, viz. to having the universe grow on you by contact with it, than scanning the “impressionistic” images or digitally defined-concepts (again primed by concerns with internal epistemology) assumed by Hume and Kant and much of our modern analytic epistemology. In metaphysics-within-the-bounds-of-epistemology, looking-at-the-world means looking inside ourselves—at “images” and “concepts”—to simulate/model-construct the world. I suspect we do not understand the world this way nor our own cognition, itself being a by-product of the unfolding of the world. We are in the country of the blind, where the concepted-model-maker is king. In any event, in a reverse metaphysics, we start with the attitude of the naturalist, the descriptive scientist of—and much rests on this tiny preposition—the Universe/ Nature. We don’t start with my cogitative-ego’s existence and my “form of intuition” and my concepts and first-order model theory as the executive arm of those concepts and the motto so dear to us—again the ubiquitous us—the limits of my language (concepts, models) are the limits of my/the world. We leave me and us alone, at least for a while, leave ourselves for later (so we can eventually really understand those items rather than invent them). And so not unlike G. E. Moore and his waved two hands, we start “naively,” but genuinely looking at the world anyway, with—the absolutely infinite universe U exists. This is not a new candidate “axiom”, for our metaphysics is not “axiomatic” (or proof theoretic). It is meant to be read as a fundamental truth (which axioms are meant to be!), the ground-zero truth. Some would call it transcendental, to signify it is a precondition of all the truths that follow. Indeed. It is the fundamental global existential truth making all other local (set theoretic) truths possible.

Appendix: The Nature versus Concept/Essence of BO and {BO} The key issue here harks back to a remark Kit Fine made a couple of decades ago in the process of explaining why the sieve of essence is stricter than that of modal necessity. His example was of singleton Barack Obama, {BO} and its relation to the man BO. Fine assumed that their existence is necessarily interconnected. But when it came to essence, an asymmetry emerged. It was of the {BO}-essence (what it is, its identity, its real definition, its concept, etc.) that it is the set whose sole member is BO; but not vice versa. One could “understand” (conceive, think

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       



about, etc.) the man BO without mentioning (quantifying, referring, thinking about, etc.) any set. One could give BO’s essence/concept without ever relating it to {BO}. This makes perfect sense on the conceptualist methodology and twice over. First, the concept of a given non-constructed intra-cosmic thing x—e.g. at the “individual” level of the man BO—would be in general nondependent on other individuals, not even those from which BO originated. Thus to the conceptualist, the condition “originated in gametes G” is not part of BO’s essence, even if it is a necessary truth about BO. This non-relationality of the essence of basic “material” individuals is all the more obvious when we look forward—rather than backward—from BO. There is nothing “in” BO to force the set {BO} or the set {{BO}} or indeed the whole absolutely infinite inbegriff of sets V(BO) constructed from BO as the ground-zero object instead of the empty set. I should like it noted that, right or wrong, these intuitions of dislocating the ingredient BO from the singleton BO play on a background assumption of split universes. That is, when it comes to abstract things like numbers or pure sets or subregions of certain spaces (e.g. Euclidean space), Fine would not allow such dislocation. Just as the successor of the successor of  has it in its essence to mention the successor of , so it is the other way round—there can be no successor of  without a further successor of successor, etc. It is because the concept of BO rules the essence-allocation that is seems that the man BO has a new “freedom” vis-à-vis his singleton. For the concept of a man (or even, this specific man) might be thought to be set-free. From the perspective of the universe-realist, Fine’s prediction about the “impure” singleton {BO} is not quite right. The judgment, if not the reason for it, given for the pure cases of numbers, sets, etc. is the right one: the “system” of finite ordinals (hereditary finite sets) comes en bloc and there is no sense of existential dependence (and genuine understanding of the pertinent existents) that breaks the symmetry—there is no  without  and no  without . Indeed all the numbers, inbegriff/frame(omega) are mutually dependent. And stronger yet, as we saw, inbegriff/frame(omega) and Obj(omega) are already mutually dependent (we don’t “construct” Obj(omega) from below). Whatever proof theoretic (or, logical) gap we may spot here—this of course depends in which derivation system we do the reverse engineering—there is no wedge to be drawn between omega (object or inbegriff) and the individual finite numbers. Existentially, the higher object makes possible (and actual!) the lower items, as its “modes” or “manifestations.” This generalizes up all the way to U and its making of inbegriff/frame(V) and thus the making of the individual sets in this concepted multiplicity. On the de-universe realist picture, the situation is not any different when instead of the pure object-ordinal  or {{Empty set}}, we start with the “impure” object Barack Obama.⁴³ BO has in it {BO}, {{BO}}, . . . indeed, all of the absolutely infinitely many sets of V(BO). And sure enough, BO has in him all of the universe’s mathematics. This is because each object is not a groundzero station but a universe-transform, a universe-image, preserving the structural invariants of the maker-universe, invariants that go into the production of any such object. So the key for the universe realist is not resting locally with Barack or the Milky Way, viz. how did this or that thing get to have set theory in it? The key lies in the universe-object that produced the local transforms, such as Barack and the Milky Way.

References Almog, Joseph. . “The What and the How.” Journal of Philosophy. Almog, Joseph. . “Nothing, Something, Infinity.” Journal of Philosophy. ⁴³ The very idea of “pure” objects rests, from the realist perspective, on the split universes methodology. There are no pure objects on the universe-realist metaphysics.

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   Almog, Joseph. . “Nature without Essence.” Journal of Philosophy. Cantor, G. . “Über unendliche, lineare Punktmannigfaltigkeiten.” Mathematische Annalen : –. Cohen, Paul. . “The Discovery of Forcing.” Rocky Mountain Journal of Mathematics (): –. Descartes, Rene. . Meditations of First Philosophy. Edited by J. Cottingham et al. Oxford: Oxford University Press. Ewald, W. B. . From Kant to Hilbert. Oxford: Oxford Univerity Press. Fine, Kit. . “Essence and Modality.” Philosophical Perspectives. Fine, Kit. . “The Logic of Essence.” Journal of Philosophical Logic (): –. Fine, Kit. . “Cantorian Abstraction: A Reconstruction and Defense.” Journal of Philosophy (): –. Gödel, Kurt. . “Remarks on Definability.” In Collected Writings, Vol. II. Oxford: Oxford University Press. Jensen, R. . “Inner models and Large Cardinals.” Bull. Symbolic Logic (): –. Kant, Immanuel. . “The Only Possible Basis for Demonstration of the Existence of God.” In Kant’s Writings –. Cambridge: Cambridge University Press. Kant, Immanuel. . Critique of Pure Reason. Kemp Smith edition. London: Macmillan and Co., . Martin, Donald A. . “Gödel’s Conceptual Realism.” Bull. Symbolic Logic (): –. Martin, Donald A., and John R. Steel. . “A Proof of Projective Determinacy.” Journal of the American Mathematical Society (): –. Reinhardt, W. a. “Set Existence Principles of Schoenfield, Ackermann and Powell.” Fundamenta Mathematica : –. Reinhardt, W. b. “Remarks on Reflection Principles, Large Cardinals, Anti Elementary Embeddings.” In Axiomatic Set Theory, Proc. Symp. Pure Math., Vol. , part . Providence, RI: AMS, –. Steel, J. . “What is a Woodin Cardinal?” Notices AMS : –. Wang, H. . Conversations with Gödel. Oxford: Oxford University Press. Woodin, H. . “The Continuum Hypothesis I.” Notices AMS (): –. Woodin, H. . “Strong Axioms of Infinity and the Quest for V.” ICM lecture. Zermelo, Ernst. . “On Boundary Numbers and Domains of Sets: New Investigations in the Foundations of Set Theory.” In Ewald .

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 Fine on Arbitrary Objects Alasdair Urquhart

 Introduction Fine’s remarkable theory of arbitrary objects has attracted less attention than it deserves. He expounded his theory in two papers (Fine, ; Fine, a), and a monograph (Fine, b). The aim of the present chapter is to provide a brief introduction to the theory, to discuss some of the historical background, and finally to sketch some connections with other areas in logic and philosophy. In the historical section, I shall try to elucidate the extent to which earlier writers anticipated Fine’s theory, while in the last section, I attempt to expand on some of Fine’s brief but suggestive remarks in his monograph.

 Fine’s Theory In this section, I present an outline of Fine’s theory, as it is expounded in his  monograph. We assume given a first-order language L containing at least one predicate symbol; it may also contain individual constants, function symbols, and the identity symbol.

. The models Let ℳ be a classical model for L; thus ℳ has the form (I , . . . ) where I is a nonempty set of individuals, and . . . indicates the interpretation in I of the non-logical symbols in L. We expand ℳ to a generic model ℳþ by adding a set A of new objects, a binary relation  between the new objects, and a non-empty set V of partial functions. If B  A, we say that B is closed if whenever a 2 B, and a  b, then b 2 B. We write [C] for the closure of C—that is, [C] is the smallest closed set containing C . The expanded model ℳþ ¼ ðI; :::; A; ; VÞ is required to satisfy a list of eight postulates: . (I , . . . ) is the model ℳ; . A is a set of objects disjoint from I ; .  is a relation on A;

Alasdair Urquhart, Fine on Arbitrary Objects In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Alasdair Urquhart. DOI: 10.1093/oso/9780199652624.003.0005

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   . V is a non-empty set of partial functions from A into I; that is to say, the domain of v 2 V, Domain(v), is contained in A, and its range Range(v), is contained in I ; . (a) (Transitivity) If a  b and b  c, then a  c; (b) (Foundation) The converse of the relation  is well founded—that is to say, there is no infinite ascending chain of objects a₁ , a₂, a₃, . . . in A so that a1  a2  a3  ::: . (Restriction) V is closed under restrictions; that is, if v is a function in V , and B  A, then vjB is in V , where vjB is the restriction of v to B; . (Partial extendibility) If v 2 V, then there is a function w 2 V such that v  w, and the domain of w contains [Domain(v)] (the closure of the domain of v); . (Piecing) If W  V is a family of functions in V with closed domain so that their union u ¼ [W is a function, then u 2 V. The postulates we have listed here are those from chapter  of the monograph (Fine, b). The version of the theory presented in Fine’s  paper (Fine, a) is slightly simplified; the set A of arbitrary objects is assumed finite, allowing simpler versions of Foundation and Piecing. For the main applications of the theory to systems of natural deduction, this is sufficient. The intended interpretation of the postulates is as follows. The set A represents the collection of arbitrary or variable individuals, while the family V represents the values that these individuals can assume. For example, if a 2 A is an arbitrary triangle, then v(a) would be restricted to triangles. The relation  represents dependence among variables; thus if a  b, the values that a can assume depend on the values assumed by b. The Foundation postulate states that if we trace back a chain of dependencies, the chain must eventually terminate in an individual that does not depend on another arbitrary individual; such an individual we call independent. In addition, following the lead of Fine (Fine, b, p. ), let us say that f is universal or value-unrestricted if the range of f is I . For the purposes of applications, the Piecing postulate is crucial; it states that a family of mutually compatible partial functions defined on A can be pieced together to make a single (partial) function.

. Generic truth At the heart of Fine’s theory is the principle of generic attribution, mentioned above in the introductory section. Using the model-theoretical machinery from §., we can formalize it as follows. Given a classical model ℳ and its expansion ℳþ ¼ ðI; :::; A; ; VÞ, we add to the language L a set of new constants, the A-names or A-letters, and assign every arbitrary object in ℳþ a unique A-letter as its name. This expanded language we denote by L*. If v 2 V, then we define the notion of truth in ℳþ relative to v as follows. Let φ ¼ φ½a1 ; :::; ak  be a formula of L*, where a₁, . . . , ak are all A-letters occurring in φ. Then: ℳþ╞ v φ if and only if a1 ; :::; ak 2 DomainðvÞ and ℳ╞ φ½vða1Þ ; :::; vðak Þ.

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   



On the right hand side, the relation ╞ is the classical notion of truth in the model ℳ. The notion of absolute truth in ℳþ can then be defined as follows: ℳþ╞ φ if and only if ℳþ╞ v φ , for any v where a1 ; :::; ak 2 DomainðvÞ. Informally stated, an arbitrary object has exactly those properties common to all of the objects in its range. However, this formulation gives rise to a problem dubbed “the argument from special properties” (Fine, , p. ). If we apply the principle to the property of being an individual object, then since all of the individual objects in the range of an arbitrary object are individual, it follows that the arbitrary object itself is an individual object, a contradiction. Fine saves himself from this paradox by restricting the principle of generic attribution to “generic conditions” (Fine, b, p. ). However, he is not very forthcoming as to what it is that makes a condition generic, other than the fact that they include all of the ordinary predicates such as “being mortal,” and that they are closed under the usual logical operations. This rather vague delineation of the distinction between generic and special predicates forms the target for one of Neil Tennant’s criticisms. He says, “ ‘Being generic’ ought to be a decidable property of conditions expressible in the language. Only then will the principle of generic attribution have application of sure axiomatic status” (Fine, , p. ). Tennant’s insistence on decidability seems rather too strong. After all, we allow notions such as validity and satisfiability in the context of classical predicate logic, that are quite clear and well determined, but not decidable. Nevertheless, the lack of a clear distinction is a little disturbing. I think we would do well to see the distinction as a semi-technical one, much akin to the distinction between internal and external sets in the context of nonstandard analysis—see for example, the textbook (Goldblatt, ). In other words, the distinction between generic and special is relative to a model and to a language. This interpretation is, I think, confirmed by some remarks of Fine himself: Even when a language is not itself generic, it will correspond to one that is. For we may reinterpret its predicates or other non-logical constants so that they are classically evaluated in their application to individuals but “generically” evaluated in their application to arbitrary objects. The new language will then be generic by definition and, over the domain of individuals, will be in semantic agreement with the original language. (Fine, b, p. )

. Definitions The final component of Fine’s theory is a theory of definitions. If we revert to our earlier example, let us suppose that we take a to be an arbitrary triangle. In this case, all we know about a is that it is an object satisfying the condition “The object a is a triangle.” Thus a can be taken to be defined by this constraint; we can formalize this as a definition, having the form (a, Triangle(x)). One arbitrary object can be defined in terms of another; for example, if b is an arbitrary triangle, we could define a as the incircle of b via the definition (a, Incircle(b)). Thus we can naturally associate a dependency relation with a system of definitions, representing the definitional dependencies between A-letters in the system.

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   In chapter  in part I of his monograph, Fine proves a series of powerful existence theorems showing that any system of definitions satisfying two natural conditions (being unequivocal and well founded) is realized in an appropriate extension of a given classical model.

 Prehistory of Arbitrary Objects In his monograph on arbitrary objects, Fine remarks that the view that there are arbitrary objects in addition to individual objects “used to be quite common, but has now fallen into complete disrepute” (Fine, b, p. ). He does not expand on these intriguing remarks, however. Let us look at some of the ideas of earlier authors about variable or arbitrary objects. As we shall see, these ideas are not necessarily direct ancestors of Fine’s ideas; it seems, rather, that there is a diversity of notions of arbitrary objects sharing a family of resemblances and connections.

. Sir Isaac Newton Fine has evoked the authority of Sir Isaac Newton in defending this theory: Within mathematics, there is the traditional explanation of variable signs as designatory of variable objects. It is important to emphasize that this was no idle frippery put out by the weak and woolly minded. It was integral to the way mathematical symbolism was conceived and used, and was endorsed by no less an authority than Newton. (Fine, , pp. –)

Newton’s ideas about variable quantities, or “fluents,” are explained in his important tract Methods of Series and Fluxions, written in the winter of –: To distinguish the quantities which I consider as just perceptibly but indefinitely growing from others which in any equations are to be looked on as known and determined and are designated by the initial letters a, b, c and so on, I will hereafter call them fluents and designate them by the final letters v, x, y and z. (Whiteside, , p. )

Just before the passage quoted above, Newton makes it plain that his fluents are functions of an independent variable, that can be considered as time, but that any other variable “increasing with an equable flow” could be considered as time: I shall, in what follows, have no regard to time, formally so considered, but from quantities propounded which are of the same kind shall suppose some one to increase with an equable flow: to this all the others may be referred as though it were time, and so by analogy the name of “time” may not improperly be conferred upon it. (Whiteside, , p. )

Thus Newton’s conception of fluents is of quantities that are a function of an independent, “equably increasing” variable. In other words, his variables v, x, y, z stand for quantities with a hidden or concealed parameter, and might be written as v(t), x(t), . . . in contemporary notation. This idea is essentially the same as the modern notion of a dependent variable; we should resist, however, the temptation to identify Newton’s notion with the modern notion of function. Newton’s conception of a fluent is more that of a quantity that varies through time. Although the notion of dependence bears some resemblance to a

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   



feature of Fine’s theory, Newton’s ideas do not appear to be clear precursors, since the connections with the notion of generality are lacking.

. Leonhard Euler Euler’s famous textbook of , Introduction to Analysis of the Infinite, contains a notion of variable quantities that is much closer to that of Fine’s arbitrary objects. At the beginning of his treatise, Euler distinguishes constant and variable quantities: Since all determined values can be expressed as numbers, a variable quantity takes on all possible numbers (all numbers of all types) . . . Variable quantities of this kind are usually represented by the final letters of the alphabet z, y, x, etc. A variable quantity is determined when some definite value is assigned to it. (Euler, , pp. –)

Here we have something much closer to the modern logical notion of the variable, and in particular to Fine’s conceptions. Variables can take on values, in Euler’s case, any numerical value (including complex values). Euler’s conception of pure mathematics is highly syntactical. This emerges clearly in his description of a function of a variable quantity as “an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities” (Euler, , p. ). This conception of function may seem unduly restrictive to a contemporary reader, but we should recall that Euler is perfectly happy to include infinitary notions such as unending decimal expansions and power series in his idea of mathematical syntax.

. Emanuel Czuber Emanuel Czuber appears as an unsung hero in Fine’s monograph, but we should remember that we are seeing him through the lens of Frege’s unrestrained polemical attack, in his essay entitled “What is a Function?” (Frege, , pp. –) published as a contribution to the  Boltzmann Festschrift. The passage from Czuber’s  textbook of differential and integral calculus that attracted Frege’s scorn reads (in my translation) as follows: By a real variable or indeterminate we understand a number, initially undefined, that, depending on the problems in which it appears, can take unboundedly many real numerical values. The totality of these values is called a value-range, and specifically the range or the domain of the variable. . . . The variable x is considered to be defined when any assigned real number can be determined as belonging to the range or not. (Czuber, , p. )

Frege represents Czuber as holding the following positions: . There are two kinds of numbers, determinate and indeterminate (Frege, , p. ); . A variable is defined completely by its range, and two variables with the same range are identical (Frege, , p. ). I very much doubt that Czuber seriously intended anything of the sort. These doubts are confirmed by the fact that in the second edition of , the first sentence quoted above is replaced by the following:

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   By a real variable or indeterminate we understand a symbol for an indeterminate magnitude, that, depending on the problems in which the indeterminate magnitude appears, can take several or unboundedly many real numerical values. (Czuber, , p. )

In other words, Czuber must have realized that his original formulation betrays symptoms of what Frege called “the mathematical sickness of our time, of confusing the sign with what is signified” (Frege, , p. ). It is not clear whether the change was in response to Frege’s criticism, but in any case, the revised edition shows clearly that Czuber did not mean to espouse the positions sketched by Frege. It’s hard to see in this passage anything that differs greatly from the remarks of Euler in , particularly if we stick to the second edition of Czuber’s textbook. In fact, Czuber is a little more careful than Euler, in that he insists on delimiting the range of his variables, whereas Euler is somewhat promiscuous in allowing his variables to take on any values, real or complex. In particular, I can see no justification for Frege’s insistence that Czuber has introduced a category of indefinite numbers. Read in context, Czuber’s remarks appear rather ordinary and unexceptionable. As in so many other cases, Frege’s polemical zeal has got the better of him. The first position listed above represents Frege’s interpretation of Czuber’s sentence: By a real variable or indeterminate we understand a number, initially undefined, that depending on the problems in which it appears, can take unboundedly many real numerical values.

However, we can just as well read Czuber’s formulation as simply a very sloppy and careless way of formulating the idea that a variable ranges over a certain domain of magnitudes; the idea that the magnitudes themselves are indeterminate seems to be due to Frege’s unsympathetic reading. The second position derives from Frege’s reading of the last sentence in the first quotation from Czuber. Again, we can give a rather dull and ordinary reading to Czuber’s formulation if we interpret him as saying, somewhat redundantly, that the variables he is considering all range over subsets of the real numbers. The Czuber who embraced a theory of indeterminate numbers appears to be a figment of Frege’s over-active imagination. Of course, it is quite probable that I am being unfair to Frege. Most likely, he knew perfectly well that his readings of Czuber’s remarks hardly corresponded to what the latter had in mind. On this view, Frege’s readings of Czuber’s textbook were simply tongue-in-cheek misinterpretations designed to highlight its careless formulations. If my milk-and-water reading of Czuber’s slipshod introductory remarks is correct, then he can hardly be considered as a precursor of Fine’s theory. However, this underlines further the original nature of Fine’s contribution.

 Enemies of the Arbitrary The enemies of arbitrary objects are legion. Fine mentions Russell, Leśniewski, Tarski, Church, Quine, Lewis, and Menger among others (Fine, b, pp. –). Here we discuss two opponents, the earliest and most significant, Bishop Berkeley, as well as a more recent critic, Nicholas Rescher.

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   



. Bishop Berkeley Bishop Berkeley’s writings against the doctrine of abstract ideas are remarkable for two reasons. First, he stated clearly the main argument against arbitrary objects; second, he gave the basic ideas behind a proof-theoretical strategy for eliminating them. The fundamental argument against arbitrary objects, though it can be given varying forms, is simple. Here is Berkeley’s version, inspired by a passage from John Locke’s Essay concerning Human Understanding: What more easy than for any one to look a little into his own thoughts, and there try whether he has, or can attain to have, an idea that shall correspond with the description that is here given of the general idea of a triangle, which is, “neither oblique, nor rectangle, equilateral, equicrural, nor scalenon, but all and none of these at once?” (Berkeley, , p. )

Every triangle is either obtuse, acute, or right-angled, so an arbitrary triangle must also satisfy this disjunctive property. However, if an arbitrary triangle is right-angled (say), then all triangles are right-angled, and so on for the other cases. Contradiction! This basic argument is dubbed by Fine the “argument from complex properties” (Fine, b, pp. –). His solution is simple: arbitrary objects do not satisfy exactly the same truth conditions as ordinary objects. For example, an arbitrary object can satisfy a disjunctive property without satisfying either disjunct. In fact, this solution is essentially forced upon us if we are to hew to Fine’s principle of generic attribution. Even more striking is Berkeley’s way of eliminating abstract objects by prooftheoretic analysis. He proposes that when we are proving a result about all triangles, then our argument proceeds by considering one particular triangle; as long as we don’t use any special features of this triangle in our proof, then the result is perfectly general. In The Principles of Human Knowledge, §, he says, When I demonstrate any proposition concerning triangles, it is to be supposed that I have in view the universal idea of a triangle; which ought not to be understood as if I could frame an idea of a triangle which was neither equilateral nor scalenon nor equicrural. But only that the particular triangle I consider, whether of this or that sort it matters not, equally stands for and represents all rectilinear triangles whatsoever, and is in that sense universal. (Berkeley, , p. )

This idea of representative generalization is a close counterpart of the restrictions in systems of natural deduction that play such a large part in Fine’s monograph, and represents keen insight on Berkeley’s part.

. Nicholas Rescher In an article (Rescher, ) of , Nicholas Rescher argued against the idea of random or arbitrary individuals in the sense of Fine. He begins with a version of the basic argument from complex properties discussed in the previous section on Berkeley. However, he also gives another and different argument against arbitrary objects. Rescher begins by observing that an arbitrary individual a should satisfy the rule of Universal Generalization (UG): φ(a), therefore 8xφ(x). He then argues that we obtain, by modus tollens, the derived mode of argument of Existential Instantiation

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   (EI): 9x:φ(x), therefore : φ(a). However, in a universe with at least one individual, the assertion 9x:ða ¼ xÞ holds true, from which, by the rule EI, we obtain :ða ¼ aÞ, a contradiction. Examining Rescher’s argument in the light of Fine’s theory, it is not hard to see where it goes wrong. The application of modus tollens to the rule UG is incorrect, since the rule UG is truth-to-truth valid in Fine’s theory, assuming that the arbitrary individual a is value-unrestricted (Fine, b, p. ); however, the rule EI is not truth-to-truth valid in general. Fine’s monograph contains an unusually detailed examination of this rule, and reveals the surprising and previously unsuspected fact that it has quite varied functions in different systems of natural deduction.

 A Model-theoretic Construction Fine’s basic model existence theorem, mentioned above in §., has a rather syntactic flavor, in which the arbitrary, or variable, objects are literally variables. In this section, we outline a more model-theoretic construction, one in which the notion of generic truth arises in a very natural way. The construction sketched here can be considered as an expansion of the brief remarks on Boolean-valued models on pp. – of Fine’s monograph (Fine, b).

. Boolean product construction In §. above, we mentioned an analogy between Fine’s theory of arbitrary objects and Abraham Robinson’s theory of nonstandard, or infinitesimal, analysis. In both cases, a classical model is enlarged to include additional entities possessing unusual properties. In particular, these new entities can be shown to satisfy certain conditions that were earlier held to be impossible or contradictory. In both theories, these contradictions can be avoided by drawing some simple distinctions. In fact, we can push the analogy even further. The usual method employed in proving the existence of models for infinitesimal analysis is the ultraproduct construction. This can be considered as a two-step procedure. Starting from a classical model ℳ we first form a Boolean product over an index set J as the first stage; in the second stage, we form a two-valued quotient model by dividing through by a nonprincipal ultrafilter over J . Let us begin by looking at the Boolean product construction. Let ℳ be a classical model for L, having the form (I, . . . ) and J a non-empty index set. Then we can form the direct product IJ in the usual manner, that is to say, IJ is defined as the set of all (total) functions defined on the index set J , taking values in I ; for f in IJ we write fj as an abbreviation for f(j). There is a natural way to define a Boolean-valued model ℳJ on the universe IJ. If φ(x₁, . . . , xk) is a formula of L containing the free variables x₁, . . . , xk and f 1 ; :::; f k 2 I J then we define the Boolean value of φ(f ¹, . . . ,f k) as follows: jjφð f 1 ; :::; f k Þjj ¼ fj 2 J : ℳ╞ φð f 1 j ; :::; f k j Þg: In other words, the Boolean value of a formula is simply the set of points j where the formula is true, after we project it down to the point j 2 J.

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   



The Boolean value of a formula can also be given in terms of a set of inductive definitions that are a direct generalization of the rules for ordinary two-valued logic: jjA _ Bjj ¼ jjAjj \ jjBjj; jj:Ajj ¼ J  jjAjj; jjⱻ x Aðx; f 1 ; :::; f k Þjj ¼ [ fjjAðg; f 1 ; :::; f k Þjj : g 2 I J g: The idea of the construction is that we form the Boolean product of the set of models fℳj : j 2 Jg (where each model ℳj is a copy of our base model ℳ) by multiplying not only the universe I, but also the space of truth-values to form the product Boolean algebra on the index set J. We can think of the complete Boolean algebra constituted by the family of all subsets of the index set J as a set of generalized truth values, as in the case of Booleanvalued models of set theory. In this algebra, the top element T is the index set J. Looking at ℳJ in this way, we can see that there is a natural way to interpret certain elements in the model as arbitrary objects. Each element f in the universe of ℳJ is a function, so that we can associate with each such element f its range Range(f). It then follows easily from our definitions that if f is a value-unrestricted element in the universe of ℳJ then for any formula φ(x), jjφðf Þjj ¼ T if and only if ℳ╞ 8xφðxÞ: This is exactly the property that we require of an arbitrary object. Furthermore, if we require that our index set J is at least as big as the universe of ℳ (that is to say, jJj  jIj), then value-unrestricted elements exist in profusion in our Boolean-valued model ℳJ . Surprisingly, though, the construction does not work! This is because the principle of generic attribution from the preceding paragraph does not extend to more than one arbitrary object. Consider the classical model ℳ ¼ ðZ; d½q, the degree of truth of the conditional should fall as the gap between antecedent and consequent rises, until at the limit, where d½ p ¼  and d½q ¼ , we have d½ p ! q ¼ . There are two notions of semantic consequence in fuzzy logic, ⊩⊤ and ⊩d. p1 ; :::; pn ⊩⊤ q means that if p₁, . . . ,pn all have degree of truth , so does q, and p1 ; :::; pn ⊩d q means that d½q  minfd½ p1 ; :::d½ pn g. DS is ⊤-valid: if d½A ¼ , then d½ : A ∨ B ¼ 1 only if d½B ¼ 1. By the lights of ⊤-validity, the disjunctive Sorites is simply unsound, for if ⊤ ≩ d½A ≩ ⊥, d½ : A ∨ B 6¼ ⊤. But DS is not d-valid: if d½A ¼ 0:5, d½B ¼ 0:4, then minfd½A; d½ : A ∨ Bg ¼ 0:5. So by the lights of d-validity, the argument relies on incorrect logic. The critique of the disjunctive Sorites in terms of ⊤-validity is like Fine’s. A difference emerges over the standard conditional formulation. Modus ponens (MP) is valid in Fine’s system, so he has to reject some of the conditional premises. The premises are supported by the Don’t Distinguish principle,¹⁵ at least so long as not distinguishing cases is understood as requiring that of any two sufficiently similar objects a and b, we either have Fa ^ Fb or we have : Fa ^ : Fb, for F the vague predicate in question. Since this would be based on LEM and deletion of inadmissible uses, Fine can say that some conditional premises are not true, and that failure to notice this is just a matter of being hoodwinked by Don’t Distinguish. In fuzzy logic, we again find that the status of Sorites reasoning varies with choice of consequence-concept: MP is ⊤-valid but not d-valid. Using ⊤-validity, the problem is as Fine says: some conditional premises are not ⊤. The reason, in terms of degrees of truth now, is that given drops of $. in salary, some of the conditionals will have ¹⁴ ∸ is cut-off subtraction, whose result is  when d½ p < d½q. ¹⁵ Fine calls his version of this principle for conditional Sorites “Tolerance”: “if two cases are sufficiently alike and the first is [F] then the second is also [F].”

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

 

a degree of truth very slightly lower than ⊤, since their consequents will be very slightly less true than their antecedents. But using d-validity, MP is to be rejected (apply (c) to d½A ¼ ., d½B ¼ .). Either way, the Sorites reasoning incorrectly detaches consequents that steadily fall in degree of truth to . However, I think fuzzy logic has an advantage over compatibilist semantics in dealing with the conditional Sorites. For Don’t Distinguish can have a stronger reading than Fine in effect gives it: it can be taken to exclude the transition from asserting “ai is well-paid” to asserting neither “aiþ1 is well-paid” nor “aiþ1 is not wellpaid.” Fine’s view of this transition is that it marks “a distinction in our response to the cases; it does not . . . commit one to a distinction in the cases themselves” (Fine : ). Presumably the same has to be said about the transition from asserting neither “aiþ1 is well-paid” nor “aiþ1 is not well-paid” to asserting “aiþ2 is not wellpaid.” Yet Fine also wants to hold that the transition from “ai is well-paid” to “aiþ2 is not well-paid” does mark “a distinction in the cases themselves.” How is this possible, when this last transition is the combination of two transitions neither of which marks a distinction in the cases themselves? This seems different from summing two insignificant changes of the same kind to get a significant change again of that kind. In terms of our running example, it instead seems that all three transitions mark distinctions in the cases themselves, since all three involve changes in use of “well-paid” answering to changes in salary. It’s just that the change is bigger comparing ai and aiþ2 . If correct, this would mean that Don’t Distinguish has a true reading that generates the conditional Sorites: if a is well-paid and b earns only $. less, you cannot politely decline to say that b is well-paid too.¹⁶ On the other hand, applied to fuzzy logic, the strong reading of Don’t Distinguish means that we should not assign different degrees of truth when salary differences are insignificant. It’s unclear why we should accept this prescription. A correct principle is that we should not assign significantly different degrees of truth unless there is a significant salary difference, which is compatible with assigning very slightly different degrees to mark insignificant salary differences. This means that assigning very slightly different degrees of truth is consistent with not distinguishing, so perhaps we should reword Don’t Distinguish to “don’t significantly distinguish cases that are not significantly different in the relevant respect.” The claim is then that assigning very slightly different degrees of truth does not signal a significant difference in cases, while going from asserting that ai is well-paid to declining to assert that aiþ1 is wellpaid does signal a significant difference.¹⁷ ¹⁶ This seems to me to be even clearer for predicates applied on the basis of how things look. If two people side by side cannot be told apart in terms of height without very sensitive measuring apparatus, what would we make of someone who, just on the basis of looking, assents to “the one on the left is tall” but refuses to assent to “the one on the right is tall” though he also accepts “I cannot see the slightest difference between them”? To make this intelligible we’d have to suppose that such an observer equivocates on “tall,” or has been persuaded by epistemicist literature and is taking a guess, or thinks some kind of trick is being played on him, or simply anticipates a Sorites paradox coming down the tracks. (With an isolated pair, Don’t Stray will not play a role. See (Raffman : –) for an argument to the effect that focus on isolated pairs explains the force of Sorites reasoning.) ¹⁷ This is not the same as objecting that the switch postulates a mysterious cut-off, i.e., a precise fact about how low “well-paid” goes. That kind of objection can also be made to fuzzy logic (see, for example, Sainsbury ), since in any interpretation there is always a lowest salary earning which makes it wholly

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 ’     



However, when we turn to negative-conjunction (:^) formulations of Sorites reasoning, compatibilist semantics diagnoses the problem more straightforwardly than fuzzy logic. Fine takes premises of the form : ðA ^ : BÞ to be true, since they reflect the version of Don’t Distinguish that he endorses, the one which rules out transitions from “ai is well-paid” to “aiþ1 is not well-paid.” However, each step in a :^-formulation carries us from A and : ðA ^ : BÞ to B, and this use of CS is invalid.¹⁸ So :^-formulations of the Sorites rest on faulty logic, by contrast with the two previous formulations, which had untrue premises on Fine’s account. The problem for fuzzy logic’s treatment of :^-formulations is not that the premises are true and CS valid in both senses. For although CS is ⊤-valid (if d½A ¼ 1, d½ : ðA ^ : BÞ ¼ 1 only if d½B ¼ 1), the premises : ðA ^ : BÞ are not in general true (⊤). The problem is rather that their degrees of truth fluctuate in an unintuitive way. For example, when d½A ¼ 0:9 and d½B ¼ 0:8, d½ : ðA ^ : BÞ ¼ 0:8, which is an acceptable result though not as plausible as ðA ! BÞ’s degree of truth, .. If d½ : ðA ^ : BÞ were stable at . whenever adjacent members of a Sorites sequence are under consideration, that would be good enough to allay Wright’s concerns (see note ). But if d½A ¼ 0:6 and d½B ¼ 0:5, d½ : ðA ^ : BÞ ¼ 0:5, which is unjustifiable: given that the degrees of truth in the two pairs are equally close, how can we have . in the one case and . in the other? This happens because by contrast with the clause for the conditional, the closeness of the two members of the sequence plays no role in fixing d½ : ðA ^ : BÞ. If d½B  0:5, then d[:B] fixes d½A ^ : B; otherwise d[A] fixes d½A ^ : B; so d[A]– d[B] is irrelevant. But ideally, we would like all the relevant instances of : ðA ^ : BÞ to have the same, high, degree of truth, because d[A]–d[B] is so small.¹⁹ To solve this problem, we could try redefining conjunction to make the magnitude of d[A]–d[B] decisive. One idea is that conjunction, now written “&,” should be defined in terms of ! in the usual way: ()

p & q ≝ : ð p ! : qÞ.

: ðA& : BÞ is d-equivalent to A ! B, and a :&-formulation of Sorites reasoning may be faulted in the same way as the conditional formulation. However, though () does produce a commutative associative connective agreeing with classical conjunction for inputs restricted to ⊤ and ⊥, the switch from ^ to & seems suspiciously ad hoc. Worse, there are respects in which & is “un-conjunction-like.” For example, A ⊮ d A & A, for whenever d½A  0:5, d½A & A ¼ 0, and though, for inputs strictly between . and , d½A & A approaches  as d[A] does, the former lags the latter,

true that you are well-paid. In (Forbes ) I respond to this problem (p. ) by appeal to Kaplan’s distinction between representational and artefactual features of models (Kaplan : ). If this distinction is helpful in the current context, Fine could also appeal to it. ¹⁸ That there is an embedded use of DN in this version of CS (“classical” CS) is not fundamental to the problem with Sorites reasoning—see Fine on “running the reasoning backwards” (: –). ¹⁹ This problem is ameliorated, but not ultimately dissolved, by the treatment of conjunction in (Edgington ).

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

 

since d½A& A ¼ d½A  d½ : A, which is always lower than d[A]. Therefore, we can assign any degree of truth strictly between  and  to A to show A ⊮ d A & A. A better approach to negated conjunction formulations of Sorites is to observe that although there is no trace of ambiguity in “and” according to whether or not it occurs in a Sorites context, there is a potential ambiguity in negation-phrases suggested by the wording of such a premise as ()

It is not the case that (ai is well-paid and aiþ1 is not well-paid).

For here the first occurrence of “not” is embedded in the context “it is . . . the case” while the second is an adjective modifier. While the second “not” is adequately captured by :, “it is the case” appears to be a synonym of “it is true.” So inserting “not” will produce an operator that semantically is sensitive only to whether or not the value of its complement (the proposition expressed by “that ai is well-paid and aiþ1 is not well-paid”) is ⊤. That is to say, the prefix “it is not the case” introduces a negation-operator , “semantic negation,” that is different from : and is governed by the following rule: ()

d½ p ¼ 1 if d½p 6¼ 1; d½ p ¼ 0 if d½ p ¼ 1.

A statement like (), therefore, will be true if taken to have the form ðA ^ : BÞ. For either “ai is well-paid” has a high degree of truth, whence “aiþ1 is not well-paid” will have a low degree of truth, or “ai is well-paid” has a middling to low degree of truth; so the conjunction in () is never , that is, never wholly true. This accounts for our intuition that all statements like () are true, as they are on compatibilist semantics. On other hand, the pseudo-CS sequent A, ðA^ : BÞ ‘ B is not even ⊤-valid, for if A is ⊤ and B is slightly less than ⊤, both premises are ⊤ and the conclusion is slightly less than ⊤.²⁰ The distinction we are appealing to, between semantic negation and what is sometimes called “fixed-point” negation (because in three-valued logic it maps the intermediate value to itself ) is of course a standard one in many-valued theories of vagueness. My conclusion about Fine’s discussion of the three formulations of Sorites reasoning, then, is that it is a very great strength of compatibilist semantics that it diagnoses a problem in each of the three, using the same formal apparatus, and other approaches which cannot do something like this are at an immediate disadvantage. However, I also think that the fuzzy logic analysis of the three formulations is overall equally as effective, in particular because it is independently plausible that embedding fixed-point negation in “it is the case” produces an operator expressing semantic negation. So the seductiveness of apparent :^-formulations is explained by revealing them to be ^-formulations.²¹

²⁰ The ⊤-invalidity of pseudo-CS is obvious if d[A] is ⊤ and d[B] is low but not ⊥. The explanation of the force of Sorites reasoning is that we only consider cases where d[B] is slightly less than d[A]. ²¹ I thank Kit Fine for helpful discussion while I was writing this chapter.

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 ’     



References Edgington, Dorothy. . “Vagueness by Degrees.” In Vagueness: A Reader, edited by Rosanna Keefe and Peter Smith, –. The MIT Press. Fine, Kit. . “Vagueness, Truth and Logic,” Synthese : –. Fine, Kit. . “The Possibility of Vagueness,” Synthese, Online First, ./s-- (page references are to the unnumbered pages of this file). Fine, Kit. . “Indeterminate Identity, Personal Identity and Fission.” In Metaphysics, Meaning and Modality: Themes from Kit Fine, edited by Mircea Dumitru, –. Oxford University Press. Forbes, Graeme. . “Thisness and Vagueness.” Synthese : –. Forbes, Graeme. . “Identity and the Facts of the Matter.” In Cuts and Clouds: Vagueness, Its Nature and Its Logic, edited by Richard Dietz and Sebastiano Moruzzi, –. Oxford University Press. Forbes, Graeme. . “Context-Dependence and the Sorites.” In Vagueness and Communication, FoLLI Lecture Notes in Artificial Intelligence, –. Springer. Kaplan, David. . “How to Russell a Frege-Church.” The Journal of Philosophy : –. Keefe, Rosanna. . Theories of Vagueness. Cambridge University Press. Parsons, Terence. . Indeterminate Identity. Oxford University Press. Raffman, Diana. . Unruly Words. Oxford University Press. Robertson, Teresa. . “On Soames’s Solution to the Sorites Paradox.” Analysis : –. Sainsbury, Mark. . “Concepts without Boundaries.” Inaugural Lecture, Stebbing Chair of Philosophy, King’s College, London. Sorensen, Roy. . Blindspots. Oxford University Press. Sorensen, Roy. . Vagueness and Contradiction. Oxford University Press. Williamson, Timothy. . Vagueness. Routledge. Wright, Crispin. . “On the Coherence of Vague Predicates.” Synthese : –. Wright, Crispin. . “Further Reflections on the Sorites Paradox.” Philosophical Topics : –.

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PART II

Modality

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 Necessary, Transcendental, and Universal Truth Steven T. Kuhn

 Introduction Thirty-nine years ago, I was a graduate student at Stanford approaching the critical time for choosing a dissertation topic. I knew I wanted to write something having to do with the modal logic I had been learning in graduate school, ideally something that would call for both technical work in logic and its application to interesting philosophical problems. I was excited that a talented and friendly young Englishman who seemed to share my interests was visiting the Department. I don’t remember exactly when I had the thesis-topic conversation with Kit, but I do remember his offering to take a look at the papers I had been writing. I showed him some of my efforts in tense logic and he pointed out that there was a kind of theme present—in every case I argued that fidelity to the phenomena for which I sought formal treatment required that formulas be sorted, so that not all connectives should apply to all formulas uniformly. Sometimes this was because it was natural to restrict the interpretations of different sorts of formulas in different ways: John swims IN the Channel should be true at a temporal interval only if it is true at all its subintervals; John swims ACROSS the Channel should be false at the subintervals of those where it is true. Sometimes it was because the sentences represented were naturally to be regarded as true or false with respect to different sets of parameters. Some sentences are naturally thought of as being evaluated at instants of time, others at intervals of time, and still others are naturally thought of as assuming truth values independently of time altogether. So, at Kit’s suggestion, I tried to gather some of these examples together with a few others as part of a general study of many-sorted modal logics. As what I imagined to be a very simple example, I briefly considered an application to the standard alethic modal logics. There were two basic sorts of formulas: one world-relative and one worldindependent, and a third sort comprised of Boolean combinations of these two. The necessity operator applied only to formulas of the first and third sort to produce a formula of the second. In retrospect, I am somewhat embarrassed by the lack of attention I paid to the metaphysical underpinnings of these distinctions. I cringe particularly at a sentence

Steven T. Kuhn, Necessary, Transcendental, and Universal Truth In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Steven T. Kuhn. DOI: 10.1093/oso/9780199652624.003.0010

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 . 

promising “The next two chapters [after the one containing these remarks about necessity] deal with more serious applications.” Many years after my dissertation was written, I watched Kit turn his attention to very similar ideas in a lecture at the University of Maryland entitled “Necessity and Non-Existence.”¹ Knowing his philosophical temperament as I do now, it is not surprising to me that he did take the metaphysical matters seriously and had a great deal to say about them. I am grateful for the opportunity now to reflect more deeply about both the metaphysical and logical aspects of these matters, and on what Kit had to say about them.² “Necessity and Non-Existence” defends the view that there is a distinction between worldly and unworldly sentences that is similar to a more familiar³ distinction between tensed and tenseless sentences. Socrates exists is true at some times and false at the others. As a consequence, Either Socrates exists or Socrates does not exist is true at all times. It is, in Fine’s terminology, sempiternal. By contrast, Socrates is self-identical and Socrates is human⁴ are true regardless of time. Fine calls these eternal truths.⁵ To conform more closely to contemporary usage it might better to use timeless here, and timely for those sentences, including the sempiternal ones, whose truth depends on time. Similarly, Socrates exists and Socrates exists or Socrates does not exist are worldly truths. They are true or false because of the circumstances of the world, because of how things have turned out. The second is true in every circumstance, that is, it is true no matter how things turn out. It is necessarily true. By contrast, Socrates is self-identical and Socrates is human are completely independent of circumstance. They are unworldly sentences and the truths they express are transcendental. Once the distinction between worldly and unworldly is acknowledged among sentences, it is natural to extend it to other domains. Predicates like is human and is self-identical, which combine with names to form unworldly sentences, are themselves unworldly; those like is snub-nosed and sits, for which similar combinations ¹ This lecture has since been published under the same title in [Fine, ], pp–. ² I am further pleased by the circumstance that this reflection, as the reader will see, has also led me back to the ideas of Kit’s teacher (and my grand-teacher?), Arthur Prior. ³ To say this distinction is familiar is not to say that it is universally accepted among philosophers. Fine argues, however, that even those whose acceptance of McTaggart’s A-theory of time leads them to reject the thesis that Socrates exists is true at a time in a way that Socrates is self-identical is not, should be willing to grant that only the former is true because of how things are at the time. Thus they should grant the distinction even if they understand it differently. ⁴ Fine’s example is Socrates is a man, which is supposed to correctly predicate a substance sortal property of an individual. I substitute human for a man because the latter may inappropriately suggest male or adult (in which case it would not name a substance sortal). Fine prefers his wording because of qualms about whether being human might indicate belonging to a biological species and whether such belonging constitutes having a substance sortal property. (See Wetzel  for an argument that it does not.) Fine (private communication) has suggested Socrates is a human being might be a better alternative than Socrates is human. I have resisted the suggestion because of worries that the former sentence is more likely to be read as having existential import. All these issues are peripheral to those addressed in the chapter and the reader should feel free to substitute any example that does correctly predicate a substance sortal of an individual. The reader who is generally skeptical about substance sortals may stick to examples like Socrates is self-identical. ⁵ In employing this terminology, Fine evokes old metaphysical distinctions. See, for example, Kneale . But in the older discussions, the labels apply at the most fundamental level to beings or things rather than truths or sentences.

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, ,   



form worldly sentences, are worldly.⁶ Properties like humanness and self-identity, expressed by unworldly predicates, are unworldly; those like being snub-nosed and sitting are worldly. Individuals like Socrates have worldly existence; numbers and sets have unworldly existence. Facts that can be expressed by worldly sentences are worldly; those that can be expressed only by unworldly sentences are unworldly. Unworldly facts, individuals, and properties belong to a transcendental realm of reality. It is one thing to acknowledge the distinction between worldly and unworldly and another to agree on a particular inventory of the transcendental realm: the “worldly” philosopher may grant that the distinction makes sense, while insisting that all facts are determined by circumstances, and therefore that the transcendental realm of reality is empty. While Fine obviously thinks the worldly philosopher is wrong, “Necessity and Non-existence” contains only the very beginnings of speculation about what kinds of facts might populate the transcendental realm. It contains just enough to address a simple-seeming puzzle whose answer, Fine tells us, “has profound implications for our understanding of the concepts of existence, identity and modality and for how these concepts connect to one another and to the world.”⁷ I believe that Fine mischaracterizes the relation between necessity and transcendental truth and that this undermines his solution to his puzzle. In § below I summarize Fine’s puzzle and his preferred solution. I explain the weakness I see in the solution and the mistaken view of the necessary/transcendental relation that underlies it and I offer an alternative view of that relation. Section  outlines a simple formal system with distinct operators for necessary truth, transcendental truth, and truth in all worlds to help clarify the alternative view. This system is tantamount to one discussed many years ago by Arthur Prior as “The System A.”⁸ In §, I pause to consider my system and Prior’s system A in more detail. In §, I consider extending my system by the addition of an “actuality” operator. That extension brings to light some additional questions about how the worldly–unworldly distinction should be

⁶ Linda Wetzel has made the interesting suggestion that sitting and snub-nosed might be unworldly predicates when applied to Rodin’s The Thinker and the Vatican’s bust of Socrates. These examples would need to be examined with some care. It is not really the sculptures that are sitting or snub-nosed but their subjects. The subject of the Vatican bust would seem to be Socrates, for whom being snub-nosed was a worldly property. As for Rodin’s subject, he appears to be sitting temporarily to facilitate contemplation, rather than suffering some incurable metaphysical paralysis. Regardless of whether these examples can be understood in a way that makes them true, they raise questions about whether predicates have stable “signatures” and, more generally, whether properties and relations should be seen as sources for the unworldliness of propositions. It is reasonable to suppose that individuals may have both worldly and unworldly properties. The worldly Socrates, for example, has the worldly property of snub-nosedness and unworldly property of self-identity. Similarly, the unworldly number  is both self-identical and the number of Adam and Eve’s children. Is there any reason why some properties might not, as Wetzel suggests, be similarly fickle? An example of a sort different than Wetzel’s might be the following: four is smaller than five is unworldly, whereas David is smaller than Goliath is worldly. Here a plausible reply is that the sentences equivocate between distinct worldly and unworldly relations. The classification of predicates mentioned above suggests that Fine believes such responses will always be forthcoming—that no such examples are possible. Nothing in his writing, however, would seem to commit him to the view that there cannot be some predicates that lack stable signatures. ⁷ Fine : . ⁸ See Prior : – (“Appendix C”).

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

 . 

made and raises other questions of philosophical interest. Section  is a brief summary and conclusion.

 Could Socrates be Both Human and Nonexistent? The puzzle serves both as an illustration and an application of Fine’s ideas on worldly and unworldly truth. Consider the following simple argument. (P) It is necessary that Socrates is human; (P) It is possible that Socrates does not exist; (C) Therefore it is possible that Socrates is human and does not exist. Taken alone, Fine thinks that the conclusion should strike us as absurd. The second premise, however, is obviously true—since it is possible that all humans, both individually and as a group, fail to exist it is surely possible that Socrates fails to exist. The first premise seems equally true—being human is a part of Socrates’ nature, and so it must be necessary that he be human. The conclusion follows, as Fine says, by “impeccable modal reasoning,” reasoning that is codified in the weakest of plausible alethic modal systems. So we have an apparently absurd conclusion following by apparently valid reasoning from apparently true premises—that is, a paradox. We might think that the paradox can be resolved by invoking distinctions commonly made in discussions of necessity and nonexistence. For example, Arthur Prior’s system Q adopts a view according to which sentences mentioning merely possible individuals lack truth value, and necessity may be strong (truth in all worlds) or weak (falsity in no world). More standard semantics for modal systems retain bivalence and distinguish between unqualified necessity (truth in all worlds) and qualified necessity (truth in all worlds containing the individuals mentioned). It is natural to think that Fine’s puzzle argument involves an equivocation between strong and weak modalities or one between qualified and unqualified modalities, but he makes persuasive arguments that such solutions are unsatisfactory.⁹ Fine’s own solution invokes the worldly–unworldly distinction. If we take necessity to be truth at all worlds or circumstances, then the first premise, that Socrates is necessarily human, is either meaningless or false. Socrates is not human in every circumstance; he is human independently of circumstances. It is quite natural, however, to extend the notion of necessity to include sentences of both types. (At the same time, we should extend the notion of possibility to include sentences that are unworldly and true as well as those true at some world.) In this extended sense, we can accept that Socrates is necessarily human. We have some reluctance to invoke the extended sense of necessity, however, so there is also some tendency to take the predicate human to mean the same as some worldly approximation—existing human being the most likely candidate. This understanding allows us to accept the first ⁹ For a defense of an “orthodox” solution based on the distinction between weak and strong modalities that attempts to overcome Fine’s objections, see Forbes : –. Notice, however, that denying P on the grounds that Socrates (and indeed all of humanity) might not have existed seems to be presupposing that Socrates is human is false in those circumstances, which is exactly what Fine denies.

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, ,   



premise even on the unextended sense of necessity. The second premise, that it is possible Socrates does not exist, is true in either the unextended or extended sense of possibility, so we can happily accept that as well. The conclusion, however, applies possibility to a hybrid sentence: Socrates is human and does not exist is a Boolean combination of worldly and unworldly parts. It is possible to further extend the notion of necessity so that it applies not only to worldly and unworldly sentences, but to these hybrids as well. Fine calls this the superextended sense of necessity, and he thinks we are naturally even less inclined to adopt it than the extended sense. The reasons for our reluctance, Fine says, are hard to articulate, but we can get a sense of the phenomenon by considering an analogy in the temporal realm. We are naturally “uncomfortable” with the idea that it will be the case that dawn breaks and two plus two equals four. We may have accommodated ourselves to the superextended interpretations of modal and tense operators, Fine suggests, because of “scientific” benefits of doing so. In applying these operators to hybrid sentences we can say things that might be difficult or impossible to do otherwise. Given these various senses of modalities, the puzzle argument can be interpreted in a variety of ways. Here are three. () We can misconstrue human as worldly and take necessity and possibility in the unextended sense throughout the argument. In that case, premise one is well formed and the argument is valid. Both premise and conclusion, however, are false. () We can understand human correctly, and take necessity and possibility in the extended sense throughout. In this case the premises are true but the conclusion is ill-formed. () We can understand the predicates correctly, but take necessity and possibility in their superextended senses. In this case the argument is sound and valid. Its air of paradox is explained by the observation that, in the absence of motivation provided by the argument, we are unlikely to interpret the necessity operator in the conclusion in the superextended sense. There are, I think, two weaknesses in this account that are closely related. The first concerns the idea that, in applying necessity to hybrids of worldly and unworldly sentences we invoke a “superextended” notion of necessity that is further removed from the ordinary notion of necessity as truth in all possible worlds and considerably more difficult for us to accept than the application of that notion to unworldly sentences themselves. Fine himself suggests that in making the transition to the extended sense we are already admitting a new, “degenerate” way that a sentence may be true at a world; it may be true at a world “simply because it is true regardless of how thing are in the world.” Once we admit that both worldly and unworldly sentences can be regarded as true or false at a world, however, there would seem to be no obstacle to so regarding their truth-functional combinations. We know very well how to compute the truth-value of a conjunction at a world when we have the truthvalues of its conjuncts at that world. Knowing that truth-values of conjunctions in possible worlds are computable, it should make perfect sense to ask whether these conjunctions are true in all or some possible world. Such combinations, moreover, would seem to clearly fit our criterion of worldliness. Their truth depends on the worldly circumstances. It is hard to imagine why one who understood that the sentence Socrates is an existent human depends for its truth on worldly circumstances would balk at the notion that Socrates exists and is

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

 . 

human should depend similarly on such circumstances. If such truth-functional combinations are worldly, then understanding how necessity and possibility applies to them should be unproblematic. Similar remarks apply in the temporal realm. It is true that Dawn breaks and += is an odd sentence. But many homogeneous conjunctions are equally odd: Socrates is self-identical and nine is less than . The universe expands and Cain killed Abel. Furthermore more natural heterogeneous compounds are not difficult to find: If that creature is traveling at thirty miles per hour, it is not human. Either Wiles made a mistake in his proof or Fermat’s Last Theorem is true. These seem perfectly reasonable things to say, requiring no special use of if and or. The more fundamental weakness is Fine’s uncritical acceptance of the common idea that necessity is truth at all possible worlds. This acceptance is somewhat surprising. In the introduction to the volume in which “Necessity and Nonexistence” was published, Fine writes that his thinking about modality has been sustained by a “deep animosity” to views of Quine and Lewis, according to which necessity, if it is an intelligible notion at all, must be seen as a kind of regularity. A necessary truth is one that is always or everywhere true; a possible truth, one that is sometimes or somewhere true. The apparent contrast between the spare modal landscape envisaged by Quine and the more florid one Lewis discerns simply reflects a disagreement about the range of the regularities permitted by their ontologies. In view of Fine’s animosity to these views, one wonders why he would take transcendental truths to be necessary only in an “extended” sense. If being human is part of Socrates’ nature, then surely he must be human in any ordinary pre-theoretic sense of must. One would think that transcendental truths like this are, if anything, more necessary than those worldly necessities whose truth derives from one set of circumstances in this world and a different set elsewhere. Indeed, the very fact that the question of their being more necessary is sensible suggests that the ordinary notion of necessity is not truth in all worlds. That thought gains credence from one of Fine’s own glosses on the notion of transcendence. We might think of the possible circumstances as being what is subject to variation as we go from possible world to another; and we might think of the transcendental facts as constituting the invariable framework within which the variation takes place. Alternatively, we might think of the possible circumstances as being under God’s control; it is what he decides upon in deciding whether to create one possible world rather than another. Thus he can decide whether Socrates exists or not and so he can do something that will guarantee that Socrates exists or does not exist. But there is nothing he can do that will guarantee that Socrates is self-identical or that + is equal to ; these are the facts that provide the framework in which he makes the decisions that he does, not the facts yet to be decided. (Fine : –)

One might imagine that by changing the right circumstances, God could make something false that would otherwise be true in all possible worlds, or turn something that had been false only in this possible world into something true in all possible worlds. Even God, however, can’t mess with the transcendental. Let us call a worldly sentence that is true at every possible world, universally true. The view Fine favors is that the more basic and natural form of necessity is universal

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, ,   



truth and that transcendental truths are necessary only in an extended sense. The view I want to defend against this is that universal and transcendental truths are two subclasses of our ordinary notion of necessary truth, and perhaps that universality is sometimes weaker or less binding than transcendence. There is evidence that Fine might be sympathetic to a view like this or even a little ambivalent between such a view and the one I attribute to him. The critical remarks about “regularity” accounts of necessity and the remarks about what lies within and without of God’s control that have already been mentioned provide some of this evidence. Further support comes from the fact that the kinds of truths labeled “transcendental” in “Necessity and Nonexistence” are said in other writings to be necessary with no indication that any extension of the fundamental notion is involved. Here is Fine on the relation between essence and necessity: Indeed, it seems to me that far from viewing essence as a special case of metaphysical necessity, we should view metaphysical necessity as a special case of essence. For each class of objects, be they concepts or individuals or entities of some other kind, will give rise to its own domain of necessary truths, the truths which flow from the nature of the objects in question. The metaphysically necessary truths can then be identified with the propositions which are true in virtue of the nature of all objects whatever. Other familiar concepts of necessity (though not all of them) can be understood in a similar manner. The conceptual necessities can be taken to be the propositions which are true in virtue of the nature of all concepts; the logical necessities can be taken to be the propositions which are true in virtue of the nature of all logical concepts; and, more generally, the necessities of a given discipline, such as mathematics or physics, can be taken to be those propositions which are true in virtue of the characteristic concepts and objects of the discipline.¹⁰

The truth of the unworldly sentence Socrates is human presumably flows from the nature of Socrates and the truth of Socrates is not identical to Plato flows from the natures of Socrates and Plato, so these both express metaphysically necessary truths as well as transcendental ones. Likewise, the unworldly 2+2=4 is true in virtue of the concepts and objects of mathematics, and so expresses a mathematically necessity as well as a transcendental one. This line of thought is maintained and refined in Varieties of Necessity. Even in “Necessity and Non-existence” there is an admission that “we are accustomed to operating with an inclusive conception of what is necessary and of what is true in a possible world and so we think of any possible world as being like the actual world in settling the truth-value of every single proposition.”¹¹ It is also maintained, however, that “we naturally operate with a more restrictive conception of what is necessary and what is true in a possible world.”¹² So, for the Fine of “Necessity and Non-existence” it seems that we operate with two conceptions of necessity. The more restrictive conception is what I am now calling universal truth, and the inclusive conception is what I believe to be the ordinary conception of necessity.

¹⁰ Fine : .

¹¹ Fine : .

¹² Fine : .

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 . 

 A Formal System We can get a clearer picture of this understanding of the universal, transcendental, and necessary by considering a simple formal language with operators representing these notions and providing truth conditions for each operator. The logic of “it is universally true that,” under this semantics, is just the logic that Arthur Prior sought as “The System A.” Our language, like Prior’s, will contain formulas of two sorts. We may think of them as the worldly and the unworldly. In keeping roughly with Prior’s conventions, capital letters from the beginning of the alphabet will be used for the former; capital letters from the middle of the alphabet for the latter. With numerical subscripts, these will be regarded as particular sentence-letters; without the subscripts, as metavariables ranging over the appropriate formulas. Letters from the end of the alphabet are meta-variables ranging over formulas of either sort. [N], [T], and [U] are unary connectives corresponding to necessary, transcendental, and universal truth. For reasons already outlined, I want to resist the idea that there is any special difficulty in forming truth functional hybrids of worldly and unworldly formulas, or that application of [U] to such hybrids, requires us to first reinterpret their unworldly constituents as, in some derivative or extended sense, true or false in a world. Accordingly, the binary truth-functional connectives are taken to form unworldly formulas when applied to pairs of unworldly formulas, and to form worldly formulas when one or more of the formulas to which they are applied are worldly. Most of Fine’s discussion of unworldly truth relies on two kinds of examples— mathematical truths, and sentences like Socrates is human predicating a substance sortal of an individual. There is, however, another class of putatively unworldly sentences that originally motivated Prior’s system A and my own consideration of the many sorted-frameworks, and that impresses Fine sufficiently to remark, “for this reason alone, the worldly view should be given up.” These are sentences that express modal facts. Consider sentences asserting that a proposition is true in all possible worlds. In Fine’s jargon they assert necessity. For me they assert universality. Their truth does not turn on the circumstances in this possible world, but rather on the nonexistence of certain other worlds. On one understanding of modal discourse, worldly sentences have an implicit unfilled argument place for possible world. The (unextended) modalities quantify over this argument place. Once such quantificational devices are applied, the resulting sentences are no longer worldly. In any event, it seems clear that the universal truth connective should apply only to worldly formulas and produce only unworldly ones. The necessity operator, on the other hand, applies to both worldly and unworldly formulas to form unworldly ones. In the case of [T] there is some choice. Do we want to say that [T](Socrates exists) is false or that it is ill-formed? For the universal operator we were trying to remain faithful to the intuition that to say of an unworldly truth that it is true in all worlds is not to say something false, but to say nothing at all. There does not seem to be the same pull in the case of transcendental truth. To say of a worldly truth that it is transcendentally true could be to say something false or to say nothing. If we wish to maintain the parallel with the universal truths, we can take [T] to apply only to unworldly sentences to form an unworldly sentence. Otherwise, we can allow [T] to apply to

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

both worldly and unworldly sentences to form an unworldly sentence. For the time being, we shall keep both options open, referring to the first as the narrow interpretation of [T] and the second as the broad interpretation. The notion of formula then is defined by the following clauses: (i) (ii) (iii)

(iv) (v) (vi) (vi0 )

For every natural number i, Ai and Pi are worldly and unworldly formulas, respectively; If A is a worldly formula then so is :A, and if P is an unworldly formula then so is :P; If one or both of X and Y are worldly formulas then so are ðX ∧ YÞ, ðX∨YÞ, ðX ! YÞ and ðX $ YÞ; If they are both unworldly formulas then so are ðX ∧ YÞ, ðX∨YÞ, ðX ! YÞ and ðX $ YÞ; If X is a formula (i.e., a worldly formula or an unworldly formula) then [N] X is an unworldly formula; If A is a worldly formula then [U]A is an unworldly formula; and (narrow interpretation) If P is an unworldly formula then [T]P is an unworldly formula, or (broad interpretation) If X is a formula then [T]X is an unworldly formula

A model stipulates truth-values of unworldly sentence-letters directly, and the truthvalues of worldly sentences at each world. More precisely, we take a model to be a triple (W,w₀,V), where W is a non-empty set (the set of possible worlds), w0 2 W (the actual world), and V is a function (the valuation) that assigns a truth value to each unworldly sentence-letter P, and a set of worlds S  W (the truth set) to each worldly sentence-letter A. The base clause of the truth definition asserts that an unworldly sentence-letter is true in a model if it is assigned truth by the valuation, and false if it is assigned falsity and that a worldly sentence-letter is true in the model at a world w if w is an element of the truth set assigned to it by the valuation and false in w otherwise. The inductive clauses extend the application of true and false to arbitrary formulas preserving the condition that unworldly sentences are true or false in a model and worldly sentences are true or false in a model at a world. This will require several clauses for each connective. In the case of conjunction, for example, we will need to specify that for a model M ¼ ðW; w0 ; VÞ and a world w 2 W M⊨P ∧ Q iff M⊨P and M⊨Q; ðM; wÞ⊨P ∧ A iff M⊨P and ðM; wÞ⊨A; ðM; wÞ⊨A ∧ P iff ðM; wÞ⊨A and M⊨P; ðM; wÞ⊨A ∧ B iff ðM; wÞ⊨A and ðM; wÞ⊨B. For universal truth, the appropriate clause in the definition for truth in a model M ¼ ðW; w0 ; VÞ is M⊨½UA iff ðM; wÞ⊨A for all w 2 W. The necessity operator requires two clauses. Applied to a worldly formula, it indicates that the formula is universally true; applied to an unworldly formula, it indicates that the formula is true.

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

 . 

M⊨½NA iff ðM; wÞ⊨A for all w 2 W; M⊨½NP iff M⊨P. The possibility operator is understood as the dual of the necessity operator in the usual way: A¼def :½N:A. Notice that if a sentence is unworldly and negation is classical then necessity and possibility are both equivalent to truth, that is, the principles ½NP $ P and P $ P are both valid. All that remains is the operator for transcendental truth. On the narrow interpretation, we need only one clause: M⊨½TP iff M⊨P. On the broad interpretation, we supplement the above clause with a second, equally simple one: Not M⊨½TA. To complete our semantics, we stipulate that A is true in M if it is true in the actual world of M, that X is valid if it is true in all models and that X and Y are logically equivalent if X $ Y is valid. It follows that validity is closed under sort-preserving substitutions for sentenceletters as well as under replacement of subformulas by logical equivalents of the same sort. Of course validity is also closed under tautological consequence. With this bit of formal machinery in place, we might pause to ask what light, if any, it might shed on Fine’s puzzle. There are two salient choices for representing the puzzle argument within our expanded modal language: taking the modalities to be about universal truth, and taking them to be about necessary truth . .

½UP1 ; A1 ⊨ðP1 &A1 Þ, and ½NP1 ; A1 ⊨ðP1 &A1 Þ,

where P₁ stands in for Socrates is human and A₁ for Socrates exists. On the first choice, premise one is just ill-formed. We are interpreting It is necessary that Socrates is human in a way that is not just false, but incoherent. That seems implausible. On the second choice, the argument is sound and valid. Here we need some explanation of our initial reluctance to accept the conclusion. Fine suggests that it is part of a general reluctance to apply possibility to hybrid sentences. I am skeptical that we exhibit such a general reluctance. One explanation for our reluctance to accept that it is possible that Socrates is human and does not exist, might come from another of Fine’s observations—that in considering this example we have a tendency to take human to mean existing human. If we succumb to this tendency we will think that the conjuncts inside the possibility operator are contraries and the assertion that their conjunction is possible is false. In that case, however, one would think that we would construe human the same way in premise one. Socrates is human would then be worldly and premise  would assert that it is universally true that Socrates is human. We are now faced with two choices. If universal truth is taken in the strong sense (true in all worlds) then premise  is false. If it is taken in the weak sense (true in all Socrates-containing worlds), premise  is false.

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, ,   



These considerations suggest that there is a simple puzzle (or a simplified formulation of Fine’s puzzle), perhaps a little less dramatic, whose solution hinges exactly on the worldly–unworldly distinction. How is it that, on the ordinary, every-day sense of necessarily, Socrates is necessarily human is true while Socrates necessarily exists is false? If the ordinary, every-day sense of necessity is truth in all worlds then the first sentence is false. If it is truth in all worlds where the objects mentioned exist then the second sentence is true. The view of necessity presented here provides a plausible explanation. Socrates is necessarily human because his being human is a transcendental truth, that is, it is true regardless of world. He does not necessarily exist because he does not exist in all worlds, that is, his existence is not a universal truth. To say that different considerations underlie the necessity of different propositions is not to say that different senses of the word necessarily are used in the expression of these propositions. Even when applied to worldly sentences, which are necessary exactly when they are true in all worlds and perhaps because they are true in all worlds, necessary does not mean true in all worlds. The situation might be contrasted with its temporal analog. Consider the sentences Socrates is always human and Socrates always exists. Again the second sentence seems false. (Indeed he no longer exists now.) To say the first sentence is true, however, is not nearly so palatable as it was in the alethic case. It is much more natural to say here that, because Socrates is human is timeless, Socrates is always human is not a coherent sentence. The reason for the divergence, I think, is that whereas word necessarily does not mean true in all worlds the word always does mean true at all times.¹³

 Prior’s System A and the Logic of Necessity, Transcendence, and Universality Once we have specified the formal language with the three special operators and provided it with a semantics it is natural to ask about the resulting logic. Let us call the language of worldly and unworldly formulas set forth in the previous section ℒNTU and set of formulas of ℒNTU that are valid according to our semantics LNTU. Given the valid principle of substitution of logical equivalents, the following valid schemes allow us to eliminate occurrences of [N]: (N) ½NA $ ½UA, (N) ½NP $ P. Furthermore, on either the broad or the narrow interpretation, [T] can be similarly eliminated. The relevant biconditionals would either be (T)

½TP $ P

¹³ Note that for this reason the temporal analog of Fine’s puzzle seems much less paradoxical. We are not so naturally inclined to accept the first premise, it is always the case that Socrates is human. If not false, that sentence seems inappropriate or ill-formed in a way that the alethic version does not.

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

 . 

alone, or that scheme together with (T) [T]A$⊥. Thus, to axiomatize LNTU it is sufficient to add a few simple biconditional schemes to the axioms for the logic of [U]. The logic of [U] is exactly what should correspond to Arthur Prior’s System A. Prior’s avowed motivation in considering System A was to represent views according to which iterated modalities are meaningless.¹⁴ Prior’s “A-formulas” and “P-formulas” are just our worldly and unworldly formulas. He points out that in such a language we can express the following formulas, all of which are theorems of S, but not of S: ) ) ) )

□ðA ! □BÞ ! □ð◊A ! BÞ, □ð◊A ! BÞ ! □ðA ! □BÞ, □ðA ! □BÞ ! ð◊A ! □BÞ, □ð◊A ! BÞ ! ð◊A ! □BÞ.

The system A is (rather tentatively) identified as the logic obtained by adding the following two rules to propositional logic: LA. If ‘ ðA ! XÞthen ‘ ð□A ! XÞ, LA. If ‘ ðP ! AÞthen ‘ ðP ! □AÞ. ◊ is taken to be defined from □ in the usual way and the system is taken to be closed under modus ponens which, in the presence of the theorems of propositional logic is equivalent to its being closed under tautological consequence. Because the rules are schematic, the theorems are also closed under substitution, but care must be taken that A-formulas and P-formulas are replaced only by others of the same sort. Prior’s confidence that this is the system for which he is searching seems to stem in large part from his ability to prove each of the four test formulas within it. His only

¹⁴ It is important to distinguish Prior’s system from other rudimentary forms of modal logic that are similarly motivated. Prior himself (: ) mentions a system he calls A0 in which the necessity and possibility formulas cannot be applied to any formula that already contains such an operator. (So we can get the language of A0 by taking the hybrid truth-functional combinations to be unworldly rather than worldly.) Although Prior was apparently unaware of it, a system like A0 is already mentioned in Parry . Parry identifies the logic S. with the theorems of S of zero and first degree modality. This logic appears again as the “basic” modal logic B in Pollock, which exhibits a complete axiomatization and a proof that it coincides with the zero and first degree theorems of KT and of all the other Lewis systems as well. In the long-gestating manuscript of Fine and Kuhn, S. is treated as a kind of stepping-stone to the system S. In addition to A and A0 , Prior considers the possibility of a system A00 , in which, using our terminology, all the Boolean connectives except negation produce unworldly sentences as well as a system A000 , in which necessity and possibility operators apply to worldly sentences to produce worldly sentences, but cannot apply to formulas whose construction has involved unworldly elements. So, for example, A00 admits as well-formed, the expression :□:A but not □:□A or □(A∨A). A000 admits □:□□A but not □(A∨P). A0 captures something like Fine’s notion of unextended necessity. It does permit the formation of the Boolean hybrids, but it does not allow necessity to be applied to such hybrids. None of the Prior systems admit □A and □P, while barring (A ∨ P ) and while barring □(A ∨ P), which would seem to be required for a system capturing extended necessity. (Superextended necessity, of course, is captured by the traditional syntax.)

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, ,   



semantic discussion of the system is the presentation of an ingeniously gerrymandered infinite-valued matrix that he plausibly conjectures to determine A. With the benefit of a half century or so of further development of modal logic, however, it is not difficult to show that A axiomatizes universal truth according to the possible worlds semantics given above. Let us first observe that we can replace Prior’s LA with the T axiom, □A ! A, with no change to the resulting logic. Since ‘ A ! X and ‘ □A ! A together imply ‘ □A ! X, LA1 is a derived rule within the resulting system. Conversely, applying LA1 to the tautology A ! A gives us the T-axiom ‘ □A ! A. Observe next that, by taking P in LA to be ðP1 ∨:P1 Þ, we derive the rule of necessitation. In addition, we can prove the K axiom. . . . . . . .

ðA ! BÞ ! ðA ! BÞ □ðA ! BÞ ! ðA ! BÞ A ! ð□ðA ! BÞ ! BÞ □A ! ð□ðA ! BÞ ! BÞ □ðA ! BÞ ∧ □AÞ ! B □ðA ! BÞ ∧ □AÞ ! □B □ðA ! BÞ ! ð□A ! □BÞ

propositional logic 1 LA1 2 tautological consequence 3 LA1 4 tautological consequence 5 LA2 6 tautological consequence

A similar derivation establishes (U)

□ðP ! AÞ ! ðP ! □AÞ

Since the rule LA is immediately derivable from U and necessitation, we have now established that Prior’s system A is can be axiomatized by axiom schemes T and U and the rules of necessitation and modus ponens. Since K is derivable in this system it can be identified, in the now standard notation, as KTU the smallest normal Kripke logic containing T and U. Completeness of KTU for the intended semantics can be established by familiar methods. Take any KTU-consistent set Γ. We construct a model M ¼ ðW; w0 ; VÞ that satisfies Γ as follows. Let w₀ be a maximal KTU-consistent extension of Γ. Let W be the set of all KTU-consistent sets u of formulas with the properties that: (i)for all unworldly formulas P, P 2w0 iff P 2 u and (ii)for all worldly formulas A, □A 2 w0 implies A 2 u. (The presence of the T-axiom ensures that W contains w₀.) For any worldly sentence-letter A and any u 2 W, let VðA; uÞ ¼ T if A 2 w and let VðA; uÞ ¼ F otherwise. For any unworldly sentence-letter P, let VðPÞ ¼ T iff P 2w0 . We can prove by formula induction that the following two conditions obtain: (i) M⊨P iff P 2w0 and (ii) for all u 2 W, ðM; uÞ⊨A iff A 2u. Here, for example, is the case where P ¼ □A. If P 2w0 , then, by the definition of W, A is a member of every set u in W. By induction hypothesis, ðM; uÞ⊨A for every u 2 W. By the truth definition again, M⊨□A, and so one direction of the equivalence has been shown. For the other direction, suppose P= 2w0 . Let u ¼ fQ : Q2w0 g [ fB : □B 2 w0 g [f:Ag.

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

 . 

Suppose, for reductio, that u- is not consistent. Then ‘ ðQ1 ∧ ::: ∧ Qm Þ ∧ ðB1 ∧ ::: ∧ Bn Þ ∧ :A ! ⊥, for appropriate Qi2w0 and Bj such that □Bj2w0 . By tautological consequence, ‘ ðQ1 ∧ ::: ∧ Qm Þ ! ððB1 ∧ ::: ∧ Bn Þ ! AÞ. By (U), ‘ ðQ1 ∧ ::: ∧ Qm Þ ! □ ððB1 ∧ ::: ∧ Bn Þ ! AÞ. Since each Qi is in w₀ and w₀ is maximal KTU-consistent, the antecedent of this formula is in w₀, and therefore □ððB1 ∧ ::: ∧ Bn Þ ! AÞ is in w₀ as well. By principles of K, this implies □B1 ∧ ::: ∧ □Bn ! □A is in w₀. Since the Bj’s were chosen so that their necessitations were in w₀, this implies that □A 2w0 , which contradicts the initial supposition. Hence u- is KTU-consistent and can be extended to a maximal KTU-consistent set u, that meets the conditions for membership in W. Since :A 2 u and u is consistent, A= 2u. By induction hypothesis, (M,u) ⊨A, and so by the truth clause for □, (M,w₀) ⊨P, as was to be shown. Thus we have established that KTU is the logic of [U] and that the logic LNTU of [N], [T] and [U] is obtained by adding a few biconditionals to this. Two alternative axiomatizations of LNTU are worth noting. The first (which was pointed out to me by Frabrice Correia) replaces U by (C)

P ! □ðA ! PÞ.

Here is a sketch of the derivations showing the equivalence of C and U. . . . .

P ! ðA ! PÞ propositional logic □ðP ! ðA ! PÞÞ 1 necessitation □ðP ! ðA ! PÞÞ ! ðP ! □ðA ! PÞÞ U P ! □ðA ! PÞ 2,3 tautological consequence

. .

ðP ! AÞ ! ððB ! BÞ ! AÞ propositional logic □ðP ! AÞ ! □ððB ! BÞ ! AÞ1 necessitation + K + consequence □ðððB ! BÞ ! PÞ ! AÞ ! ð□ððB ! BÞ ! PÞ ! □AÞ K P ! □ððB ! BÞ ! PÞ C □ðP ! AÞ ! ðP ! □AÞ 2,3,4 tautological consequence

. . .

tautological

To obtain the second alternative, note that we could view the schema N as a device for eliminating [U] in favor of [N] rather than the reverse. Thus to axiomatize LNTU it would be sufficient to add a few biconditionals to the logic of [N], and we might expect that logic to be just the familiar S. A proof that the logic of [N] is indeed S can be given in two parts. First, one checks that every instance of an S axiom is provable and that necessitation (i.e., prefixing with [N]) preserves provability. For example, a check of the K axiom requires showing that the schema ½NðX ! YÞ ! ð½NX ! ½NYÞ is provable in each of the four cases where X and Y are worldly or unworldly. That establishes that the fragment contains at least S5. To show that it cannot contain any more, we can use the fact that the only normal extensions of S5 contain some “domain-size” schema ◊X1 ∧ ::: ∧ ◊Xn ! ∨i 2 Γ∗ . But also, by rule Act, ½A> 2 Γ∗ , which would violate the consistency of Γ*. So [A](Γ*) is consistent, as was to be proved. Next, prove that [A](Γ*) is maximal. Suppose X= 2½AðΓ∗ Þ. There are two cases. If X is unworldly then, by definition, it can’t be in Γ*. Since Γ* is maximal, :X2Γ∗ and so by definition :X 2 ½AðΓ∗ Þ, as required. If X is worldly, then by definition of [A](Γ*), ½AX= 2Γ∗ , and so :½AX 2 Γ∗ . By Neg0 [A], ½A:X 2 Γ∗ , and, by definition of [A](Γ*), :X 2 ½AðΓ∗ Þ, as required. Now we construct our model for Γ as before with [A](Γ*) playing the role that Γ* had played before. Let w0 ¼ ½AðΓ∗ Þ and let W be the set of all maximal consistent sets u that contain the same unworldly formulas as w₀ as well as every worldly formula whose necessitation is in w₀. Since our logic contains the T-axiom, it follows that w02W. It remains only to verify the truth lemma. The □ case is as before. The [A] case requires that we prove M⊨½AA iff ½AA2w0 . Suppose first that M⊨½AA but not ½AA2w0 . Then :½AA2w0. By Neg[A], ½A:A2w0. Since [A]:A is unworldly, it is also in Γ*. By the definition of w₀ then, :A2w0 , and so not A2w₀. By induction hypothesis. (M,w₀)⊨A, and so by truth definition, M⊨[A]A. This contradicts our initial supposition and so this direction is proved. Suppose next that ½AA2w0 . Since [A]A is unworldly, then it is also in Γ*. By definition of w₀, A2w0 . By induction hypothesis, ðM; w0 Þ⊨A. By truth definition M⊨A, as required. It follows from the truth-lemma that ðM; Γ∗ Þ⊨Γ: At first glance our logic KTU-A appears to lack correlates of several of the characteristic axioms of Crossley and Humberstone’s SA. Conspicuously missing are axioms resembling the following (with notation slightly changed to better conform to ours): (A) (A) (A)

½Að½Ap ! pÞ, □p ! ½Ap, ½Ap ! □½Ap.

Within the sorted framework, the □ in A can only be understood as [N], and on this understanding A is provable from our N. The other two axioms require a little more thought. Below are sketches of derivations establishing correlates of A and A.

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 . . . . . . . . . .

 .  ½UA ! A ½Að½UA ! AÞ ½UA ! ½AA :½AA ! ð½AA ! AÞ ½Að:½AA ! ð½AA ! AÞÞ :½AA ! ½Að½AA ! AÞÞ A ! ð½AA ! AÞ ½AðA ! ð½AA ! AÞÞ ½AA ! ½Að½AA ! AÞ ½Að½AA ! AÞ

T 1 Act 2, U[A] tautological consequence tautology 1 Act 2, U[A] tautological consequence tautology 4 Act 5, K[A] tautological consequence 3,6 tautological consequence

So, on closer examination, it turns out that the sorted system does have theorems corresponding closely to the axioms of SA. The sorted formulation again seems to allow for simpler axioms (and a simpler completeness proof) than the more standard one.

 Conclusion Let us briefly review Fine’s thoughts on necessity and transcendence and the emendations and refinements proposed here. Sentences that ascribe to an individual a property that it has by its very nature, according to Fine, are unworldly, and the propositions that they express are transcendental. Such propositions are necessary or possible only in an extended sense and, when these sentences are combined with ordinary worldly ones, the results are necessary or possible only in a superextended sense. This explains why someone who accepts the necessity of Socrates being human and the possibility of his not existing is still likely to balk at the possibility of his being human and failing to exist. I have argued that the ordinary, every-day sense of necessity encompasses Fine’s extended and superextended necessities as well as those propositions true in all worlds. This undermines the idea that Fine’s puzzle provides evidence for the worldly–unworldly distinction. I have suggested, however, that a slightly different puzzle (or a slightly different version of Fine’s puzzle) does provide such evidence: Why do we ordinarily judge Socrates is human to be necessary, but Socrates exists to be contingent? A plausible explanation is that sentences like Socrates is human are both necessarily true and true regardless of worldly circumstances. Even if this line of thinking does not convince us about sentences like Socrates is human, it is still reasonable to think that other kinds of sentences should fit those two descriptions. One such comprises sentences expressing mathematical truths. Another comprises sentences that truthfully state that a sentence is universally true, that is, true in every worldly circumstance and those that state that a sentence is true in some worldly circumstances. Somewhat surprisingly, perhaps, this category is naturally understood to include sentences that truthfully state that a sentence is true in particular worldly circumstances, for example those circumstances that obtain in this, the actual, world. A simple formal system that reflects and elucidates the worldly–unworldly and the necessary–universal distinctions has formulas of two sorts. Models determine truth

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, ,   



values of worldly formulas at possible worlds; they determine truth values of unworldly formulas simpliciter. The logic of the operator [U], for truth in all worlds, is exactly what Arthur Prior once sought under the name System A. Prior’s conjectured axiomatization, it turns out, is correct. It is equivalent to a simpler one obtained by adding to a two-sorted version of KT the axiom schema □(P!A) !(P!□A) (where A and P are worldly and unworldly formulas). To this we can add an actuality operator that accords with our understanding that sentences asserting the truth of worldly sentences at particular worlds are themselves unworldly. The logic that results is again shown to have a pleasantly simple axiomatization. The argument that Fine’s puzzle, as originally formulated, provides evidence that Socrates’ humanness is a transcendental, rather than a worldly, fact assumes that we are more reluctant to attribute necessity to propositions true because of the nature of individuals than to propositions true because of worldly circumstances, and that we find it still more difficult to attribute necessity to “hybrid” sentences. A simpler version of the argument, however, reaches the same conclusion under a more plausible understanding of the relation between necessary and transcendental truth. We may question Fine’s emphasis on his two-premise puzzle and even, perhaps, his principal example of a transcendental truth. It is difficulty to deny, however, the interest and importance of the worldly–unworldly distinction and Fine’s thought about it. We should be grateful to him for bringing this, as many other neglected topics in metaphysics, into focus.

References Adams, Robert (), “Theories of Actuality,” Noûs : –. Crossley, John, and Lloyd Humberstone (), “The Logic of Actually,” Reports on Mathematical Logic : –. Fine, Kit (), “Essence and Modality,” Philosophical Perspectives, Vol : Logic and Language, –. Fine, Kit (), Modality and Tense: Philosophical Papers, New York: Oxford University Press. Forbes, Graeme (), “Critical Notice of Kit Fine’s Modality and Tense: Philosophical Papers,” Philosophical Review, : –. Hanson, William (), “Actuality, Necessity and Logical Truth,” Philosophical Studies : –. Hazen, Allen (), “The Eliminability of the Actuality Operator in Propositional Modal Logic,” Notre Dame Journal of Formal Logic : –. Kamp, Hans (), “Formal Properties of ‘Now’,” Theoria : –. Kaplan, David (), “Afterthoughts,” in J. Almog, J. Perry, and H. Wettstein (eds.), Themes from Kaplan, Oxford: Oxford University Press. Kneale, Martha (), “Eternity and Sempiternity,” Proceedings of the Aristotelian Society, New Series : –. Lewis, David (), “Anselm and Actuality,” Noûs : –. Nelson, Michael, and Edward Zalta (), “A Defense of Contingent Logical Truths,” Philosophical Studies : –.

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 . 

Parry, William (), review of Oskar Becker, Untersuchungen über den Modalkalkül, Journal of Symbolic Logic : –. Pollock, John (), “Basic Modal Logic,” Journal of Symbolic Logic : –. Prior, Arthur (), Time and Modality, Oxford: Oxford University Press. Wetzel, Linda (), “Is Socrates Essentially a Man?,” Philosophical Studies : –. Zalta, Edward (), “Logical and Analytic Truths that are Not Necessary,” Journal of Philosophy : –.

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 What is Normative Necessity? Gideon Rosen

 The Least Controversial Thesis in Metaethics You are reading this chapter now, and that’s OK.¹ In reading this chapter you do nothing wrong. Of course your act could have been wrong. If things had been different—if you had made a solemn promise not to read the chapter, or if reading it might have had disastrous consequences—then it would have been wrong for you to read it. Still as things stand, your act is fine. The permissibility of your act is thus a contingent moral truth. The moral truths we bother with in daily life are almost all contingent in this way. And yet we are powerfully inclined to think that in the vicinity of this and every other contingent moral truth, there must be deeper moral truths—perhaps unknown to us—that hold of absolute necessity. As an example, call your act A and let D(A) be a complete specification of its nonnormative features. D(A) specifies A’s intrinsic nature, its causes and effects, the intentions with which it was done, and so forth, insofar as these can be specified in nonnormative terms.² Now consider the conditional:

¹ The text published here is the text delivered at the NYU symposium on Kit Fine’s work in January . Some (but not all) subsequent developments are noted in footnotes. I am grateful to audiences at Columbia University, the University of Texas, Austin, the University of Nebraska, NYU, and the Chapel Hill Metaethics Workshop for their constructive incredulity. Thanks in particular to Emad Atiq, Alisabeth Ayars, Selim Berker, Aaron Bronfman, John Collins, Jonathan Dancy, Janice Dowell, Jamie Dreier, Kit Fine, Reina Hayaki, Thomas Hofweber, Paul Horwich, Jennan Ismael, Philip Kitcher, Isaac Levi, Tristram McPherson, Joe Mendola, Kate Nolfi, Ian Rumfitt, Geoff Sayre-McCord, David Sosa, Mark van Roojen, and David Velleman. ² I assume a distinction between normative properties and relations on the one hand and nonnormative properties and relations on the other, with moral permissibility as paradigmatically normative and the properties and relations of physics paradigmatically nonnormative. I leave it open how that line is to be drawn, but see Rosen a for a proposal. I further assume that propositions are structured entities, built from objects and properties and the rest roughly as a sentence is built from words. A proposition is normative if it has a normative constituent. A fact is a true proposition, so a normative fact is a true proposition with a normative constituent. When I speak of nonnormative “descriptions” of acts, these should be understood as structured propositions involving the act and any number of wholly nonnormative constituents. Gideon Rosen, What is Normative Necessity? In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Gideon Rosen. DOI: 10.1093/oso/9780199652624.003.0011

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() If D(A) then A is morally permissible, or its generalization (): ()

For any action x, if D(x) then x is morally permissible.

Whenever an action has a normative property there will be associated “fact-norm conditionals” of this sort. And with a small handful of exceptions, philosophers agree: these conditionals inevitably hold of absolute necessity if they hold at all.³ This is a consequence of the least controversial thesis in metaethics: Strong Supervenience: If two metaphysically possible entities are alike in every nonnormative respect, they are alike in every normative respect. Metaethicists disagree about most of the big questions. They disagree about whether morality is objective or subjective, absolute or relative, factual or nonfactual, and so on. But when it comes to Supervenience they do not disagree. Indeed, Supervenience is as close to common ground as we find anywhere in philosophy, and it entails that in the vicinity of every contingent moral truth there are necessary truths like () and (). Supervenience is in turn a consequence of a widely held (if slightly more controversial) view about the explanatory structure of normative reality. On this view, whenever a particular act has a normative feature F, there is always an explanation of this fact that cites its relevant nonnormative features ϕ together with a general principle according to which whatever ϕs is F. This general principle may be explained in turn by further principles, perhaps together with contingent facts. But in the end we reach bedrock with a class of pure normative principles that cannot be grounded in this way. These bedrock principles are the basic laws of ethics, the holy grails of moral theory. The compelling view under consideration has it that every non-fundamental moral truth, contingent or otherwise, is ultimately grounded in natural facts together with pure general principles of this sort. This entails Supervenience and the absolute necessity of conditionals like () and () provided we also accept The Non-Contingency of Pure Moral Principles: ciples of ethics are metaphysically necessary.

The basic explanatory prin-

This is more controversial than Supervenience, since some philosophers deny the existence of exceptionless explanatory principles of the envisioned sort (Dancy ).⁴ But it is not controversial that pure moral principles hold of metaphysical necessity if they exist at all.

³ One recent exception is Hattiangadi (). ⁴ The debate over particularism is mainly a debate over the existence of snappy moral principles that hold without exception, but which are also sufficiently accessible to play a role in moral practice. Moderate particularists reject such principles but concede that when an act is (say) wrong, there is always some explanation of this fact that cites only natural features of the act together with a general, though perhaps infinitely complex, conditional connecting those features to normative features. The metaphysical particularist denies even this, holding that singular normative facts of the form A is N are sometimes brute.

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

My first aim in this chapter is to suggest that Supervenience and Non-Contingency should be more controversial than they are. Fine () has argued against these claims and his position is worth taking very seriously. My larger goal, however, is to articulate the truth in the vicinity of Supervenience and Non-Contingency. As Fine notes (: §), even if () and () are not absolutely necessary, they are clearly necessary in a sense. It was no accident that your act A was permissible given its nonnormative profile. Any act with that profile would have been permissible—would have had to be permissible—even if there are remote possible worlds where such acts are wrong. On the replacement view advanced by Fine, basic moral principles and factnorm conditionals⁵ like () and () enjoy a sui generis modal status: they are normatively, though not metaphysically, necessary. The larger aim of this chapter is to provide an account of this notion: to say what it is for a proposition to hold of normative necessity, and to explain why pure moral principles are necessary in this sense.

 Against Supervenience: A First Pass The next section presents an argument against Supervenience (basically Fine’s). But experience has taught me that philosophers otherwise earnestly devoted to the truth will reflexively construe any such argument, not as a welcome challenge to received wisdom, but as a reductio of its premises. A softening-up exercise is therefore in order before we get down to business. The first thing to say is that Supervenience is not a commonsensical view, at least not in any literal sense. The man on the street has never heard of it and cannot understand the terms in which it is stated. (The man on the street does not distinguish metaphysical necessity from necessity of other sorts.) What’s more, as we will see, there is no sense in which ordinary moral practice assumes or presupposes Supervenience. So the principle cannot derive its authority from our practices. Supervenience rather functions in recent philosophy as a technical axiom. It is an esoteric principle, stated in recherché terms. And yet despite its esoteric character, no one bothers to defend it since everyone who understands it finds it completely obvious. Confronted with a claim of this sort, it can be instructive to stage an encounter with a clearheaded interlocutor who does not find the proposition obvious. So consider the impeccably trained Professor Z. She takes in your innocuous act A with its complete nonnormative profile D(A) and judges correctly that A is permissible. We then invite her to pronounce on the modal status of the conditional “If D(A) then A is permissible,” and she says emphatically: “The conditional is obviously contingent. It’s true in the actual world, but there are possible worlds in which D(A) is true and A is wrong.” Z concedes that she cannot imagine these worlds. But that counts for nothing, she thinks. Normative features are invisible, so of course we can’t I assume that moderate particularists regard the highly specific principles they posit as necessary truths, and so accept Non-Contingency. The metaphysical particularist may be a fictional character. ⁵ This label is shorthand and should not mislead. I assume moral/normative realism in what follows, so for me normative statements are fully factual in every sense. A so-called “fact”-norm conditional is a conditional whose antecedent specifies the wholly nonnormative facts (or features of some particular act) in complete detail and whose consequent is normative.

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visualize a moral difference without some further difference.⁶ Still, we can readily conceive of a situation that would falsify the conditional, and that’s what matters. As she puts it, Consider a world w that is just like the actual world in nonnormative respects but in which Act Utilitarianism is true. Your act A would have been wrong if w had been actual. After all, no matter how much benefit the universe derives from your scrutiny of this paper, you would have done more good licking stamps for Oxfam instead. So we have a world w in which D(A) is true and A is wrong. Together with the actual world—where D(A) is true and A is not wrong—this yields a counterexample to Supervenience.⁷

How to reply? It is no use denying that there is such a world as w. For present purposes, a world is not a physical universe. It is a class of propositions or something of the sort: a maximal consistent world-description.⁸ D(A) is, or corresponds to, a class of propositions: the nonnormative propositions that are in fact true. If we pretend —for simplicity—that moral permissibility and impermissibility are the only normative properties on the scene, the logical closure of D(A) and Act Utilitarianism (AU) is a world: a class of propositions that specifies the nonnormative facts in complete detail and then assigns a normative status to every act. So the question is not whether w is a world. It is whether w is a metaphysically possible world. Professor Z thinks it is because it strikes her as possible, and also because it shows all the marks of metaphysical possibility. W is logically consistent. It is also analytically and conceptually consistent in any reasonable sense of these terms. After all, some philosophers believe that w is actual, and however misguided these utilitarians may be, their mistake does not smack of linguistic or conceptual incompetence. Moreover, we can reason smoothly about how things would have been had w been actual without tying ourselves up in any of the knots we encounter when we try to reason about how things would have been if Hobbes had squared the circle or if Fred had been a married bachelor. As she reflects, Professor Z can see no obstacle to the possibility of w, no reason why things could not have been as w says they are. Indeed, she cannot see how there could be such an obstacle. So she concludes that w is possible after all. Of course Z knows that conceivability is a fallible guide to possibility. Let H be a hydrogen atom and let Δ(H) be a complete description of H at the level of elementary particles—quarks and electrons. Now consider a world w* that includes Δ(H) together with the proposition that H is a lithium atom. This world is logically and analytically consistent, and a chemical ignoramus might be able to reason smoothly about how things would have been had w* been actual. But w* is not metaphysically possible, and we know why. It lies in the nature of lithium that lithium atoms have three protons. H has only one. So w* is impossible—despite its superficial consistency—because it is inconsistent with an essential truth (Fine a). ⁶ Compare the invitation to picture a difference in objective, single-case probabilities without a difference in the non-probabilistic facts. However difficult this may be, it has no tendency to show that the probability facts supervene on the rest. ⁷ Since AU is presumably incompatible with the actual moral principles, this is also a counterexample to the Non-Contingency of Moral Principles. ⁸ See Kment (: ch. ) for the conception of worlds I take for granted.

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

Professor Z should therefore be open to the following reply: You think the Act Utilitarian world w is genuinely possible, but that’s only because you don’t know enough about the essences. If you knew more, you would see that an act with nonnormative features D(A) could not possibly have been wrong and that AU could not possibly have been true (the natural facts being as they are).

And she is open to such an argument. Z will gladly change her mind if we can produce the essential truth with which w is inconsistent, or at least make it plausible that there must be some such truth. It’s just that as things stand she finds this unlikely. It’s as if someone were to say: “Even though it seems to you that there could have been blue swans, the existence of such things is incompatible with the (unknown) essential truths.” Our ignorance of essences is profound, so it’s always possible that some unknown essence rules out w and other worlds with different basic moral principles. It’s just that as things stand we have no reason to take this possibility seriously, or so says Z. She is thus left with the undefeated judgment that w is possible and that Supervenience as we have formulated it is therefore false. If you still find Supervenience obvious, you must think that Z has missed something important. So tell us: What is she missing, and what would it take to get her to see it?

 Against Supervenience: An Argument We can turn these rough considerations into an argument against Supervenience. The first premise is the essentialist account of metaphysical modality due in relevant form to Fine (a). The account starts with the premodal notion of essence. Given any item x—object, property, relation, and so on—we have the truths that obtain in virtue of x’s nature or identity. It lies in the nature of the number  to be a number, to follow , and so on. And so we have, in Fine’s notation: □₂ □₂

 is a number  follows 

By contrast, even though it is a necessary truth that the number  is not the moon, it does not lie in the nature of the number , considered by itself, to be distinct from the moon. Intuitively, one might know everything there is to know about the identity of this number—about what makes  the number that it is—without knowing the first thing about astronomy. Consider, by contrast, the set {, the moon}. It presumably lies in the nature of this set, not just that the moon and the number  are its only members, but also that it is a two-membered set. To know the nature of a set is to know (inter alia) its cardinality. And so we have: □{, the moon}

 6¼ the moon,

but not: □₂

 6¼ the moon.

Just as we can speak of the essences of individual items, we can speak of the essences of things in the plural. If you and I are both essentially human, then, while it does not lie in your nature to be my species-mate (since your essence makes

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no reference to me), nor vice versa, it does lie in our essence to be members of a single species. □you, me

There is a species to which you and I both belong (if we exist).

If there exist one or more items, the Xs, such that it lies in the nature of the Xs that p, we say that p is an essential truth.⁹ We could argue about whether this notion is clear enough for serious use, but I’m going to suppose that it is. Even though it is officially primitive, it can be explained by examples, verbal glosses, and ultimately by an explicit theory (Fine a; ). The notion is vindicated to the extent that this theory is compelling and fruitful elsewhere in philosophy. Fine’s main application of the fine-grained notion of essence is a definition of metaphysical necessity: The Essentialist Account of Metaphysical Necessity For p to be metaphysically necessary is for there to be some items X such that □Xp For a proposition to be metaphysically necessary just is for it to be an essential truth (or a logical consequence of essential truths if essential truths are not automatically closed under logical consequence, as I shall suppose they are). A proposition is metaphysically possible when it is logically consistent with the essential truths, or equivalently, when no collection of essential truths constitutes a logical obstacle to its truth. (For a related view, see Peacocke : ch. .) Recall the world w*, which contains both the complete description of a hydrogen atom H at the level of fundamental physics and the proposition that H is a lithium atom. This world is logically inconsistent with an essential truth, viz., the proposition that lithium nuclei contain three protons. According to the essentialist, that is why things could not have been as w* says they are. The friend of Supervenience says that D(A) + AU is likewise impossible. So given the essentialist framework, the challenge is to identify the items whose natures rule it out.

⁹ Here I flag a subtle question about these plural essences, viz., whether such essences are separable: Separability: If □a, b, . . . p then there are propositions A, B, . . . such that □aA, □bB, . . . and A, B, . . . logically entail p. The examples that motivate appeal to plural essences are generally consistent with this principle, and it simplifies the theory significantly to assume it, so I shall. The main examples that pull against separability concern nonidentities. If a and b are distinct, they are necessarily distinct. If every necessary truth must be a logical consequence of essential truths, as Fine maintains (see immediately below), then we need to find some items X such that □X a 6¼ b. When the essences of a and b are logically incompatible, a and b together will do the trick. (If it lies in the nature of Socrates to be an animal and in the nature of Mt. Rushmore to be inanimate, then these natures taken together will entail that Socrates 6¼ Mt. Rushmore, and so ground the necessity of their nonidentity. But if a and b are two electrons, say, whose natures are logically compatible and perhaps even identical, their natures will not entail their distinctness as a matter of standard logic. We could invoke the set {a, b}, whose nature presumably does entail that a 6¼ b. But it sounds quite wrong to say that a and b are necessarily distinct because {a, b} essentially has two members. (That sounds backwards.) We cannot resolve this issue here, and I shall assume separability in what follows. (The alternative seems to me to involve a mystical holism in the theory of essence.) If separability is false, the argument of this section does not go through.

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The culprits cannot lie exclusively among the nonnormative properties and relations in D(A). Say everything there is to say about what it is for a thing to be a proton or an elephant or a toothache: if these notions are genuinely nonnormative, you will never find yourself mentioning moral permissibility or the like.¹⁰ The essences of nonnormative things are silent about moral permissibility, so these essence are consistent with DðAÞ þ AU. This is not a conjecture. We have not said how the line between the normative and the rest is to be drawn, but one clear constraint on the enterprise is this. If the essence of some item nontrivially involves a paradigmatically normative property like moral permissibility, that item must itself be reckoned normative. Since D(A) is a wholly nonnormative account of A, it follows that a complete elaboration of the essences of the items that figure in it will not mention moral permissibility (except trivially), and will thus place no constraint on the extension of permissibility. The essences of nonnormative things therefore cannot rule out DðAÞ þ AU. If DðAÞ þ AU is impossible, it must therefore be inconsistent with the essence of moral permissibility. (The culprit might be some other normative item that does not figure explicitly in AU, but we are pretending for now that permissibility is the only normative property on the scene.) And of course this is exactly what one sort of ethical naturalist maintains. To a good first approximation, ethical naturalism is the thesis that every normative item (property, relation, etc.) admits of real definition in wholly nonnormative terms. I emphasize real definition to make the familiar point that naturalism is not a thesis about moral language or moral thought. Present-day naturalists mostly concede the Hume/Moore point that moral words and concepts cannot be defined in nonnormative terms, while maintaining that every moral predicate nonetheless “picks out a natural property.” But a natural property is a property that can be defined in wholly nonnormative terms.¹¹ So the naturalist’s distinctive claim (again, to a first approximation) must be this: For each normative property M, there is a correct account of the form: For a thing to be M just is for it to be φ, Or Being M consists in (reduces to) being φ, where φ is a condition composed entirely of wholly nonnormative ingredients.

¹⁰ Except trivially. If essences are closed under logical consequence, as I assume, then it lies in the nature of elephant that everything an elephant does is either permissible or not permissible. But that will only be because for any property F, it lies in the nature of elephant that everything an elephant does is either F or not-F. Following Fine (b), say that G occurs nontrivially in X’s essence iff □X ( . . . G . . . ), but it’s not the case that □X 8F( . . . F . . . ). Then the claim in the text is the claim that if an item is nonnormative, normative items figure only trivially in its essence. ¹¹ More exactly, a definable natural property is a property that can be defined in wholly nonnormative terms. Indefinable natural properties count as natural for some other reason. I suppose it could turn out that some normative property is also an indefinable natural property, in which case this characterization of ethical naturalism would need revision, but this possibility is too far-fetched to bother with. (Thanks to Emad Atiq for pressing this point.)

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This characterization may seem unnecessarily metaphysical. But if this or something like it is not the view, then I don’t know what ethical naturalism is supposed to be. Ethical naturalism is not the view that the normative supervenes on the nonnormative. G. E. Moore and his twenty-first-century followers, including Parfit () and Scanlon, all accept Supervenience, but they are not ethical naturalists just for that.¹² The nonnaturalist’s distinctive claim is that even if there are true propositions that specify naturalistic necessary and sufficient conditions for each moral feature—as there must be given Supervenience if we allow infinitary conditions—these principles do not tell us what it is for an action to be, say, right. They are rather metaphysically synthetic laws that connect a normative feature, moral rightness, with utterly distinct nonnormative features, the so-called “rightmaking” features. The naturalist’s distinctive claim must therefore be that not only do such principles exist: they tell us what it is for an action to be right.¹³ For present purposes, a simple account of real definition will suffice. Say that φ defines F iff (a) (b)

□F 8xðFx

$ φxÞ The essences of the constituents of φ make no nontrivial reference to F.¹⁴

On this account, ethical naturalism is the thesis that the natures of the normative properties and relations, taken one at a time, determine naturalistic necessary and sufficient conditions for their application. (A more relaxed form of naturalism would hold that the natures of the normative properties collectively determine such conditions.) The alternative, ethical non-naturalism, is the view that in at least one case, the essence of a normative property fails to determine naturalistic necessary and sufficient conditions for its instantiation. So we can state the second premise in the argument against Supervenience as follows: Non-naturalism. There is a normative property M that does not admit of real definition in wholly nonnormative terms. Given some simplifying assumptions, it can be shown that these premises—the essentialist account of modality and non-naturalism—are incompatible with Supervenience. As is familiar, Strong Supervenience entails that each moral property is necessarily equivalent to some nonnormative condition. Take moral rightness as our example. Let a, b, . . . be all of the metaphysically possible right acts, and let Da(a), Db(b), . . . be their respective nonnormative descriptions in the worlds in which they are right. Strong Supervenience then guarantees that

¹² Moore’s position in Moore () can be read, following Fine (), as rejecting the metaphysical supervenience of the ethical on the natural. For a discussion of Scanlon on supervenience, see Rosen (b). ¹³ I now favor a ground-theoretic account of ethical naturalism, according to which ethical naturalism is the thesis that for each moral property M, atomic facts of the form [Mα] are always metaphysically grounded without remainder in the nonnormative facts. The account given in the text, according to which each moral property must be definable in nonnormative terms, is an account of reductive naturalism, not of naturalism tout court. See Rosen a. ¹⁴ This is Kment’s account of real definition (Kment : –). See Rosen () for an alternative.

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   ? ()

□8xðx



is right $ ðDa ðxÞ _ Db ðxÞ _ :::ÞÞ

Given the essentialist account of metaphysical necessity, this is equivalent to ()

9Y □Y 8xðx is right $ ðDa ðxÞ _ Db ðxÞ _ :::ÞÞ

Now as we’ve seen, it can’t possibly lie in the natures of the nonnormative properties, individually or collectively, that right should have some particular extension. (The essences of the nonnormative properties don’t mention right, except trivially, and so cannot constrain its extension.) So if we assume, as we have thus far, that right is the only normative property on the scene, we can conclude that () is true because () is true. ()

□right;P;Q;::: 8xðx

is right $ ðDa ðxÞ _ Db ðxÞ _ :::ÞÞ

That is, the nature of right, possibly together with the natures of various nonnormative properties P, Q, . . . , entails that right is equivalent to the long disjunction on the right-hand side. Now if the list of additional properties P, Q, . . . is empty, () is a real definition of right in naturalistic terms and we’re done. We’ve derived ethical naturalism from Supervenience in the essentialist framework. But of course it is highly implausible that this list will be empty. The various Dis are complete, fully detailed, nonnormative descriptions of acts. So they will mention things like quark color and elephants. But the nature of right has never heard of such things, or so it’s natural to suppose. (A definition of right that mentions quarks is like a definition of house that mentions bricks: too specific to be correct.) So the proponent of Supervenience must therefore think that it is the nature of right together with the natures of various nonnormative properties and relations that grounds the equivalence of right and the long nonnormative disjunctive condition. We can complete the argument by invoking the principle called separability in note  above. This principle tells us that when a truth is grounded in the essences of several things collectively, it is a logical consequence of the essences of those things taken individually. Given this assumption, () entails that the equivalence of right and ðDa ðxÞ _ Db ðxÞ _ :::Þ must follow from two propositions, one grounded in the nature of right alone, the other grounded in the natures of various nonnormative items: P, Q, and so on. But that means that (4) must be a consequence of two principles: (a) (b)

□right 8xðx

is right $ φxÞ, and $ ðDa ðxÞ _ Db ðxÞ _ :::ÞÞ,

□P;Q;::: 8xðφx

where φ is wholly nonnormative. But then again, we’re done, since (a) is a real definition of right in wholly nonnormative terms. The argument shows that given our simplifying assumption that right is the only normative property on the scene, Supervenience entails naturalism within the essentialist framework. Things are less straightforward when we relax this simplifying assumption. Suppose, for example, that there are two normative properties, M and N, neither of which is definable in terms of the other. Supervenience entails the existence of suitable biconditionals giving naturalistic necessary and sufficient conditions for each.

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

 

(a) (b)

□8xðMx □8xðNx

$ ϕxÞ $ ψxÞ

Given the essentialist account, these in turn entail (a) (b)

□M;X 8xðMx □N;Y 8xðNx

$ φxÞ $ ψxÞ

where X and Y are lists of items whose natures play a role in grounding the relevant equivalence. If X and Y are empty, or if they contain only nonnormative ingredients, we’re done. These are real definitions of M and N in wholly nonnormative terms. The trouble is that we cannot rule out the possibility that, for example, the nature of N plays an ineliminable role in grounding the equivalence of M and φ. ()

□M;N;X 0 8xðMx

$ φxÞ, but not: □M;X 0 8xðMx $ φxÞ.

Supervenience entails naturalism as we have defined it only if this cannot happen. I note that this problem arises only if the natures of M and N make no nontrivial reference to one another. For if they do, we can invoke a principle in the logic of essence, here called chaining, to rule out cases of the sort described in (). Chaining:

If □xp and x figures nontrivially in y’s essence, then □yp

The rationale for chaining is clear enough. Suppose it lies in the nature of knowledge that S knows that p only if S believes that p, and that it lies in the nature of belief that beliefs have a certain functional role. Then it follows intuitively that someone who knows everything there is to know about what it is to know something must also know that beliefs have the functional role in question. If he doesn’t, he hasn’t seen all the way to the bottom of the nature of knowledge. But that is just to say that it must lie in the nature of knowledge that beliefs have the functional role in question.¹⁵ If M’s nature makes nontrivial reference to N, it follows from □M;N 8xðMx $ φxÞ that □M 8xðMx $ φxÞ, and hence that cases like (8) cannot arise. But suppose M’s nature makes no reference to N. In that case I do not know how to derive a naturalistic real definition of M from the assumption that the natures of M and N together determine naturalistic necessary and sufficient conditions for M.¹⁶ The ¹⁵ Fine (b) distinguishes several notions of essence. Chaining holds only for the notion there called mediate essence. But for independent reasons, that is exactly the right notion of essence for the account of real definition (Rosen ). So it is not ad hoc in the present context to assume chaining. ¹⁶ An artificial move might close the gap. Suppose □M ;N;::: 8xðMx $ φxÞ. We can then ramsify, replacing the normative words in the formula with variables, and identify M as follows M = the property X such that there are properties Y, etc., such that $ φxÞ.

□X;Y;::: 8xðXx

This can then be converted into what appears to be a naturalistic real definition of M in our format: □M 8x (Mx $ x possess the property X such that there are properties Y, etc., such that □X;Y;::: 8xðXx

$ φxÞ).

This would be a nonstandard real definition, in which the defined property is picked out, in the definiens, by means of a description that embeds the essentiality operator. Whether such definitions should be admitted is unclear.

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   ?



possibility strikes me as bizarre, and I can think of no remotely plausible cases that would illustrate it. But I cannot rule it out from first principles, and so cannot show that Supervenience entails ethical naturalism as we have defined it. And yet Supervenience does entail a view that deserves to be called ethical naturalism, namely: Ethical Naturalismalt: For each normative property M there is a nonnormative condition φ such that □M;N;::: 8xðMx $ φxÞ, where M, N, . . . are all of the normative properties and relations taken together. The naturalist’s fundamental thought, it seems to me, is not that each normative property is separately definable in nonnormative terms. It is rather that the normative facts are fixed by the wholly nonnormative facts (e.g., facts of physics and psychology) together with the natures of the normative properties and relations. On this sort of view, anyone who knows the nonnormative facts is in a position to derive the ethical facts provided she also knows what it is for an act to be right, good, rational, and so on. The non-naturalist’s distinctive commitment, by the same token, is that someone who knows the natural facts and the essences might still be in the dark about the synthetic principles that connect the normative facts to their nonnormative grounds. Even if Supervenience does not entail the separate definability of the normative properties, it does entail naturalism in this somewhat more capacious sense. So we have a proof of ethical naturalism from Supervenience and the essentialist account of metaphysical modality. Equivalently: We have a proof whose premises are Ethical Non-Naturalism and the Essentialist Account of Metaphysical Necessity, and whose conclusion is that Strong Supervenience is false.¹⁷

 The Status of the Premises Of course what we really have is an inconsistent triad: Strong Supervenience The Essentialist Account of Metaphysical Necessity Ethical Non-Naturalism. If any two are true, the third is false. I have called this an argument against Supervenience. But of course others will see things differently. So let me be clear about the status of this argument as I see it. In my view, the essentialist account of modality and ethical non-naturalism are both (separately and together) serious contenders. Apart from Lewis’s modal realism, the essentialist account has no clear rival as a reductive account of metaphysical necessity; and while I would not object to taking the metaphysical modal notions as ¹⁷ It is often said, following Blackburn (), that non-naturalists have trouble accounting for Supervenience, and that this provides a powerful objection to the view. See McPherson () for a recent defense of this position. My claim is stronger: if the Finean essentialist is right about the nature of metaphysical possibility and necessity, then non-naturalists must reject Supervenience. (As I will go on to show, this is a feature, not a bug.)

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

 

primitive if there were no alternative, the essentialist account promises to ground and explain the facts of metaphysical modality in a compelling way. If nothing else, the essentialist account strikes me as a permissible sharpening of the perhaps somewhat unclear notion of metaphysical necessity, and indeed as the most fruitful such sharpening yet proposed.¹⁸ As for ethical non-naturalism, I claim only that in the absence of credible reductive proposals or reasons to think that some naturalistic reduction must be possible, the question whether the normative facts reduce to natural facts remains wide open. We simply do not have clear enough view of the natures of the moral properties and relations to say whether these natures determine naturalistic necessary and sufficient conditions for their application. We should therefore be prepared for the possibility that some ethical properties are non-natural, and hence for the possibility that Supervenience is false. Fine himself takes a harder line, arguing on general grounds that ethical naturalism is clearly false (Fine : §; also Parfit : v. , §). But it seems to me that Fine’s argument misses a possibility. Fine shows persuasively that if there is some essential equivalence between a normative property like right and a natural property like being such as to maximize happiness, it is neither a conceptual equivalence nor an a posteriori equivalence of the sort favored by naturalists who take reduction in the sciences as their model. But there is room for another view according to which truths of the form □M 8xðMx $ φxÞ are neither conceptual nor empirical, but rather the sort of truth one comes to know through philosophical reflection. Suppose we sit down to noodle out the single true morality, testing hypotheses against our judgments about cases, and suppose that at the end of the day we find ourselves endorsing a counterfactual-supporting universally quantified biconditional: 8xðx is right $ φxÞ, where φ is nonnormative through and through. One sort of theorist—Parfit certainly, Fine perhaps—will contemplate this culmination of centuries of philosophy and say: “This principle may be necessary in a sense. But it does not tell us what it is for an action to be right. It is rather a synthetic moral law connecting rightness with something else.” However, another sort of theorist will say: “I didn’t see this coming, but now that we have the principle in front of us, it seems to me that we have attained the sort of understanding of the nature of rightness that physicists have long have of nature of (say) heat and to which philosophers aspire when they ask ‘What is F?’ questions in other areas. This knowledge is not analytic or conceptual in any useful sense, nor is it simply knowledge of a synthetic law. Rather ethical reflection has delivered a non-obvious but recognizably correct real definition of rightness cast entirely in naturalistic terms.” Can we know in advance—without seeing φ—that it would be wrong to respond in this way?

¹⁸ I argue for the unclarity of the received notion of metaphysical necessity, and for the availability of the essentialist sharpening, in Rosen (). For an alternative sharpening, see Kment (: ch. ).

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   ?



Fine does not address the possibility, but his remarks against conceptual naturalism are apposite. Extracting the sound point in Moore’s Open Question Argument, Fine writes, Perhaps a more satisfactory way to formulate the objection is as follows. If there is a correct analysis of good, say, as what promotes pleasure over pain, then something’s being good must consist in nothing more than its promoting pleasure over pain. But we have a strong intuition that it does consist in something more. Here we are not relying on the purported epistemic status of a correct analysis, as is Moore, but on its metaphysical consequences. (Fine : )

We can take this as a prediction that even if we were presented with a compelling principle connecting good with some nonnormative condition, we would not respond in the second way canvassed above, treating the equivalence as real definition, but rather in the first, treating the equivalence as a synthetic law.¹⁹ For what it’s worth, my hunch is that Fine is right. But I don’t think we’re entitled to much confidence on the point, since we’ve seen very few serious candidate analyses. As an antidote to this intuition, consider Schroeder’s account of reasons for action (Schroeder ). Schroeder takes up the metaphysical question: What is it for R to count as a reason for X to do A? The inquiry is thoroughly a priori. Putative accounts are tested against hypothetical cases and refined in the armchair. The first upshot is a plausible counterfactual-supporting biconditional: For all propositions R, agents X and actions A, R is a reason for X to do A iff there is some proposition P such that X has a desire whose object is P and the truth of R is part of what explains why doing A promotes P. (Schroeder : )

After defending the biconditional, which is meant to be wholly nonnormative on the right, Schroeder makes the stronger claim that the account yields a real definition of reason. Of course we can doubt the extensional correctness of the account, but suppose we don’t. Fine’s objection to naturalism, applied to this case, would run as follows: If this is a correct analysis, then something’s being a reason for X to do A must consist in nothing more than its explaining why A would promote the satisfaction of one of X’s desires. But we have the strong intuition that it does consist in something more.

But do we have this intuition? I can’t say that I do. More exactly, I can’t tell whether I have the Finean intuition of a metaphysical gap between the normative relation on the left and the natural condition on the right, or instead the irrelevant Moorean intuition that the two formulae have different meanings. If I thought that something like this could be done for every normative property and relation, I would be optimistic about the prospects for reductive naturalism in ethics. As things stand, I’m pessimistic. But that’s just an expression of my state of mind. As a matter of philosophy, the question remains open.

¹⁹ Fine’s claim is an expression of what Enoch calls the “just too different” intuition (: , n. ). See McPherson () for discussion.

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

 

So I do not say that naturalism is false, or that we have strong grounds for rejecting it. I say only that we have no strong ground for accepting it,²⁰ and that we should thus prepare for the possibility that naturalism is false. Put another way: we should not assume Supervenience as a premise in metaethics, since it stands and falls with naturalism, and naturalism is unsettled. The argument thus far is meant to motivate this stance.

 Normative Necessity and Fact-Independence Suppose the oracle of philosophy tells us that Fine is right: right about metaphysical modality, right in thinking that at least one normative property cannot be defined in nonnormative terms; and hence right in rejecting Strong Supervenience. This means that some true conditionals of the form 8xðDðxÞ ! NxÞ are contingent truths, where D(x) is the complete nonnormative profile of some act, and N is normative. Still, as Fine notes, we are powerfully inclined to think that given that an act has certain nonnormative features, it has to be (say) wrong. It is no accident that it was wrong for me to bribe the judge, given that my act had the natural features that it had. Any act with those features would have been—would have had to be—wrong. But that is just to say that even if the relevant conditional is not metaphysically necessary, it is not an accidental generalization either. In this respect it resembles a law of nature: necessary in a sense, but not absolutely necessary. The same goes for the general moral principles that moral philosophers typically seek: exceptionless laws that specify the conditions under which it is permissible to break a promise or cause harm or do anything at all. Such principles are elusive. But everyone agrees that if we found them they would not be accidental regularities, true only because every actual act of promise breaking in such and such circumstances is permissible. They would support counterfactuals. If it’s a true moral principle that it’s okay to break a promise in conditions C, it follows that if Bob had broken a promise in C, his act would have been okay. Fine says that principles of this sort—the detailed fact-norm conditionals and general moral principles if they exist—possess a sui generis species of necessity: they are normatively necessary. The true supervenience claim is not the standard one, formulated in terms of metaphysical necessity, but rather the weaker claim that the natural facts determine the normative facts as a matter of normative necessity (Fine ). That is: For each fully determinate constellation of nonnormative facts D, there is a constellation of normative facts N, such that as a matter of normative necessity, if D obtains then N obtains. Fine argues that normative necessity cannot be defined in terms of metaphysical necessity (or natural necessity) and leaves it open whether more can be said about its nature. ²⁰ “But isn’t the plausibility of Supervenience evidence for the truth of naturalism?” No, it is not. As I suggest below, once we distinguish standard Supervenience theses from weaker versions, we will see that the plausible thesis in the vicinity, and the thesis that moral theory and practice tacitly take for granted, is not strong Supervenience, but a weaker claim that is consistent with non-naturalism.

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   ?



I believe that more can be said. Start by focusing on the complete fact-norm material conditionals of the form D  N, where D is a complete and consistent specification of a way for the nonnormative world to be, and N a complete assignment of normative properties and relations to the items that would exist if D were actual. Some of these conditionals are false. Throw those out. Some are true, but only because the antecedent is false. More exactly, some are such that the counterfactual, “If D had been the case, then N would have been the case” is false. Throw those out as well. What remains are the true fact-norm conditionals that would still have been true if their antecedents had been true, or in other words: the fact-norm conditionals that would have been true no matter how the nonnormative facts had been. These are precisely the fact-norm conditionals that we are inclined to regard as necessary—as metaphysically necessary if we are incautious, as normatively necessary if we’ve seen the light. Let D specify in complete detail the natural features of a world rather like this one in which Fred tortures a cat for kicks. There are countless true conditionals of the form D  N, where N is a complete specification of the normative facts. But we are powerfully inclined to think that for each complete specification of the nonnormative facts D, there is only one true counterfactual of the form, “If D had been the case then N would have been the case,” and that the consequent of this conditional tells us not only how things would have been had D been true, but how things would have to have been in such a case. This counterfactual-supporting fact-norm conditional is normatively necessary. The rest are at best accidentally true. We can generalize the point. Say that p is fact-independent if p is the case and would have been the case no matter how things had been in wholly nonnormative respects.²¹ Fact-independence is clearly an important feature. If a proposition is factindependent, it would still have been true no matter what we had done, no matter how hard we had worked to falsify it, no matter how history had transpired, no matter how the initial conditions of the physical universe had been configured, no matter what the laws of nature had been, and so on. Moreover, these are all marks of necessity. They are features of the essential truths²² and of all of the truths we normally regard as metaphysically necessary. But they are also features of the pure moral laws even if those are not metaphysically necessary. All of which suggests a proposal: for a proposition to be normatively necessary just is for it to be fact-independent.²³ Normative necessity defined. For a proposition p to be normatively necessary at w is for p to be a proposition true at w such that for any wholly nonnormative proposition q, the counterfactual “if q had been the case, p would still have been the case” is true at w.²⁴ ²¹ Again, since I assume that normative claims are fully factual (as distinct from non-factual), this terminology is shorthand. “Fact independence” is really “wholly nonnormative-fact-independence.” ²² Pace Salmon (), who argues that ordinary material objects would have had different essences if they had been made from slightly different matter. For discussion see Leslie (). ²³ There is room for a slightly weaker proposal, viz., that the normative necessity of p is always grounded in, or explained by, the fact that p is fact-independent. Nothing in what follows will turn on the distinction between this view and the view defended in the text, viz., that normative necessity just is fact-independence. ²⁴ Given the conventional assumption that counterfactuals with metaphysically impossible antecedents are vacuously true, this entails that metaphysically necessary truths are all normatively necessity, and is thus compatible with a picture according to which the sphere of normatively possible worlds is nested

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

 

 Normative Necessity is the Only Necessity that Ethics Needs The idea that moral principles are normatively necessary in this sense coheres well with the case-based methodology of moral theory. Suppose we’re studying promissory morality and want to know, for example, when it’s morally permissible to break a promise. On the table is a principle that says: It’s permissible to break a promise if C. According to the usual case-based methodology, we test such principles by entertaining hypothetical cases in which C holds and then asking whether it would be permissible to break a promise in such a case. (Note the counterfactual in this formulation.) If we find a case of this relevant sort in which breach of promise would be wrong, then as we normally think, we have a counterexample that sinks the principle. But suppose a proponent of the principle hears the case and shrugs: “Oh, I agree that the principle would have been false if the facts had been as described; but that doesn’t mean it is false, since the facts are not as described.” This response is confused, and it’s easy to say why. The principle was proposed as a putative moral principle. Even if it sounded like an unadorned conditional something stronger was intended, viz.: As a matter of (some sort of) necessity: It’s permissible to break a promise if C. Any moral principle worth the name must apply, not just in the actual world, but across a range of possible worlds. And so we face the question: What sort of necessity must attach to moral principles if our case-based methodology is to make sense? And I claim: Normative necessity as we have defined it would do quite nicely. Of course in some parts of ethics we’re content with principles that would have been false if the facts had been different in certain ways. In medical ethics, we can’t refute a candidate principle governing the allocation of kidneys by pointing out that it would have been false if kidneys had grown on trees. But there is a part of moral theory—pure moral theory—in which purported principles are vulnerable to the most far out counterexamples. In this part of ethics, if a putative principle would be false if death were temporary (like sleep), or if people spawned like salmon, then that principle is false as stated. Unlike the empirically grounded precepts of applied ethics, the principles of pure moral theory are supposed to hold, as it were, on moral

strictly within the sphere of metaphysically possible worlds. If we relax this assumption, as is sometimes convenient (Nolan ), then some metaphysically necessary truths will be normatively contingent. For example, the proposition that 2 þ 2 ¼ 4 may be metaphysically necessary, but it will fail the test for normative necessity, since it would have been false if (per impossibile) there had been no numbers. This yields a two-way dissociation of metaphysical possibility and normative possibility. There are metaphysically impossible worlds that are normatively possible—worlds at which the normative principles are just as they are but (say) numbers don’t exist. But there are also metaphysically possible worlds that are normatively impossible: worlds at which act utilitarianism is true, for example. Given these assumptions, the class of normatively possible worlds and the class of metaphysically possible worlds cannot both be spheres around the actual world in the sense of Lewis (). See Lange  for an argument that this would undermine the view I have been defending.

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   ?



grounds alone. The case-based methodology of normative ethics seeks to identify these principles roughly as follows: We consider hypothetical cases and ask whether some putative principle would have been false if the facts had been as specified. If so, we have a counterexample and the principle takes a hit. If we look hard for counterexamples and fail to find them, the principle is to some degree confirmed. All of this makes perfect sense if the aim of pure moral theory is to identify principles that hold of normative necessity as we have defined it. When all goes well our methodology works to screen out principles that would have been false had the facts been different, and so to certify principles that would have been true no matter how the facts had been. But note: As we ordinarily think, it is no objection to a putative principle of promissory morality that it would have been false if act utilitarianism had been true. Perhaps this is because we take it for granted that act utilitarianism could not possibly have been true, and more generally, that pure moral principles are metaphysically non-contingent. However no such assumption is needed to make sense of our practice. For that purpose it is enough to suppose that we expect pure moral principles to exhibit fact-independence. Philosophers who self-consciously accepted the metaethical view I have been discussing and who sought pure moral principles that were normatively necessary but metaphysically contingent would conduct their first order inquiry in moral theory just as we actually do. Nothing in the practice of moral theory assumes or presupposes that the pure moral principles we seek hold of absolute necessity. The proposal also makes sense of the role of moral principles in ground-level moral practice. When a conscientious person makes a moral judgment about a case, she brings her (often tacit) knowledge of principles to bear. Inevitably this involves the assessment of unchosen options—options that have not been taken or will not be taken. The conscientious decision-maker needs to know in advance which of her options is permissible; but more exactly, she needs to know which of her options would be permissible if she took it. The conscientious moral judge needs to know whether things would have been better or worse had the agent acted differently, and so on. For all of these purposes we need modally robust moral principles: principles that are true and would still have been true had the agent acted differently. Normatively necessary principles would suffice, since their fact-independence guarantees that they would still have been in place no matter what anyone had done. And yet it is quite irrelevant for these purposes if such actions are permissible in remote possible worlds where the moral laws are different. Even if there are worlds of this sort, our practices of choice and assessment never lead us to consider them.²⁵ When we rely on moral principles in practice we assume that they are robust enough to be held fixed in reasoning about how things would have been had the nonnormative facts been otherwise. But we do not assume, and need not assume, that our principles could not have been false under any circumstances. Moral practice simply takes no stand on whether counter-moral worlds are possible. If we have determined that act ²⁵ Unless, of course, we are uncertain about the moral principles, in which case deliberation and assessment may lead us to consider words which are, in fact, worlds whose moral principles differ from the actual moral principles.

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

 

utilitarianism is false, and that it would not have been true no matter how the facts had been, it simply does not matter for practical purposes whether remote worlds in which it is true get classified as “possible” in some other sense. This last point amounts to the claim that nothing in moral practice assumes or presupposes Strong Supervenience. I do not claim to have established it. But I honestly cannot think of anything in our use of moral principles that cares about whether remote scenarios that would supply counterexamples to Supervenience get classified as metaphysically possible or impossible. The scenarios are “coherent,” and so we can entertain them if we like, no matter how we classify them. We take it for granted that the actual moral principles, whatever they may be, would still have been in place no matter what we had done, no matter how things had gone, and so on, and hence that Supervenience holds within the sphere of worlds that we can arrive at by starting with the actual world and varying the nonnormative facts by counterfactual supposition. If we assume something stronger, I can find no trace of that assumption in the uses to which moral principles are put. These remarks are meant to suggest that the metaphysical view I have been discussing is non-revisionary with respect to moral theory and practice. If this is right, then received ideas like Supervenience and Non-Contingency cannot derive authority from these sources.

 The Explanatory Structure of Normative Reality This account of the modal status of moral principles suggests a picture of normative reality that we should make explicit. According this picture, every world includes a class of fact-independent normative propositions.²⁶ These are the normative laws of w, the propositions that hold at w as a matter of normative necessity. In many cases these principles are metaphysically contingent: they do not hold in virtue of the natures of their constituents, so there are metaphysically possible worlds in which they vary. But they are necessary in the sense that they are not contingent on the nonnormative facts. The domain of pure moral principles is heterogeneous. It includes fully detailed counterfactual-supporting fact-norm conditionals D ! N, where D gives a fully detailed specification of a way for the nonnormative facts to be. But it also includes principles that look more like general, action-guiding moral precepts—principles that apply across a range of cases. These pure principles, moreover, are not all on a par. Some ground others and are in that sense more fundamental.²⁷ For one sort of particularist, the explanatorily basic principles are the detailed fact-norm conditionals in all of their infinite complexity and multiplicity; principles of ²⁶ The class can be empty, to allow for worlds relative to which normative properties and relations are altogether alien. ²⁷ According to me, the grounding relation among normative principles is the same metaphysical grounding relation that holds in other areas (Rosen ). In a slightly different context, Fine () posits a distinct grounding relation that holds between the natural features of an act and its normative features. I say instead that in such cases, the normative features of the act are grounded in its natural features together with a bridge law that holds of normative necessity.

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general applicability hold only because these more determinate principles are as they are. The more conventional moral theorist, by contrast, holds out for a small number of fundamental general principles that underlie the detailed fact-norm conditionals and any number of mid-level principles. She may even hold out for a single fundamental law—the “supreme principle of morality”—that grounds every other normative principle. No matter how this issue is resolved—and it may be resolved differently in different worlds—the pure moral principles are all normatively necessary, even though some are to be explained in terms of others, just as the truths of mathematics are all metaphysically necessary, even though some are to be explained in terms of others. Suppose there is a single pure moral principle that explains the rest. We can then field the question: Why is this the supreme principle of morality when it might just as well have been . . . act utilitarianism? It is a feature of the framework we have assumed that this question can have no answer. The supreme principle can’t be explained by reference to deeper principles, since by hypothesis there are none; and it can’t be explained by reference to contingent natural facts, since it would have been just as it is no matter how those facts had been. If ethical naturalism is true, it will be grounded in the natures of the properties and relations that figure in it, and pointing this out would yield a maximally satisfying answer to our question.²⁸ But if ethical naturalism is false, then the supreme principle of morality can only be a brute, inexplicable metaphysical contingency. If there are many such basic principles, the ethical facts will ground out in a plurality of brute contingencies.²⁹ This is a consequence of the picture of moral reality we have been discussing, not an objection to it. There is no denying that ethical naturalism would be a more satisfying package in this respect. When a series of why-questions terminates in an answer of the form “p because it lies in the nature of x that p,” the question is well and truly answered with no residual mystery. If ethical naturalism is false, however, it simply follows that some basic normative principles are inexplicable.³⁰ In this respect they would resemble the fundamental laws of nature as pictured by non-Humean views on which the laws are grounded, neither in the mosaic of particular fact nor in the essences of the properties that figure in them. Such views may be objectionable, but not because they posit laws that cannot be explained. There is no reason whatsoever to expect, or to insist, that every law or principle must be explicable. It cannot be stressed enough that this commitment to inexplicable moral principles is a feature of any non-naturalist view, not just mine. The conventional non-

²⁸ The explanation would be of the form: p because it lies in the nature of (say) moral permissibility that p, by analogy with: water is H₂O because water is essentially H₂O. Some people deny the explanatory force of these remarks and hear them instead as rejections of the demand for explanation. Either way, ethical naturalism would ensure that our question—Why is the supreme principle of morality as it is?—gets an answer if it needs one. ²⁹ For a discussion of brute contingencies in another context, see (Hale and Wright , ) and (Field ). ³⁰ I have ignored the remote possibility that there might be infinite descending grounding chains in ethics, where principle P is explained by Q, which is explained by R, etc., ad infinitum. To my knowledge, no one has ever seriously suggested that ethics might have this structure, but it would good to know whether this is a genuine (metaphysical) possibility.

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naturalist who embraces Supervenience and Non-Contingency (and so rejects the essentialist account of metaphysical modality) must agree that basic moral principles are inexplicable. Of course she will add that these principles are metaphysically necessary truths. But if challenged to explain why these basic principles include p but not q, she will fall silent. It is sometimes said that the demand for explanation arises only for contingent truths, and hence that conventional non-naturalist can legitimately shrug it off in a way that my sort of non-naturalist cannot. But this is just not so. There are genuine explanations in mathematics and metaphysics. We give them all the time. Even in ethics the conventional non-naturalist is presumably happy to entertain explanations of more specific moral principles in terms of more general ones. So even she will have to acknowledge that on her view, some truths of a sort that sometimes admit of explanation cannot be explained. This is just a fact of life for the non-naturalist, and no one should be embarrassed by it.

 Explaining the Normative Necessity of Moral Principles Given a fact-independent moral principle p, it is one thing to ask, “Why is p the case?” and another to ask, “Why is p fact-independent?” Questions of the first sort do not always have answers, as we’ve just seen. But questions of the second sort can always be answered, and seeing why will provide a clearer view of the metaethical picture I’ve been discussing.³¹ To say that p is fact-independent is to say that for any wholly nonnormative proposition q, if q had been the case, p would still have been the case. Given the standard account of counterfactuals, this amounts to saying that for any nonnormative proposition q, the q-worlds most similar to the actual world are always p-worlds. So we can reformulate our question as follows: Given some pure moral principle p, why are the nearest worlds in which the nonnormative facts are different always worlds in which p remains in place? To see the point of the reformulation, consider a schematic case. Here you are in the actual world with nonnormative facts D and a battery of moral principles M contemplating some action A. The actual moral principles prohibit A, and since you’re a good egg, you don’t do it. But if you had, the nonnormative facts would have been different in various ways. Suppose for simplicity that there is a single determinate way these facts would have been had you done A. Call it D*. Now there are D*-worlds in which the actual moral principles M are all in place, and in those worlds you act wrongly by doing A. But there are also D*-worlds in which those principles are different and your act is right. To say that the actual moral principles are factindependent is just to say that D*-worlds of the first sort are always closer to ³¹ This question is analogous to a familiar question in the philosophy of science: “Given that L is true, why is L a law? Why does this fact exhibit this particular species of necessity?” The corresponding question is rarely asked in ethics, presumably because we’ve been hazy about the distinctive species of necessity that distinguishes pure moral principles from mere contingent moral facts. Now that we have a clearer view of this distinctive species of necessity, the question is ripe for the asking.

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actuality than D*-worlds of the second sort; or in other words, that whenever we have three worlds: @: w: w:

D, M D*, M D*, M*

w₁ is more similar to @ than w₂ is. The challenge is to explain why this is so. This may seem trivial. After all, w₁ and w₂ are exactly alike in nonnormative respects, so they are equally similar to @ along that dimension. But w₁ also resembles @ in its basic moral principles, whereas w₂ departs gratuitously from @ in this respect. So isn’t it automatic that w₁ is more like @ than w₂ is? No, this is not automatic. For all we’ve said, there may be respects in which w₂ matches @ but w₁ does not. These points of similarity will not be wholly nonnormative, and they will not concern pure moral principles. Rather they will concern mixed normative facts. Here’s an example. Suppose that in the actual world @ with nonnormative description D and moral principles M, you perform cosmetic surgery on Sam with his consent, turning a frog into a prince. This act A is permissible, let’s suppose. (You’re a credentialed plastic surgeon.) So M and D together entail that A is permissible. Now ask how things would have been if you had performed the surgery without Sam’s consent. Suppose again, just for simplicity, that there us a single way the facts would have been if you had done this, D*. There is a D*-world—call it w₁—in which the actual moral principles M are still in place, and in w₁ your act is wrong. But there is also a D*-world, w₂, in which act utilitarianism is true. And it may well be that in that world A is permissible (because the world is happier overall with Sam’s appearance improved, his lack of consent to surgery notwithstanding). Now we are confident that if you had not secured consent, it would have been wrong to perform the surgery, or in other words, that w₁ resembles @ more closely than w₂ does. But it is not immediately obvious why this should be so. The two worlds are exactly alike in nonnormative respects. w₁ agrees with @ about the pure moral principles M, but disagrees with @ about the permissibility of A; w₂ agrees about the permissibility of A (and about many other particular moral facts), but disagrees about the moral principles M. So if we are simply counting points of similarity and difference, neither world is clearly closer than the other. And yet it seems quite clear that similarity with respect to moral principles counts for quite a lot, whereas similarity with respect to particular moral facts counts for very little in this case. To explain why the moral principles in M are fact-independent is to explain why this is so. The idea we need comes from papers by Boris Kment and Ryan Wasserman (Kment ; Wasserman ). Recall Fine’s famous “future similarity” objection to Lewis’s theory of counterfactuals (Lewis ; Fine ). Nixon did not press the button in , but if he had there would have been a nuclear conflagration. In the Lewisian framework, this amounts to the claim the possible worlds most similar to the actual world in which Nixon pushes the button are worlds in which the missiles fly and the post- future is radically different from the actual post- future. But of course there are worlds in which Nixon pushes the button and the signal dies

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in the wire, and in some of those worlds the future plays out almost exactly as it actually did. These latter worlds are intuitively much more similar to the actual world than any world in which the human race is wiped out by a nuclear holocaust in . So why don’t we accept the counterfactual, “If Nixon had pressed the button, the signal would have died in the wire”? Lewis’s response is that the theory of counterfactuals requires a standard of comparative similarity that is otherwise unfamiliar, one according to which near perfect conformity to the laws of nature matters a great deal, whereas approximate match of particular fact matters not at all (Lewis ). Lewis’s account can be rigged to give the right answers in an impressive range of cases, and it could be extended to cover our case by stipulating that match of moral laws counts more than match of particular moral fact. But crucially, Lewis does not explain why some respects of similarity matter more than others, and in the absence of such an account, this sort of ad hoc extension of the theory would be unilluminating. Kment and Wasserman provide a compelling principle at just this point. Let w* be a nearby world in which Nixon pushes the button and the signal dies in the wire. w* has much in common with the actual world. In both, for instance, you are reading this chapter now. But note: Although the worlds have this fact in common, the explanation of this fact is different in the two worlds. In the actual world, the fact that you are reading this chapter now has nothing to do with a fluky electronic glitch in Nixon’s bomb-launching apparatus, while in w* this glitch plays a crucial role in explaining this fact. Kment and Wasserman take this to be the telling point. As Kment puts the principle: (C) If some fact f obtains in both of two worlds, then this similarity contributes to the closeness between the two worlds if and only if f has the same explanation in the two worlds. (Kment : ) The future similarities between w* and actuality do not satisfy this condition, so they do not count. That is why the most similar world in which Nixon presses the button is not w*, despite these future similarities, but rather a world in which the missiles fly.³² With this in mind, return to Sam’s cosmetic surgery. We have three relevant worlds:

@ w w

Nonnormative facts

Pure principles

Particular normative facts

D D* D*

M M AU

A is permissible A is not permissible A is permissible

We want to know why w₁ is more similar to @ than is w₂, and the Kment/Wasserman theory tells us why. @ and w₁ agree about pure moral principles. This similarity counts because these principles have the same explanation in both worlds. The basic ³² Of course (C) by itself does not explain why worlds in which the missiles fly are closer. For that we need other principles that tell us how the similarities that do matter are to be weighted. See Kment (: ch. ) for a derivation of those principles.

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principles are not explained at all, and the derivative pure principles are grounded in the basic principles in the same way in both worlds. By contrast, even though @ and w₂ agree about the permissibility of the surgery, the explanation for this particular fact is very different in the two worlds. In @ it is explained by the fact that Sam consented to the surgery together with pertinent moral principles in M. In w* it is explained by the good consequences of the surgery together with the principle of utility. Since the facts have different explanations, this point of similarity counts for nothing. These considerations can be generalized to show that the nearest world at which the natural facts are different in some specified way is always a world that resembles the actual world in respect of pure normative principles. The argument depends on a substantive assumption about the structure of the normative domain, viz., that all normative facts are ultimately grounded in pure normative principles together with nonnormative facts.³³ Given this assumption, we can explain why pure normative principles are fact-independent, hence normatively necessary, by pointing to the deeper fact that pure normative principles are explanatorily independent of the natural facts. In fact it would be best to say that this is what renders the pure principles pure. It is then a substantive thesis that the pure moral principles are counterfactually independent of the nonnormative facts. The Kment/Wasserman idea then helps us to explain this substantive fact, and so to explain why the pure normative principles are, if not absolutely necessary, nonetheless necessary in a sense.

 Recapitulation In the first part of the chapter I argued that for all we know, the pure principles of morality are not metaphysically necessary. They would be metaphysically necessary if, but only if, the natures of the various normative properties and relations determined naturalistic necessary and sufficient conditions for their instantiation, and we simply do not know enough about these natures to know that this is so. We should therefore be prepared for the possibility that some pure normative principles are contingent, and hence for the possibility that the normative fails to supervene on the natural. It is nonetheless a datum that normative principles are necessary in a sense. It is no accident—no mere contingent truth—that an act with the natural features of the act you are performing now is a permissible act. Fine claims that these principles are necessary in a special sense, and we have been exploring that possibility. I have suggested that the normative necessity of moral principles consists in their fact-independence. The pure normative principles of are true and would still have been true no matter how the nonnormative facts had been. A fortiori, they would still ³³ This is highly plausible, but substantive. It would be false if it were possible for an individual act to be (say) wrong, not because it had some feature (qualitative or otherwise) that combines with a pure principle to ground its wrongness, but rather brutely. It is very hard to believe that this is possible. It seems to me to lie in the nature of normative properties and relations that facts of the form a is N, for normative N, are always grounded in nonnormative features together with general principles (Rosen c). If that is not so, the argument of this section will need rethinking.

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have been in place no matter what we had done or thought, no matter how society had been organized, and so on. That is a kind of necessity and (I claim) the only necessity that ethics needs. The challenge then was to explain why pure normative principles are necessary in this sense. The proposed explanation turns on the fact that pure moral principles are explanatorily independent of the natural facts. (We took this as a definition of purity.) Given this assumption, the Kment/Wasserman point about counterfactuals entails that the nearest worlds in which the natural facts are different are worlds in which the pure moral principles are as they are. We have thus derived the factindependence of pure moral principles from their purity, and so explained why such principles are necessary in their distinctive way. With this in mind, return to the question with which we began. If you are like most philosophers you used to find it perfectly obvious that the normative strongly supervenes on the natural—so obvious that no argument for this slightly arcane thesis seemed necessary. If the account I have sketched is cogent, you can now entertain an alternative to Strong Supervenience: the view that the normative supervenes on the natural as a matter of normative necessity. (This follows from the normative necessity of pure moral principles, together with the further assumption that all normative facts are ultimately grounded in pure principles plus nonnormative facts.) These views disagree about the modal classification of remote worlds— worlds in which act utilitarianism is true and the natural facts are just as they are, for example. Everyone agrees that such worlds are impossible in a sense. The open question concerns the interpretation of the modal word in this formulation. Having seen the alternative, are you confident that these remote worlds must be metaphysically impossible—ruled out by the natures of the moral properties and relations? If so, you see more deeply into these natures than I do.

 Challenges Needless to say, the metaethical picture I have been painting raises questions. If one grants that the pure moral principles are metaphysically contingent, one might want to know, for example, how much variation is (metaphysically) possible. Suppose we have a possible world like our world, in which act utilitarianism is false, and a world like w in which the nonnormative facts are just as they are but act utilitarianism is true. Does it follow—as it might, given a familiar principle of recombination—that there is a possible world with the same natural facts in which act utilitarianism is true until December ,  and false thereafter? Are there worlds that falsify weak Supervenience—worlds in which nonnormative duplicates differ in some normative respect?³⁴ Are there possible worlds just like our world in nonnormative respects in which the only impermissible act is the act you are performing now? Are there metaphysically worlds in which Caligula’s moral principles are true?

³⁴ For example, a world of two-way eternal recurrence in which the natural history of the world is repeated endlessly down to the last nonnormative detail, but in which an act utilitarian epoch is followed a Kantian epoch.

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The short answer is: no one knows. These bizarre worlds are impossible if and only if they are ruled out by the essences of normative properties and relations, and our insight into these essences is sadly limited. That said, it is consistent with my view that these normative features have essences that impose stringent constraints on which propositions involving them are possible. It presumably lies in the nature of permissibility that permissibility is a property of actions. That rules out worlds in which the sun is morally permissible. A more interesting possibility is that it lies in the nature of the normative properties and relations that particular facts involving them are always grounded in general principles. The idea would be that for each (basic) normative feature N it lies in the nature of N that if x is N, x is N in virtue of fact that x is D (where D is wholly nonnormative), together with the fact that for all x, whatever is D is N.³⁵ If we add that these general principles must be qualitative, not mentioning particular individuals, places, and times, that would rule out violations of weak Supervenience and certain patchwork worlds as well. Another potentially interesting sort of constraint is epistemic. It is sometimes said that normative principles must be knowable, or more modestly, that since it is of the essence of normative principles to guide the conduct of rational creatures, it lies in the nature of (say) moral permissibility that the principles of permissibility in w must be principles that rational creatures in w could reasonably believe. Consider the world that is naturalistically like this one, but in which the moral principles shifted imperceptibly at the turn of the last millennium. It seems fair to say that if there were such a shift, it would be radically undetectable: no one could ever have reason to believe that the moral principles exhibited this patchwork character. If that is so as a matter of metaphysical necessity, then the epistemic constraint on moral principles would brand this patchwork world impossible. It might even rule out vile worlds, like the world in which Caligula’s moral principles are true. But note: this constraint can only have these consequences if two conditions are satisfied: not only must it lie in the nature of (say) moral permissibility that the principles in which it figures must be rationally accessible; it must lie in the nature of epistemic rationality or some related notion that there is no possible world in which people have reason to believe these bizarre principles. And it is unclear (to say the least) whether such principles are plausibly built-in to the natures of the epistemic notions. That said, I should emphasize that it would not be a disaster, or even a serious objection to the view, if there were no strong essentialist constraints of this sort on moral principles. The bizarre worlds we have been discussing appear to be consistently describable. To call them metaphysically possible is just to say that this appearance would survive perfect knowledge of the essential truths. But so far as I can see, nothing hangs on how these worlds are ultimately classified. Here is an analogy. Focus on the physical probabilities of ordinary events. If we are nonHumeans about the laws of chance, the actual mosaic of particular fact is consistent with a wide range of assignments of probabilities to events, including bizarre assignments on which the half-life of U²³⁸ changed imperceptibly on December , ³⁵ Added : This simple proposal is clearly wrong. It may be that particular moral facts are always partly grounded in general laws; but those laws are not simple universally quantified conditionals or anything of the sort. See Rosen d for discussion.

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. It’s an open question about whether a bizarre law of this sort is ruled out by the essential truths. But what would hang on this determination? The metaphysical classification of worlds that include patchwork laws of chance would make no difference to physics or to more down-to-earth questions in the philosophy of physics. It may be interesting in its own right, but it’s not the sort of issue on which the credibility of a metaphysical theory of chance should turn. I take the same view about the classification of worlds with bizarre moral laws. At this point the reader will protest that the metaphysical classification of these bizarre worlds matters for moral epistemology. The various worlds we have been imagining amount to skeptical scenarios: worlds in which we form our moral beliefs just as we actually do (since the facts of belief-formation are natural facts, and these are held fixed) and yet our methods lead us wildly, massively astray. It’s not news that our moral methods are fallible. But surely it should have some impact on our confidence to be told that there are genuinely possible worlds—worlds we have never considered and never paused to exclude—in which we diligently employ our best methods and yet come nowhere near the truth, or so the objection runs. Now there are versions of this challenge that we can easily shrug off. The mere metaphysical possibility of undetectable error should not undermine our confidence and does not prelude knowledge. That is the great lesson of twentieth-century antiskeptical epistemology. The countermoral worlds we’ve been discussing are to moral epistemology what evil demon worlds and the like are to the epistemology of perception. Their existence shows that our methods are fallible and nothing more. There is, however, a version of this challenge that is harder to shrug off. If the metaphysical picture I have been advancing is correct, our beliefs about pure moral principles are not sensitive to the pure moral facts. Suppose P is a true moral principle which we have come to believe as a result of the most scrupulous philosophical enquiry. Now ask what we would have believed had P been false. My view entails that there are worlds of this sort in which the nonnormative facts, including the facts about our beliefs, are just as they are, and these worlds are closer to the actual world than worlds in which P is false and the natural facts are also different. (This follows from our framework on the assumption hat the normative facts play no role in explaining the natural facts.) And that means that our beliefs in pure moral principles would have been just as they are even if the pure moral facts had been quite different. So far this is just an observation. It amounts to an objection only if it suggests that our beliefs in moral principles must be unjustified, or fail as knowledge, if they are insensitive in this way. This is an enormous question that we cannot address here fully, so I’ll be brief. First, even if it is plausible on reflection that there is a sensitivity constraint on knowledge or justified belief in many areas, it is a step to hold that such a constraint applies across the board. A perfectly general sensitivity constraint entails (given the metaethical view I have been defending) that even when someone accepts a true moral principle for what we normally regard as good reasons, he does not know (or justifiably believe) that principle. Confronted with such a case, we will face a choice: accept the across-the-board sensitivity requirement and deny that the case is a case of knowledge, or treat the case as a counterexample to the across-the-board sensitivity

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requirement. And to put the point as concessively as possible, it is at least unclear which of these theoretical options is to be preferred. Philosophical theories of knowledge must be liable to refutation by counterexample. If some theory tells me that I can’t know that it is wrong to cause intense pain in a non-consenting victim for fun (because my belief in this fact is not sensitive to the fact that makes it true), I am inclined to say: so much the worse for your philosophical theory of knowledge, since I certainly do know/justifiably believe this moral fact.³⁶ Second, the sensitivity challenge does not just arise for my view, but also for what I take to be its closest competitor, the sort of non-reductive normative realism that embraces Supervenience and non-contingency (and so rejects the essentialist account of metaphysical modality). These familiar views hold that pure normative principles are metaphysically necessary. But we can still apply a nontrivial sensitivity test in this domain, provided we allow for nontrivial counterfactuals with impossible antecedents. If we do, then it is highly plausible that the non-reductive realist will have to say that had the pure moral principles been different (per impossibile), our beliefs would have been just as they are. So either the problem sinks both sorts of realism or it sinks neither. Moreover, the verdict that it sinks neither will be more secure in this case, since the cost of maintaining an across-the-board sensitivity requirement will be quite high. The truths of pure mathematics are presumably metaphysically necessary truths, but we can coherently suppose many of them to be false by considering worlds in which there are no mathematical objects of any sort, worlds in which all sets are finite, and so on. Many of our mathematical beliefs will then fail the sensitivity test: if there had been no numbers (or infinite sets), these beliefs would have been just as they are. Does this show that our ordinary mathematical beliefs are unjustified, or that they don’t amount to knowledge? Perhaps. But it is at least equally plausible to say that it shows instead that there is no across-the-board sensitivity requirement in epistemology. Of course this companions-in-guilt gambit can be blocked by insisting that counterfactuals with metaphysically impossible antecedents are trivially true. But the fact that the test can have no application given this assumption suggests that the sensitivity requirement itself is fishy. There is a good epistemological question about how we can know whether mathematical objects exist, or whether God exists, or whether some putative moral principle is true. But it is hard to see how our epistemic tasks in these areas becomes easier when the fact that interests us is non-contingent. So long as the truth in question is not a conceptual truth, we still face the challenge of determining which of two (or more) epistemic possibilities is actual on the basis of whatever indirect evidence we can muster. There is no a priori guarantee that our method will succeed. So even when we have employed it correctly we can always wonder whether the actual world and all of the other possible worlds are worlds in which our method leads to false belief. It’s hard for me to see how it can be comforting to be told that if we have gotten it right, our belief is (trivially) sensitive to the facts. It remains the case that our method is not tracking the facts:

³⁶ This is analogous to the response to epistemological challenges to Platonism based on the causal theory of knowledge in Burgess and Rosen ().

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 

In these non-empirical domains we do not believe what we believe because the facts are as they are. And my strong inclination is to say that either this or something like it is a general constraint on knowledge, in which case every form of nonreductive realism about causally inert features of reality is undermined, or it is not, in which case there is no special epistemological problem for any such view, including the view of the normative facts we have been discussing.

References Blackburn, Simon. . Supervenience Revisited. In Essays in Quasi-Realism. Oxford University Press. Burgess, John, and Gideon Rosen. . A Subject with No Object. Oxford University Press. Dancy, Jonathan. . Ethics without Principles. Oxford University Press. Enoch, David. . An Outline of an Argument for Robust Metanormative Realism. Oxford Studies in Metaethics : –. Field, Hartry. . The Conceptual Contingency of Mathematical Objects. Mind : –. Fine, Kit. . Critical Notice. Mind : –. Fine, Kit. a. Essence and Modality. Philosophical Perspectives : –. Fine, Kit. b. Senses of Essence. In Walter Sinnott-Armstrong, Diana Raffman, and Nicholas Asher (eds.), Modality, Morality and Belief. Essays in Honor of Ruth Barcan Marcus. Cambridge University Press. Fine, Kit. a. The Logic of Essence. Journal of Philosophical Logic (): –. Fine, Kit. b. Ontological Dependence. Proceedings of the Aristotelian Society : –. Fine, Kit. . Semantics for the Logic of Essence. Journal of Philosophical Logic (): –. Fine, Kit. . The Varieties of Necessity. In T. Gendler and J. Hawthorne, eds. Conceivability and Possibility, –. Oxford University Press. Fine, Kit. . Guide to Ground. In Fabrice Correia and Benjamin Schnieder, eds. Metaphysical Grounding. Cambridge University Press. Hale, Bob, and Crispin Wright. . A Reductio ad Surdum? Field on the Contingency of Mathematical Objects. Mind : –. Hattiangadi, Anandi. . Moral Supervenience. Canadian Journal of Philosophy (–): –. Kment, Boris. . Counterfactuals and Explanation. Mind : –. Kment, Boris. . Modality and Metaphysial Explanation. Oxford University Press. Lange, Marc. . What Would Normative Necessity Be? Journal of Philosophy (): –. Leslie, Sarah-Jane. . Essence, Plentitude and Paradox. Philosophical Perspectives : –. Lewis, David. . Counterfactual Dependence and Time’s Arrow. Noûs  (): –. Lewis, David K. . Counterfactuals. nd ed. Wiley-Blackwell. McPherson, Tristram. . Ethical Non-Naturalism and the Metaphysics of Supervenience. Oxford Studies in Metaethics : –. Moore, G. E. . The Conception of Intrinsic Value. In Moore, Philosophical Studies. Routledge and Kegan Paul. Nolan, Daniel. . Impossible Worlds: A Modest Approach. Notre Dame Journal of Formal Logic : –.

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Parfit, Derek. . On What Matters. Oxford University Press. Peacocke, Christopher. . Being Known. Oxford University Press. Rosen, Gideon. . The Limits of Contingency. In F. MacBride, ed. Identity and Modality. Oxford University Press. Rosen, Gideon. . Metaphysical Dependence: Grounding and Reduction. In Bob Hale and Aviv Hoffman, eds. Modality: Metaphysics, Logic and Epistemology. Oxford University Press. Rosen, Gideon. . Real Definition. Analytic Philosophy (): –. Rosen, Gideon. a. Metaphysical Relations in Metaethics, in T. McPherson and D. Plunkett, eds. The Routledge Handbook of Metaethics. Routledge. Rosen, Gideon. b. Scanlon’s Modal Metaphysics. Canadian Journal of Philosophy (): –. Rosen, Gideon. c. Ground by Law. Philosophical Issues (): –. Rosen, Gideon. d. What is a Moral Law? Oxford Studies in Metaethics : –. Salmon, Nathan. . The Logic of What Might Have Been. Philosophical Review (): –. Schroeder, Mark. . Slaves of the Passions. Oxford University Press. Wasserman, Ryan. . The Future Similarity Objection Revisited. Synthese (): –. Wright, Crispin, and Bob Hale. . Nominalism and the Contingency of Abstract Objects. Journal of Philosophy (): –.

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 The Problem of de re Modality Bob Hale

 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 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—for example ◇Fa, or 9x□ðx ¼ 172 Þ, as contrasted with □9xðx ¼ 172 Þ. 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 “ is necessarily prime”) or left unspecified (as in “Some number is necessarily the square of ”).

¹ My title is intended to remind readers of one of the two papers by Kit Fine on this topic (Fine  and ), rereading which led me to write the present chapter. I have 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 s. 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 chapter to this volume in his honor. Bob Hale, The Problem of de re Modality In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Bob Hale. DOI: 10.1093/oso/9780199652624.003.0012

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 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 de dicto is relatively unproblematic.² 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 : )

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, □A(t), will be a nontrivial de re necessity only if what it says holds by necessity of its object, t, is not, as a matter of necessity, true of all objects whatever—that is, only if it is not the

² This is true even of the most prominent and influential critic of modality de re, W. V. Quine.

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case that □8xA(x)—and similarly for a true existentially quantified proposition, 9x□A(x).³ According to the broadly linguistic conception, □A(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 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, “□(t is red ! t is red)” and “□(t is red ! t is colored)” express de re de re necessities according to the linguistic conception—but merely trivial ones, because □8x(x is red ! x is red) and □8x(x is red ! x is colored)). 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 nontrivial 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.⁴ If, as I believe, we ³ There is an analogous conception of what it is for a property ϕ to be nontrivially essential—this is so if 9x□ϕðxÞ∧9x:□ϕðxÞ. Cf. Marcus 1971, p. 62 ⁴ Plantinga  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. ). In his view (pp. –), 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 if 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  has the property of being prime essentially, and we assume that “” is a proper name of , and “being prime” of the property of being prime, then—on Plantinga’s proposal— is essentially prime if and only if  is prime and the proposition expressed by “ 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 skeptic 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, nontrivial de re claims of the sort advanced by essentialists—admit of no such saving reconstruction. There is, for example, no prospect of reinterpreting 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

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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 remain? Is there any further, independently compelling, reason to doubt that de re modalities are intelligible?

 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:⁵ . If 9x□A(x) or 8x□A(x) is to be meaningful, it must make sense to say of an object that it satisfies the open sentence □A(x). . 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. . Co-referential terms are not interchangeable salva veritate in □A(x). ∴ . It makes no sense to say of an object that it satisfies □A(x). ∴ . Neither 9x□A(x) nor 8x□A(x) can be meaningful. This argument is plainly valid. Premise  is indisputable, provided that—as Quine intends—the quantifiers are interpreted as objectual (Quine has separate objections to substitutional quantification). Premise  is supported by well-known examples, such as the alleged failure of the substitution of “the number of solar planets” for “” in “□( > )” to preserve truth, in spite of the identity of the number of solar planets with . Clearly the crucial question is why we should accept premise . We shall discuss that shortly. First I should defend my claim that this is Quine’s central and basic argument.

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 skeptic, 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. ⁵ This formulation closely follows Fine :  and Fine : . But see subsequent discussion.

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 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, 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: . If 9x□A(x) or 8x□A(x) is to be meaningful, it must make sense to say of an object that it satisfies the open sentence □A(x). . It makes sense to say of an object that it satisfies the open sentence □A(x) only if it makes sense to say of an object that it necessarily satisfies the non-modal open sentence A(x). . 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. ∴ . It makes no sense to say of an object that it satisfies □A(x). ∴ . Neither 9x□A(x) nor 8x□A(x) can be meaningful The two arguments are similar, but they diverge over premises  and . 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.⁶

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.

 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 ⁶ Fine : . 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 : ).

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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 (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 8xA(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 FðxÞ⊃FðxÞ, 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 in which a1 ¼ a2 may be correctly said to logically satisfy “x ¼ y,” even though the universal closure of this open sentence, 8x8y x ¼ 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 nontrivial 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,

⁸ Fine : –.

⁹ Fine : .

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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 Phosphorus, or being constituted predominantly of H₂O molecules if it is water, and so on. 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  necessarily has the property of being prime that it is necessarily true that  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 skeptic 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.

 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 ¹⁰ One might also ask why we should accept premise  in the metaphysical argument. Why accept that it makes sense to speak of an object satisfying □A(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 □A(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 □A(x) at a given possibility w iff it satisfies A(x) at every possibility w 0 (or perhaps every w 0 accessible from w).

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

necessary fulfilment of a condition, rather than fulfilment simpliciter. There is no obvious reason to deny that, say, the number of solar planets (“in and of itself”) fulfils the condition x > . 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 > ,” for example, is true, it is not necessarily true, that is “□(the number of solar planets > )” is not true, even though “□(>)” and “ = 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—that is 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  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—that is 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 “” and “the number of solar planets” in the sentence: ()

□(>)

produces a falsehood, it makes no sense to say of the object to which these terms refer that it satisfies “□(x>).” That is, we cannot understand () as expressing a necessity de re—that is as saying, of the object  that it is necessarily greater than . At most, only a de dicto reading of the sentence is intelligible. “,” as it occurs in () 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 “,” is itself non-referential, 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, “9x□ðx >)” makes no sense—for it asserts, in effect, that there is an object which satisfies the open sentence “□(x>),” and that is something which—if Quine is right—no object can do.

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 

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: (a) 2:2 ¼ 4

(b) Eve’s elder son was Cain

2¼1þ1 ∴ 2:1 þ 1 ¼ 4

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 ¼ 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 coreferential 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 “” as it occurs in () is replaced by “the number of solar planets,” the result: () □(the number of solar planets > ) is arguably ambiguous between a reading on which it implies “□(There are more than  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 .” One may then claim that the step from the apparent failure of substitution to the conclusion that “” does not occur purely referentially in () depends on tacitly resolving the ambiguity in favor of the first reading of (). But a parallel insistence

¹¹ Under what I believe are standard arithmetical conventions, we read “.+” 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”).

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

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 descriptions, the ambiguity of sentences such as () 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 eyewitness, 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: ()

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

¹² Cf. Smullyan ; Fitch . Quine (: , fn.) claims that Smullyan requires a modified version 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 *.. The point is made by Smullyan himself—see Smullyan : . 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 () comes from () by substitution of a co-referential singular term is in any case entirely illusory.

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 

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 (), the description, and so the information encoded in it, may or may not lie within the scope of the modal operator. If it does, () asserts, falsely, that as a matter of necessity, some single number both numbers the solar planets and exceeds . If it does not, () asserts, concerning the number which, as a matter of presumably contingent fact, numbers the solar planets, that it— that very number—necessarily exceeds . 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 () 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 () 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 “” in () 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 ƒ(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 √-, 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 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. ¹⁵ We assume here that 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

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

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 (). On one reading, 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 ()’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 coreferential 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 ¼ 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 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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 

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 (), 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 “□(x > ),” thereby begging the very question at issue. But it seems to me that this objection simply mislocates 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 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 substitution-failure the argument adduces as proof that open sentences like “□(x > )” cannot be true (or false) of objects simply, but only relative to ways of specifying them, can be explained without drawing that conclusion.¹⁶

References Fine, Kit. . “The Problem of De Re Modality.” In Fine : –. Fine, Kit. . “Quine on Quantifying In.” In Fine : –. Fine, Kit. . Modality and Tense: Philosophical Papers. Oxford: Oxford University Press. Fitch, Frederic B. . “The Problem of the Morning Star and the Evening Star.” Philosophy of Science : –. Linski, Leonard. . Reference and Modality. Oxford: Oxford University Press. Marcus, Ruth Barcan. . “Essential Attribution.” The Journal of Philosophy : –; page references to the reprint in Marcus : –. Marcus, Ruth Barcan. . Modalities. New York and Oxford: Oxford University Press. Plantinga, Alvin. . The Nature of Necessity. Oxford: Oxford University Press. Quine, W. V. . “Reference and Modality.” Essay VIII, in Quine : –. Quine, W. V. . From a Logical Point of View. Cambridge, MA: Harvard University Press,  and ; reprinted by Harper & Row, . Smullyan, Arthur F. . “Modality and Description.” The Journal of Symbolic Logic (): –; page references to the reprint in Linsky : –.

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 , for helpful discussion of an earlier version of this chapter, 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.

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 Can Metaphysical Modality Be Based on Essence? Penelope Mackie

 Introduction In his hugely influential paper “Essence and Modality” (), Kit Fine argued that the then orthodox view that essence can be understood in terms of metaphysical modality is fundamentally flawed. He proposed, in its place, the view that all metaphysical modality has its source in the essences or natures of things, where the notion of a thing’s essence or nature can be understood in terms of a broadly Aristotelian notion of real definition. It is fair to say that if Fine is correct, this has revolutionary consequences for the theory of metaphysical modality. And if Fine is correct, his theory seems to promise an answer to the first of the two fundamental questions about modality famously proposed by Michael Dummett when he said, “the philosophical problem of necessity is twofold: what is its source, and how do we recognise it?” (: ). In spite of the attention that it has received, however, one aspect of Fine’s revolutionary theory seems to me to have been surprisingly neglected. The theory appears to require that the relevant conception of real definition can itself be isolated without appeal to metaphysical modality. And I do not see how this requirement can be met. Hence I am genuinely puzzled about how an “essence-based” theory of metaphysical modality is possible. In this chapter, I explain my reasons for skepticism about this issue. I then briefly consider the implications of my argument for the relation between essence and metaphysical modality.

 Fine and the Priority of Essence Fine () rejects what he calls the “modal account” of essence. Until Fine’s attack on it, this was the standard modern account of the notions of essence and essential property. This modal account defines essence and essential property in terms of de re modality. It equates the notion of an essential property with the notion of a (metaphysically) necessary property. The modal account can be characterized by the following principles: Penelope Mackie, Can Metaphysical Modality Be Based on Essence? In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Penelope Mackie. DOI: 10.1093/oso/9780199652624.003.0013

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

 

(M) Being (an) F is an essential property of x =df being (an) F is a necessary property of x. (M) The essence of x is the set or sum of x’s essential properties, as defined by (M).¹ “Necessary property” as it occurs in (M) is to be understood as follows: being (an) F is a necessary property of x if and only if x could not have existed without being (an) F.² In addition, the “could not” and “necessary” here are intended to represent metaphysical modality.³ Fine famously argued that the modal account’s equation of essential properties with necessary properties is untenable. Not all necessary properties of a thing should be classified as essential properties of the thing, in a sense of essential property that is of interest to metaphysics. Fine’s proposed counterexamples to the modal account included Socrates’ property of being distinct from the Eiffel Tower, Socrates’ being such that there are infinitely many prime numbers, and Socrates’ belonging to the set singletonSocrates: the set that has Socrates as its sole member. In all these cases, there is a property that Socrates could not have existed without having—a property that he has in all possible worlds in which he exists—and yet which, according to Fine, is not properly classed as an essential property, because it does not pertain to the essence or nature or identity of Socrates. Fine made a further move, which might be labelled “the priority of essence,” or “the ‘essence-first’ approach”: . . . far from viewing essence as a special case of metaphysical necessity, we should view metaphysical necessity as a special case of essence. . . . [E]ach class of objects . . . will give rise to its own domain of necessary truths, the truths which flow from the nature of the objects in question. The metaphysically necessary truths can then be identified with the propositions which are true in virtue of the nature of all objects whatever. (Fine : ; my italics)⁴,⁵

To take stock: Fine rejects the modal account of essence. But he does not reject it simply on the grounds that it “over-generates essential properties.”⁶ He also thinks ¹ Cf. Lowe : . ² Cf. Fine : . For simplicity, I ignore the alternative version of the modal account that defines “necessary property” in an “unconditional” way: i.e., that says that a necessary property of an object is one that it “possesses in all possible worlds” simpliciter, rather than “in all possible worlds in which it exists” (see Fine : –). ³ Throughout this chapter, when I use the terms “necessity,” “possibility,” etc. I shall mean metaphysical necessity, possibility, etc., unless otherwise indicated. ⁴ “ . . . [D]ifferent essentially induced truths [roughly: truths that are based on essences] may have their source in the identities of different objects—Socrates being a man having its source in the identity of Socrates,  being a number having its source in the identity of  . . . [It] is true in virtue of the identity of singleton Socrates [that Socrates is a member of singleton Socrates], but not of the identity of Socrates” (Fine : ). ⁵ The implicit equation, in the passage I have quoted and elsewhere in his article, of “essence” with “nature” is explicitly embraced in later work by Fine: “In general, I shall use the terms ‘essence’ and ‘identity’ (and sometimes ‘nature’ as well) to convey the same underlying idea” (b: , n.). See also Fine a: passim. ⁶ As Lowe has put it (: ).

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       ?



that it goes wrong more fundamentally, in attempting to explain the notion of essence in terms of that of metaphysical modality. This gets things the wrong way round, according to Fine. Before proceeding, it is important to note the following. As we have seen, Fine holds that not all necessary properties are essential properties. He thus rejects the following implication, (M), of the modal account: (M) If being (an) F is a necessary property of x, then being (an) F is an essential property of x. Crucially, though Fine accepts what I shall call the Necessity Principle (NP) (the converse of (M)), to the effect that any essential property is also a necessary property: (NP) If being (an) F is an essential property of x, then being (an) F is a necessary property of x. For example, he holds that, if it is part of the essence of Socrates (in his sense of essence) to be a man, then Socrates could not have existed without being a man. Fine explicitly endorses (NP) when he says: I accept that if an object essentially has a certain property then it is necessary that it has the property (or has the property if it exist[s]). (: )⁷

Moreover, it’s not just that Fine thinks that the Necessity Principle (NP) is something that can be salvaged from the wreckage of the modal account. Retaining the principle (NP) is obviously crucial to his project of grounding metaphysical modality in essence. For example, if it is a (metaphysically) necessary truth that Socrates is a man (or at least that he is a man if he exists), then, according to Fine, this is a case of a necessity that is grounded in essence. This particular metaphysically necessary truth would be one that is grounded in the essence of Socrates. Clearly, though, the principle (NP) is at work here. Why is it the case (if it is) that it is a necessary truth that Socrates is a man (if he exists)—or, in other words, that Socrates is necessarily a man (if he exists)? According to Fine, assuming that this is so, it is because Socrates is essentially a man—that is, because it is part of his essence to be a man. Of course, the example of Socrates’ being a man represents only the simple case. It is part of the essence-based account that not every necessary truth about Socrates is grounded in Socrates’ essence—and hence that not every necessary truth about Socrates is grounded in Socrates’ essential properties. That is the main point of some of Fine’s proposed counterexamples to the modal account. It is, however, part of the essence-based theory of metaphysical modality that every necessary truth about Socrates is grounded in the essences of some entities, and hence that the properties that those entities have “in virtue of ” their essences—their essential ⁷ Another example of Fine’s endorsement of (NP): “Certainly, there is a connection between the two concepts [essence and necessity]. For any essentialist attribution will give rise to a necessary truth; if certain objects are essentially related then it is necessarily true that the objects are so related (or necessarily true given that the objects exist)” (: ).

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

 

properties, on this conception of essential properties—must be necessary properties. So the point still stands that without (NP), it seems that it would be impossible to base metaphysical modality on essence, since the appropriate connection between essence and modality would be lacking.

 A Rival “Non-modal” Account of Essence? It is useful to distinguish the two aspects of Fine’s claims about the priority of essence over metaphysical modality (his “essence-first” approach). On the one hand, metaphysical modality is to be explained in terms of essence. On the other hand, essence is not to be explained in terms of metaphysical modality. Given this second point, it follows that if essence is to be explained at all, it must be given what, for obvious reasons, I shall call a “non-modal” explanation. E. J. Lowe, who endorses both Fine’s rejection of the modal account and his advocacy of the priority of essence,⁸ is explicit in undertaking a commitment to the provision of a non-modal account of essence: [If we are to] try to explicate the notions of metaphysical necessity and possibility in terms of the notion of essence, rather than vice versa, . . . we need, of course, to provide a perspicuous account of the notion of essence which does not seek to explicate it in modal terms. (Lowe : )⁹

But what about Fine himself ? Fine has been credited with the view that essence is a primitive notion.¹⁰ But in that case, one might suppose that he thinks that no account of essence can be given at all, and that, a fortiori, no non-modal account can be given. Not so. Fine explicitly maintains that, even if no analysis of the notion of essence can be given, some kind of explanation or clarification can be provided of the notion (: , in the passage to be quoted in § below).¹¹ To summarize: if, as the “essence-first” approach demands, metaphysical modality is to be explained in terms of essence, rather than essence in terms of metaphysical modality, any explanation of the notion of essence must be a non-modal one, in the ⁸ “I am persuaded by Fine’s objections to the modal account of essence and accept the lesson that he draws: that it is preferable to try to explicate the notions of metaphysical necessity and possibility in terms of the notion of essence, rather than vice versa” (Lowe : ). Like Fine, Lowe also explicitly embraces the Necessity Principle (NP): “any essential truth is ipso facto a metaphysically necessary truth, although not vice versa . . . —understanding an essential truth to be a truth concerning the essence of some entity” (Lowe : ; italics in the original). ⁹ Lowe’s commitment to providing a non-modal account of essence is also evident in an earlier paper, where he objects to the modal account by saying that it “puts the cart before the horse” when it attempts “to characterize essence in terms of antecedently assumed notions of possibility and necessity” (Lowe : ). ¹⁰ E.g. Correia : ; Wildman : –, ; Hale : , n.. I have not been able to find, in Fine’s own writings, an explicit claim that essence is to be taken as a primitive notion. He does, however, treat the locution “true in virtue of the nature of ” (or sometimes “true in virtue of the identity of”) as primitive in some of his writings (: ; a: ). ¹¹ See also the following passage from a slightly later paper: “A property of an object is essential if it must have the property to be what it is . . . . I doubt whether there exists any explanation of the notion [of an essential property] in fundamentally different terms. But this is not to deny the possibility of further clarification” (Fine b: ).

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

sense of an explanation that makes no appeal to metaphysical modality. This leaves two possibilities: that no explanation of the notion of essence is to be given, or that an explanation is to be given, but in non-modal terms. Fine rejects the “no explanation” option—an option which in any case seems an implausible one. Hence he is committed to giving a non-modal account of essence (although he is not committed to giving a non-modal account that provides an analysis of the notion).

 Essence and Real Definition How, then, could a non-modal account of essence be given? According to Fine, the key to the alternative, non-modal account of essence is the notion of a real definition—a definition that says what a thing is, as opposed to a verbal definition (a definition of a word that says what the word means). Having claimed that “the notion of essence which is of central importance to the metaphysics of identity is not to be understood in modal terms,” he continues: I shall . . . argue that the traditional assimilation of essence to definition is better suited to the task of explaining what essence is [than is the contemporary assimilation of essence to modality]. It may not provide us with an analysis of the concept, but it does provide us with a good model of how the concept works . . . . [M]y overall position . . . sees real definition rather than de re modality as central to our understanding of the concept [of essence]. (: ; my italics)

This “traditional assimilation of essence to definition” is characterized earlier in Fine’s paper as follows: . . . essence [has] been conceived on the model of definition . . . just as we may define a word, or say what it means, so we may define an object, or say what it is. The concept of essence has then [been] taken to reside in the “real ” or objectual cases of definition, as opposed to the “nominal” or verbal cases. (: ; my italics)

In a slightly later paper, Fine says: . . . in my paper “Essence and Modality,” I argued against the modal construal of essence and proposed . . . a conception of essence as a form of definition. (a: ; my italics)¹²

In this appeal to real definition Fine is, again, followed by Lowe. According to Lowe, in answering the challenge to provide a non-modal account of essence, [a] key notion . . . is that of a real definition . . . A real definition of an entity, E, is to be understood as a proposition which tells us, in the most perspicuous fashion, what E is—or, more broadly, since we do not want to restrict ourselves solely to the essences of actually existing things, what E is or would be. This is perfectly in line with the original Aristotelian understanding of the notion of essence, for the Latin-based word “essence” is just the standard

¹² The passage continues: “Under this alternative conception, each item would give rise to its own sphere of truths, the truths that had their source in the identity of the item in question. Thus the proper expression of the claim that x essentially ɸ’s would not be that it is necessary that x ɸ’s if it exists, for some amorphous notion of necessity, but that it is true in virtue of the identity of x that it ɸ’s, or that it ɸ’s if x exists” (Fine a: ).

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

 

translation of a phrase of Aristotle’s which is more literally translated into English as “the what it is to be” or “the what it would be to be.” (: ; italics in the original)

Elsewhere in the same paper Lowe speaks of an appeal to: . . . essences, understood in a neo-Aristotelian fashion, according to which essences are expressed by “real definitions.” (: )

But now I am puzzled. I do not see how the notion of real definition to which Fine and Lowe appeal can do the job that is required of it in an essence-based theory of metaphysical modality. I do not see how it is possible to isolate a notion of real definition that will generate a conception of essence that delivers the result that essential properties are necessary properties (as the Necessity Principle [NP] demands), unless we appeal to modal notions—in particular, to the notion of metaphysical necessity—in explaining what the relevant notion of real definition is. It looks as if the account of essence in terms of real definition is intended to deliver a modal rabbit out of a non-modal hat. And I do not see how this can be done.

 Real Definition without Modal Assumptions To clarify what is at issue, I introduce the term of art “D-essence,” short for “Real Definition essence.” I characterize it as follows: “D-essence” (my term): Essence as understood (a) via the notion of real definition, but (b) without appeal to modal notions (or, at any rate, without appeal to notions of metaphysical modality). On the basis of the remarks about the connection between essence and real definition provided by Fine and Lowe, I suggest the following as statements that should be acceptable to the proponent of the conception of essence for which I am using the label “D-essence”: • To specify the D-essence of a thing is to give a real definition of the thing. • A real definition of a thing specifies what that thing is, or what it is to be that thing. • So: the D-essence of a thing is what that thing is, or what it is to be that thing. • A thing’s D-essence can be described as its “nature.”¹³ To this list it is plausible to add: • A thing’s D-essence (at least in the case of the D-essences of kinds) is what the thing fundamentally is.¹⁴ ¹³ For the association of “essence” with “nature,” see Fine : passim; also the quotation from Fine cited in note  above. ¹⁴ Plausible, given that the project involves attempting to isolate a notion of “what a thing is” that has its roots in Aristotle’s theory of substance and essence. The reference to the fundamental is also in accord with the following passage from Lowe: “what Aristotle and the Scholastics understand by an account of the essence of some kind of substance is an account of what that kind of substance fundamentally is” (: ; italics in the original).

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       ?



Corresponding to the notion of D-essence, we shall need the notion of a “D-essential property.” I characterize this (admittedly roughly), as follows: D-essential property: (Roughly:) a D-essential property of a thing is a property that is part of its D-essence, or which it has in virtue of (which flows from) its D-essence.¹⁵

The features I have attributed to D-essence in the bullet-pointed list are intended both to be relatively uncontroversial and to give us some help in understanding the notion of D-essence and the related notion of a D-essential property. Of course, the list includes expressions that obviously cry out for clarification: what a thing is, what it is to be a thing, a thing’s nature, and what the thing fundamentally is. Crucially, however, since D-essence is, by definition, a “non-modal” conception of essence— that is, a conception of essence that is to be explained in non-modal terms¹⁶—it must not be the case that we have to appeal to notions of metaphysical modality in interpreting them. Nevertheless, some legitimate help is at hand, not only via the Aristotelian origins of the concept of real definition, but also from some of Fine’s remarks indicating how he thinks the “what is it?” question that is relevant to the topic of essence should be understood, including his examples of properties that do not count as part of a thing’s essence or nature: . . . one of the central concerns of metaphysics is with the identity of things, with what they are. But the metaphysician is not interested in every property of the objects under consideration. In asking “What is a person?,” for example, he does not want to be told that every person has a deep desire to be loved, even if this is in fact the case . . . . (Fine : ; the italics here and in the quoted passages that follow are mine) It is no part of the essence of Socrates to belong to the singleton [i.e., singleton-Socrates] . . . . There is nothing in the nature of a person . . . which demands that he belongs to this or that set or which demands, given that the person exists, that there even be any sets. (: –) . . . it is not essential to Socrates that he be distinct from the [Eiffel] Tower; for there is nothing in his nature which connects him in any special way to it. (: ) . . . it is no part of Socrates’ essence that there be infinitely many prime numbers . . . (: ) . . . can we not recognize a sense of nature, or of “what an object is,” according to which it lies in the nature of [singleton-Socrates] to have Socrates as a member even though it does not lie in the nature of Socrates to belong to the singleton? (: )

¹⁵ This characterization of “D-essential property” is rather rough, since the expression “has in virtue of its essence” needs clarification. If it were taken to imply that every property that an object has that is a logical consequence of a property that is part of its D-essence is a property that it has in virtue of its essence, then the resulting notion of D-essential property would be likely to be too undiscriminating, for reasons akin to the reasons that underlie Fine’s original objections to the modal account of essence. For example, since it is a logical consequence of Socrates’ being a man that he is either a man or a mountain, the latter, disjunctive property will count if the former does, on the “logical consequence” interpretation of “has in virtue of.” Similarly, Socrates’ being such that either he is a man or  +  =  would seem also to count, on the “logical consequence” interpretation. Important though this issue may be to a precise characterization of a theory of D-essential properties, it will not matter for my purposes. I shall work with an intuitive notion of “has in virtue of its D-essence” that is intended (somehow) to exclude such “unwanted” properties. ¹⁶ Remember that by “modal” I am referring to metaphysical modality, unless otherwise indicated (see § of this chapter).

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I shall assume, in the argument that follows, that the notion of a D-essence is a tolerably clear one, even though I have not attempted to give an analysis of the notion.¹⁷ And this is all that I need for my purposes.¹⁸

 Is the (Non-modal) D-essence Account an Adequate Account of Essence? I shall now argue that there are conceptions of a thing’s “nature” that appear to satisfy the criteria for being a D-essence (for saying “what a thing is” in the D-essence sense) and yet do not entail that the properties that a thing has in virtue of its D-essence—its D-essential properties—are necessary properties of the thing. In other words, I shall argue that the satisfaction of the criteria for being a D-essence does not appear to license the principle (NPD), which is the Necessity Principle (NP) applied to D-essence: (NPD) If being (an) F is a D-essential property of x, then being (an) F is a necessary property of x (i.e., x could not have existed without being (an) F). However, if the D-essence account does not license the principle (NPD), then it is a failure as a basis for the project of grounding metaphysical modality on a non-modal account of essence. For (as I have emphasized), it is crucial to this project that the non-modal account of essence should license the Necessity Principle (NP) (cf. § above).¹⁹ Let us use “adequate account of essence” to describe an account of essence that licenses the principle (NP), as well as being adequate in other respects. (For example, if Fine’s objections to the modal account of essence are cogent, then an adequate account of essence must not treat all of a thing’s necessary properties as essential properties.) Evidently, what Fine and Lowe are seeking is an account of essence that is adequate in this sense. Let us also use “essence proper” as a label for the notion of essence provided by such an adequate account. Then the problem that I find with the D-essence account may be stated simply as follows: If the D-essence account is an adequate account of essence (an account of “essence proper”), it must license the principle (NPD). The D-essence account does not license (NPD). So: The D-essence account is not an adequate account of essence. ¹⁷ In not attempting to provide an analysis of the notion of D-essence, I take myself to be in accord with Fine: see the passage quoted from Fine :  in § above. And although Lowe appears to be more optimistic about the prospects of providing an analysis of the notion, he suggests that an adequate grasp of it is possible in advance of that: “I do not attempt to offer here a semantic analysis of expressions such as ‘what X is,’ ‘what it is to be X’ or ‘the identity of X,’ although that is no doubt an exercise that should be undertaken at some stage in a full account . . . I assume that our practical grasp of the meaning of such expressions is adequate for a preliminary presentation of the approach of the sort that I am now engaged in” (Lowe : , n.; italics in the original). ¹⁸ I do not claim that the notion of D-essence is sufficiently precise to avoid all disputes about whether some feature counts as part of a thing’s D-essence. On the other hand, I shall rely on examples which seem to me intuitively plausible as candidates for D-essences. ¹⁹ The account of essence must be non-modal in its basis, and yet have modal implications.

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       ?



I shall present two examples which, I claim, show that the criteria for being a D-essential property may be satisfied even if the relevant D-essential properties are not necessary properties.

 Case 1: Lockean Real Essences of Natural Kinds My first example concerns the essences of natural kinds. I shall argue that the following two theses are consistent with one another: (L) (L)

Lockean real essences of natural kinds are D-essences. Lockean real essences of natural kinds do not satisfy (NPD).

However, given that (L) and (L) are consistent with one another, we can conclude: (C)

Being a D-essence is not sufficient for being an “essence proper,”

and: (C)

The D-essence account is not an adequate account of essence,

(where the expressions “essence proper” and “adequate account of essence” are to be understood as explained in the previous section).²⁰ To introduce this example, I begin with some historical background. In his Essay concerning Human Understanding, Locke said that the notion of essence in what he called its “proper and original signification” is: “the being of any thing, whereby it is what it is” (III..).²¹ As Fine notes, in giving this characterization of essence Locke was, so far, in accord with the (Aristotelian and Scholastic) tradition of associating essence with real definition.²² Having characterized essence in this way, Locke immediately adds that essences, in this sense, may be supposed to be “the real internal . . . constitution of things, whereon their discoverable qualities depend”— which he describes as “real essences,” and which he says, in the case of Substances, are generally unknown (Essay III..; cf. III..–). So far, then, it appears that Locke, at least, thought that his real essences—internal constitutions—insofar as they represent “the being of anything, whereby it is what it is”—are good candidates for what I have been calling D-essences. Did Locke also think that these real essences represent necessary properties of the natural kinds

²⁰ Since I argue only for the consistency of (L) and (L), and not for their truth, my example does not take the form of a direct counterexample to (NPD). If successful, however, it is a direct counterexample to the necessitation of (NPD): it shows that (NPD) is not a necessary truth. The same is true of the example concerning sortal concepts that I present in § below. But this is, I think, sufficient for my purposes. It seems to me that if the D-essence account is adequate, it should be a necessary truth, not a merely contingent truth, that satisfying the criteria for being a D-essence is sufficient to license (NPD). Thanks to Gonzalo Rodriguez-Pereyra for alerting me to this issue. ²¹ Locke supports the claim that this is the “proper and original signification” by pointing to the etymological connection between the English word “essence” and the Latin “essentia,” translated as “being.” ²² Fine : . See also Lowe :  and Lowe : .

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

 

whose real essences they are? This is a difficult question, and not one that I wish to explore here.²³ What interests me here is not the interpretation of Locke’s own view about this issue, but the following point. I think that there is a very good case for saying that Lockean real essences of natural kinds (internal constitutions) satisfy the criteria for being D-essences, regardless of whether they also represent (metaphysically) necessary properties of the natural kinds whose real essences they are. Now, of course, the view that Lockean real essences of natural kinds do represent necessary properties (metaphysically necessary properties) of the kinds whose real essences they are is widely held, as a result of the advocacy of this position by Kripke and Putnam. Kripke and Putnam hold that the discoveries that water has the chemical composition H₂O, and that gold has the atomic number , for example, may be treated not merely as the discoveries of the real essence of water and gold in Locke’s sense of real essence— the internal constitution of these kinds of thing, on which their more readily observable properties depend—but also as discoveries of the necessary properties of the natural kinds in question, where the necessity is metaphysical necessity.²⁴ In spite of its popularity, though, the view of Kripke and Putnam is not compulsory, nor universally held. Lowe, for example (in a paper whose main argument is quite independent of his advocacy of the “essence-based” account of modality), has argued that the Kripke–Putnam view is mistaken (Lowe ). According to Lowe, although it is indeed true that the “internal constitution” of water is H₂O, this does not represent the discovery of a necessary property of water, in the sense of a metaphysically necessary property—a property that water has in every possible world in which it exists. At most, Lowe holds, this Lockean real essence of water represents a feature that water has in all possible worlds that share relevant laws of nature with the actual world (: –). Now, suppose that one takes Lowe’s side in this debate, and holds that Lockean real essences of natural kinds do not represent metaphysically necessary properties of the kinds in question. Could one still consistently maintain that Lockean real essences satisfy the criteria for being D-essences—remembering that the notion of a D-essence is (by definition) to be understood in terms that do not appeal to metaphysical modality? I think that the answer to this question is emphatically “yes.” I think that Lockean real essences are prime candidates for the role of D-essences, regardless of whether they represent (metaphysically) necessary properties of the kinds whose real essences they are. Even if it is merely physically necessary that all samples of water are composed (primarily) of H₂O molecules, “being H₂O” still looks like a respectable candidate for an answer to the question what water is, or ²³ Famously, Locke held that, in the case of natural kinds of substance—including kinds of stuff such as gold and water and kinds of animal and plant—these real essences (internal constitutions) could play no role in classification. On the other hand, he held (apparently mistakenly) that if the real essence (in his sense) of a natural kind such as gold were known, then one would be able to deduce the other properties of the natural kind (Essay IV..). For a discussion of Locke’s own views about the association between real essence and necessity, see Lowe : §IX. ²⁴ The view of Kripke and Putnam referred to here is, of course, the view expressed in Kripke  and , and in Putnam  and . As noted by Lowe (), Putnam subsequently changed his mind about this issue (Putnam ).

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       ?



for what it is to be water. It looks like a respectable candidate for something that specifies “the nature” of water, even if we think that water could (in another possible world) have had a different nature. And being H₂O looks like a respectable candidate for saying what water fundamentally is—even if, again, what it is fundamentally is not a property that belongs to it necessarily. If this is right, however, then it is at least consistent to hold both (L): (L)

Lockean real essences of natural kinds are D-essences,

and (L): (L)

Lockean real essences of natural kinds do not satisfy (NPD).

Now, it is clear that Lowe himself would deny that, if the Lockean real essence of water does not represent a metaphysically necessary feature of water, it can still count as the essence or nature of water in the “real definition” sense. Lowe implies that if water is not necessarily H₂O, then “H₂O” (or “composed of H₂O molecules”) is not an appropriate answer to the question “what is water?” (or “what is it to be water?”), construed as a demand for a “real definition” of the kind (: ). However, I do not see how this claim could be justified without appeal to modal notions (specifically, to the notion of metaphysical necessity) in the explanation of what it is for something to be a genuine real definition. But if it is conceded that the notion of metaphysical necessity needs to be invoked in order to distinguish the appropriate sense of real definition from the inappropriate sense (the sense in which Lockean real essences pass the test even if they do not represent (metaphysically) necessary properties), then this is to give up on the project of providing a genuinely non-modal account of essence in terms of real definition.²⁵ It is, in effect, to admit that the non-modal notion of D-essence is not an adequate account of essence, given that an “adequate account of essence” must license the principle (NP).²⁶

²⁵ Lowe asserts that in the Aristotelian tradition, “there is a clear and uncontentious connection between the notions of essence and necessity, arising from the fact that on the Aristotelian view the essences of substances are revealed by their real definitions” (: ; italics in the original). This claim, however, presupposes that a conception of essence based on real definition must satisfy the necessity principle (NP). And this is precisely the assumption that, I maintain, cannot consistently be combined with the view—to which I take Lowe to be committed—that the notion of real definition can be explained in non-modal terms. ²⁶ For an example of a philosopher who holds a conception of real essence, associated with the notion of real definition, which is intended to be free of modal implications, consider Quine: “There is . . . in science a different and wholly respectable vestige of essentialism, or of real definition . . . It consists in picking out those minimum distinctive traits of a chemical, or of a species, or whatever, that link it most directly to the central laws of the science. Such definition . . . is of a piece . . . with the chemical or biological theory itself [and] . . . conforms strikingly to the Aristotelian ideal of real definition, the Aristotelian quest for the essences of things. This vestige of essentialism is of course a vestige to prize” (Quine : ; quoted in part in Wiggins : , n.). The “vestige of essentialism or . . . real definition” that the archanti-essentialist Quine here commends as “wholly respectable” is a conception of essence that, I claim, appears to satisfy the criteria for being a D-essence, even if, as Quine insists, it does not imply that the distinctive traits in question are metaphysically necessary features of the chemical or biological kinds to which they belong. Cf. Cowling (), who emphasizes the possibility of a non-essentialist, Quinean, conception of a thing’s “nature.”

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

 

 Case 2: Substance Sortals That Are Not Ultimate Sortals My second example comes from discussions of sortal concepts and criteria of identity. Consider the question: Is it possible for a horse to cease to be a horse and become a cow? Let us interpret this as a question, not about whether a certain transformation is a causal possibility, but about whether it is a metaphysical possibility. Philosophers—even those who agree in holding a broadly Aristotelian approach to issues of identity and individuation—take different views about the answer to this question. Many, however, take horse and cow to be substance sortals, where a substance sortal represents a property that is “necessarily permanent”: one that belongs to an object throughout its existence if it belongs to it at any time in its existence.²⁷ Rather more formally: A sortal S is a substance sortal if and only if, necessarily (if an individual falls under S at any time in its existence it falls under S throughout its existence).

Thus, if being an F is a substance sortal property, then nothing can change over time from being an F to not being an F, or vice versa (without going out of existence).²⁸ If horse and cow are substance sortals, the answer to the question raised above is “no”: once a horse, always a horse: so no horse, however much it came to resemble a cow, could continue to exist while ceasing to be a horse. This view (that horse and cow are both substance sortals) may, however, be combined (and has been combined) with the view that horse and cow supply the same criterion of identity or principle of individuation.²⁹ If so, then horse and cow would not be what David Wiggins has called ultimate sortals, where the notion of an ultimate sortal may be characterized as follows: A sortal S is an ultimate sortal if and only if S is the most general sortal corresponding to some principle of individuation (or criterion of identity).³⁰

Once this distinction between a substance sortal and an ultimate sortal has been made, however, a question arises about the connection between substance sortals and the necessary properties of the things to which they apply. If we suppose that horse

²⁷ I take the term “substance sortal” from Wiggins, who distinguishes sortals that are also what he calls “substance concepts” from sortals (including “phased sortals” such as boy or tadpole) that are not. See Wiggins : ff. and : ff. Wiggins describes sortals that are substance concepts as “substancesortals” at : , n. and : , n.. The use of the term “substance” is, of course, deliberately Aristotelian. The expression “necessarily permanent property” I borrow from Parsons : . ²⁸ “According to whether ‘x is no longer f ’ entails ‘x is no longer,’ the concept that [a sortal] . . . predicate stands for is in my usage a substance concept” (Wiggins : ). ²⁹ As explained below, Wiggins suggests that this is a possibility, although he does not commit himself to the claim that it is exemplified by the sortals horse and cow. This combination of views has, however, been explicitly endorsed by Dummett. While maintaining that “the very same criterion of identity that is used for horses is also used for, e.g., cows, and, if not for quite all animals, at any rate for all vertebrates,” Dummett claims that it is a consequence of this shared criterion of identity that something that is at one time a horse cannot become a cow, and holds that horse and cow have the characteristics of what I have called substance sortals (Dummett : ). ³⁰ Cf. Wiggins : , n.; Wiggins : .

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       ?



and cow (although distinct substance sortals) supply the same principle of individuation or criterion of identity, there is a question whether we should say that the “metaphysical barrier” to a horse’s ceasing to be a horse and becoming a cow is also a metaphysical barrier to a horse’s having been a cow throughout its existence. Wiggins has presented an account of individuation that implies that it is not. The view expressed in Wiggins  is that it is only ultimate sortals that represent necessary properties of the things that fall under them: that only ultimate sortals are what we might call “necessary sortals,” where: A sortal S is a necessary sortal if and only if the things that fall under S could not have existed without falling under S.

Wiggins is committed to holding both (S): (S)

There may be substance sortals that are not ultimate sortals,

and: (S)

Only ultimate sortals are necessary sortals.³¹

Clearly, Wiggins’s is not the only possible view here. One might hold, in opposition to Wiggins, that all substance sortals are necessary sortals, even if they are not ultimate sortals. It is important to note, however, that it does not follow, merely from the definition of a substance sortal, that substance sortals are necessary sortals. The notion of a substance sortal is indeed a modal notion. But the modality involved is not the modality de re that is involved in the concept of a necessary sortal, but a version of de dicto modality.³² The definition of a substance sortal implies that if a thing falls under a substance sortal S at some time in its existence in a possible world w, it falls under S throughout its existence in world w. The definition does not imply that if a thing falls under S in world w, it falls under S in every possible world in which it exists—that it could not have existed without falling under S. Moreover, this point—that there is a logical gap between being a “necessarily permanent” property and being a necessary property—is an uncontroversial one, and explicitly ³¹ I think that this is a clear consequence of the views expressed in Wiggins , although not one that Wiggins himself emphasizes. According to Wiggins’s “individuative essentialism,” the sole reason for saying that a sortal is a necessary sortal is that it is an ultimate sortal: “The whole justification of our criterion for essential properties is the claim that there can be no envisaging this or that particular thing as having a different principle [of ] individuation . . . from its actual principle” (: ; my italics). But Wiggins also commits himself to the possibility that not all substance sortals are ultimate sortals (: –). Thus I do not see how Wiggins could be hospitable to an argument that all substance sortals are necessary sortals. Given his commitment to the possibility of substance sortals that are not ultimate, an argument that all substance sortals are necessary sortals would be in tension with his argument that all ultimate sortals are necessary sortals. For since all ultimate sortals are substance sortals, we should then have a puzzling overdetermination in the explanation of why ultimate sortals are necessary sortals. Although Wiggins  is less explicit in its commitment to the combination of (S) and (S), I see no evidence in the text of a change of theory on this issue between Wiggins  and Wiggins . For more discussion of Wiggins’s version of sortal essentialism, see my : ch. . ³² At least on one standard construal of the distinction between the de re and the de dicto. What really matters, however, is not these labels, but the distinction between the “intra-world” modal principle that defines the concept of a necessarily permanent property, and the “inter-world” (or “cross-world”) modal principle that defines the concept of a necessary property.

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

 

acknowledged by many writers on this topic.³³ It would appear, then, that Wiggins’s advocacy of (S) and (S) represents a position that is at least a consistent one.³⁴ But now, I want to claim, assuming that (S) and (S) are consistent, it is also consistent to combine them with (S): a thesis to the effect that all substance sortals pass the test for being D-essential sortals—that is, for specifying the D-essences of the things that fall under them: (S)

All substance sortals are D-essential sortals.

I propose that not only is (S) independently plausible, but it remains so even in conjunction with (S) and (S). The case for treating substance sortals as D-essential sortals is—I suggest—independent of the question whether they are also necessary sortals. Prima facie, at least, a substance sortal is an eminently respectable candidate for saying what something is, or what its nature is, in the sense of those expressions that is associated with the Aristotelian notion of real definition. On the assumption that horse is a substance sortal, to say, of a horse, that it is a horse (as opposed, say, to saying that it is brown, or neighing, or in the stable, or an Ascot winner) appears to be an eminently plausible answer to the (Aristotelian) questions “what is it?,” “what is it to be the thing that it is?,” even if we think that the horse could have existed without being a horse. Substance sortals seem to be admirable candidates for the role of D-essences, regardless of whether they are necessary sortals. Moreover, this view appears to be in accord with Wiggins’s own. It is evident that Wiggins’s notion of a substance sortal is intended to provide an adequate answer to Aristotle’s “what is it?” question.³⁵ But it is also evident that Wiggins believes that his account of substance sortals can be defended, as an answer to the Aristotelian question, without regard to whether substance sortal properties also represent necessary properties of the things to which they belong. It is only after the development of his theory of individuation (the theory that gives a major role to the concept of a substance sortal) that Wiggins turns to questions concerning metaphysical modality, introducing the new topic, under the heading “Independence from the explicitly modal of the foregoing theory of individuation,” with the following words: We have made little or no use up to this point of the notion of necessity, and have resisted the idea that a theory of individuation must be a set of judgements about all possible worlds, or try to occupy itself with the problems that are special to the making of statements of explicit necessity de dicto or de re. (: ; repeated almost verbatim at : )

It could, I think, hardly be clearer that Wiggins thinks that the notion of a substance sortal can be explained, and justified as providing an answer to the Aristotelian “what is it?” question, quite independently of issues about the de re modal implications of substance sortals.³⁶ ³³ For more discussion, see my : chs.  and , especially pp.  and –. ³⁴ I shall discuss a challenge to the consistency of the position later in this section. ³⁵ See Wiggins : chs.  and , passim; Wiggins : chs.  and , passim. ³⁶ In Wiggins’s discussion, we find remarks that suggest the even more radical thesis that any sortal— even one that is not a substance sortal—will satisfy the criteria for saying “what a thing is” in the Aristotelian sense that is relevant to real definition. See, for example: Wiggins : , n., and the following remark: “what will count as knowing what a is in the Aristotelian sense? What counts as a

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       ?



If this is correct, however, it undermines the thesis that D-essential properties must be necessary properties. If (S), (S), and (S) together form a consistent set, then: (C) Being a D-essential property is not sufficient for being a “proper” essential property, where a “proper” essential property is one that conforms to the notion of “essence proper” introduced at the end of §. And if (C) is true, then: (C)

The D-essence account is not an adequate account of essence,

where an “adequate account of essence” is one that licenses the principle that all essential properties are necessary properties (cf. § above). Before concluding this section, however, I need to consider an objection. The argument I have just given depends crucially on the claim that the combination of (S) with (S) represents a consistent position. But this may be challenged. One may argue, as Lowe has done, that (although a substance sortal is not by definition a necessary sortal)³⁷ the only explanation for something’s being a substance sortal is that it is also a necessary sortal. If this is right, then clearly Wiggins’s combination of (S) and (S) is not, after all, a consistent one. And if the combination of (S) and (S) is not consistent, then my “Case ” argument for the inadequacy of the nonmodal conception of essence collapses. But does Lowe’s argument succeed? I claim that it does not, because it begs the question—that is, the question whether something’s D-essence represents a (metaphysically) necessary property of the thing. Lowe has presented his argument as follows: Why is it—if indeed it is the case—that Aristotle . . . cannot cease to be a man without ceasing to exist [which is an implication of man’s being a substance sortal]? Surely, the obvious reply is that “A man” is the correct answer to the question “What is Aristotle?,” where this is understood precisely as an inquiry as to his essence or nature. Only this, it seems, explains why we can be so confident that by ceasing to be a man, Aristotle must cease to exist. He must cease to exist because, on ceasing to be a man, there would be nothing that he could then be. But if that is the correct explanation, then it follows also that he could not have failed to be a man altogether. (Lowe : ; italics in the original)

My response is, I suppose, predictable. My objection is that, when Lowe here moves, from the claim that if being a man specifies what Aristotle is in the sense of giving his essence or nature, to the claim that there is nothing else for Aristotle to be except to be a man, he is making an appeal to the assumption that a thing’s essence or nature should specify something that is necessary to its existence. But however reasonable this assumption may seem, it is not one that it is legitimate to make about

sortal concept for a continuant?” (: ). I take it, however, that in these remarks Wiggins must be speaking loosely, and that he would not count what he calls “phased sortals” (e.g., boy or kitten) or other non-substance sortals such as philosopher or astronaut as representing adequate answers to the Aristotelian “what is it?” question. ³⁷ For the reasons explained earlier in this section.

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

 

D-essence—about a conception of essence (and the associated conception of a thing’s nature) that is required to be explicable in non-modal terms. Once this is recognized, it opens up the possibility that although the reason why Aristotle cannot cease to be a man is indeed that (the substance sortal) man answers the question “what is Aristotle?” by specifying his essence or nature, this essence or nature is a D-essence that does not represent a necessary property of Aristotle. I conclude, then, that my Case  objection stands. We have been given no adequate reason to deny that substance sortals may be D-essential sortals even if they are not necessary sortals. Hence, it appears, the criteria for being a D-essential property are insufficient for being a necessary property, and hence are insufficient to license (NPD).

 Conclusion I have presented two examples intended to tell against the viability of an account of essence in terms of real definition that is both non-modal and yet yields the result that essential properties are also necessary properties. I think that such examples could be multiplied, although I shall not argue for that here. I have argued that the conditions for being a D-essence—that is, essence as characterized in terms of real definition, but without appeal to metaphysical modality—are not sufficient for an adequate conception of essence, if an adequate conception of essence is one that (inter alia) guarantees that essential properties are necessary properties (cf. § above). My principal challenge, therefore, is to ask: If my examples are not credible as examples that show that D-essential properties need not be necessary properties, why not? Where have I gone wrong? If there is some crucial feature of the notions of D-essence and D-essential property that I have overlooked and that rules out my examples, what is this crucial feature? Suppose, though, that I am right in thinking that the notion of real definition, understood in non-modal terms, cannot provide an adequate account of essence. And suppose that no adequate non-modal account of essence can be given, either in terms of real definition or in terms of anything else. What exactly follows from this? First, this would certainly not be a vindication of the original modal account of essence. Remember that the “modal account” is (by definition) one that equates the notions of essential property and necessary property (§). And nothing that I have said, in questioning whether a non-modal account of essence can be given, suggests that every necessary property of a thing can properly be regarded as an essential property of that thing. The most that I have established is that an adequate account of essence and essential property must be an at least partly modal account. So my argument allows that Fine’s proposed counterexamples to the modal account may indeed be genuine counterexamples. My claim is that, even if they are, Fine has drawn the wrong moral, from the counterexamples, when he supposes that it is “real definition rather than de re modality” that is central to our understanding of the concept of essence (Fine : ; my italics). Second, if I am right, the situation with regard to a non-modal account of essence is in important respects similar to the situation that Fine complains of with regard to the modal account of essence. Fine complains that the modal account of essence does

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       ?



not give sufficient conditions for being an essence proper, because it includes necessary properties that are not essential properties. I complain that a non-modal account of essence does not give sufficient conditions for being an essence proper, since it includes essential properties that are not necessary properties. For anyone persuaded by both Fine’s complaint and mine, the appropriate response would appear to be to conclude (in opposition to both the modal account and its essencebased rival) that, of the notions of essence and metaphysical modality, neither is more basic than the other. Third, if, as I have maintained, the notion of essence must be explained partly in terms of metaphysical modality, could it still be true that metaphysical modality has its source in the essences or natures of things?³⁸ And if so, might this serve as a vindication of the spirit of the proposal that metaphysical modality can be based on essence? I do not know the answer to these questions. But I hope that this chapter will at least promote further debate on the fascinating issues raised by Fine’s work concerning the relation between the concepts of essence and metaphysical modality.³⁹

References Correia, F. . “On the Reduction of Necessity to Essence.” Philosophy and Phenomenological Research : –. Cowling, S. . “The Modal View of Essence.” Canadian Journal of Philosophy : –. Dummett, M. . “Wittgenstein’s Philosophy of Mathematics.” The Philosophical Review : –. Reprinted in Dummett, Truth and Other Enigmas (London: Duckworth, ), –. Dummett, M. . Frege: Philosophy of Language. London: Duckworth. Fine, K. . “Essence and Modality.” In J. Tomberlin, ed., Philosophical Perspectives : Logic and Language. Atascadero, CA: Ridgeview, –. Fine, K. a. “Ontological Dependence.” Proceedings of the Aristotelian Society : –. Fine, K. b. “Senses of Essence.” In W. Sinnott-Armstrong, D. Raffman, and N. Asher, eds., Modality, Morality, and Belief: Essays in Honor of Ruth Barcan Marcus. Cambridge: Cambridge University Press, –. Fine, K. . “Semantics for the Logic of Essence.” Journal of Philosophical Logic : –. Hale, B. . Necessary Beings: An Essay on Ontology, Modality, and the Relations between Them. Oxford and New York: Oxford University Press. Kripke, S. . “Naming and Necessity.” In D. Davidson and G. Harman, eds., Semantics of Natural Language. Dordrecht: Reidel, –.

³⁸ Bob Hale has recently presented what he calls an “essentialist theory” of metaphysical modality, while claiming to differ from Fine in denying that metaphysical modality can be reduced to essence or nature (Hale : , n.; also , n.). ³⁹ Versions of this chapter have been given to audiences at Oxford and the Universities of Durham, Bucharest, St. Andrews, and Hamburg. I thank all the participants in the discussion on those occasions. Special thanks are owed to Martin Lipman, the respondent to a version of the paper given at an Arché conference at the University of St Andrews in November , to Kit Fine for his participation in the discussion at the ECAP conference in Bucharest in August , and to Robert Frazier for comments on several drafts. Finally, I would like to thank Mircea Dumitru and Kit Fine for giving me the opportunity to contribute the paper to this collection celebrating Kit Fine’s work.

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Kripke, S. . Naming and Necessity. Oxford: Basil Blackwell (expanded monograph version of Kripke ). Locke, J. . An Essay concerning Human Understanding, ed. P. Nidditch. Oxford: Oxford University Press. Originally published in . Lowe, E. J. . Review of Mackie, How Things Might Have Been. Mind : –. Lowe, E. J. . “Two Notions of Being: Entity and Essence.” In R. Le Poidevin, ed., Being: Developments in Contemporary Metaphysics: Royal Institute of Philosophy Supplement : –. Cambridge: Cambridge University Press. Lowe, E. J. . “Locke on Real Essence and Water as a Natural Kind: A Qualified Defence.” Proceedings of the Aristotelian Society, Supplementary vol. : –. Lowe, E. J. . “What is the Source of Our Knowledge of Modal Truths?” Mind : –. Mackie, P. . How Things Might Have Been: Individuals, Kinds, and Essential Properties. Oxford: Clarendon Press. Parsons, J. . “I Am Not Now, Nor Have I Ever Been, a Turnip.” Australasian Journal of Philosophy : –. Putnam, H. . “Meaning and Reference.” The Journal of Philosophy : –. Putnam, H. . “The Meaning of ‘Meaning’.” In Putnam, Mind, Language, and Reality: Philosophical Papers, vol. . Cambridge: Cambridge University Press, –. Putnam, H. . “Is Water Necessarily H₂O?” In Putnam, Realism with a Human Face. Cambridge, MA: Harvard University Press. Quine, W. V. . The Ways of Paradox and Other Essays. Cambridge, MA: Harvard University Press. Sider, T. . Writing the Book of the World. Oxford: Oxford University Press. Wiggins, D. . Sameness and Substance. Oxford: Blackwell. Wiggins, D. . Sameness and Substance Renewed. Cambridge: Cambridge University Press. Wildman, N. . “Modality, Sparsity, and Essence.” The Philosophical Quarterly : –.

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 More on the Reduction of Necessity to Essence Fabrice Correia

“Essence and Modality” has had a considerable impact on subsequent philosophical thinking about essence and modality. The paper argues that the traditional view that essence reduces to metaphysical modality is wrong. Many have found these arguments convincing and have accordingly abandoned the view. The paper also argues that the reduction goes the other way around, that is, that it is metaphysical modality which reduces to essence. Twenty years after the publication of “Essence and Modality,” Kit Fine’s reductive view has become widely recognized as one of the main contenders for a reductive account of metaphysical modality. There are several ways in which such a reductive account can be spelled out. In Correia , I raised objections against one natural way of doing so, and developed an alternative account—the “rule-based” account—which relies on a brief suggestion Fine makes in Fine a. The aim of the present chapter is twofold: first, I wish to strengthen the case for the rule-based account by criticizing alternative accounts, including some accounts based on other suggestions made by Fine, or inspired by material one can find in his work; and second, I wish to discuss certain objections to the rule-based account and suggest how they can be met.

 A Short Précis of “Essence and Modality” Fine’s objections to the view that essence reduces to metaphysical modality are well known, as well as his positive views about essence and its connections to modality, but it may nevertheless be convenient to briefly run through the basics of his  piece. The initial targets of Fine’s objections are the following two modal accounts of essentiality: Conditional account: For an entity x to be essentially F is for it to be the case that as a matter of metaphysical necessity, if x exists, then x is F. Categorical account: For an entity x to be essentially F is for it to be the case that as a matter of metaphysical necessity, x is F. Given the conditional account, each instance of the following schema must be taken to be true: Fabrice Correia, More on the Reduction of Necessity to Essence In: Metaphysics, Meaning, and Modality: Themes from Kit Fine. Edited by Mircea Dumitru, Oxford University Press 2020. © Fabrice Correia. DOI: 10.1093/oso/9780199652624.003.0014

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() x is essentially F if as a matter of metaphysical necessity, if x exists, then x is F, and similarly, given the categorical account each instance of the following schema must be taken to hold: ()

x is essentially F if as a matter of metaphysical necessity, x is F.

Fine’s objections to the two accounts are objections to the effect that some instances of () and some instances of () can, with plausibility, be taken to be false. His attack against () consists in arguing that the following four views hold, or at least have a great plausibility: ) Every entity is such that as a matter of metaphysical necessity, if it exists, then it exists. Yet some entities (e.g. Socrates) do not essentially exist. ) For any metaphysically necessary truth p, every entity is such that as a matter of metaphysical necessity, if it exists, then it is such that p is true. Yet there are metaphysically necessary propositions (e.g. the proposition that 2 þ 2 ¼ 4) such that some entities (e.g. Socrates) are not essentially such that these propositions are true. ) For any two distinct entities, as a matter of metaphysical necessity, if one of them exists, then it is distinct from the other. Yet there are distinct entities (e.g. Socrates and the Eiffel Tower) such that none is essentially distinct from the other. ) Every entity is such that as a matter of metaphysical necessity, if it exists, then so does the set whose sole member is that entity. Yet there is at least one entity (e.g. Socrates) which does not essentially belong to its singleton-set. Against (), Fine invokes the last three views or slight modifications thereof (the details are somewhat straightforward). The conditional and the categorical accounts are only two particular modal accounts of essence, and it may be thought that even if the previous objections are effective, some other modal account might still be correct. Fine doubts it: “it seems to be possible,” he says, “to agree on all of the modal facts and yet disagree on the essentialist facts” (: ). Two main retaliatory strategies are available in response to Fine’s attack. One is to attempt to save one of the original modal accounts by rejecting the modal-essentialist views Fine puts forward against it, for instance on the grounds that they lack the plausibility he attributes to them, or on the grounds that they are simply false (even though perhaps to some extent plausible). Yet I take it—with the vast majority, I trust—that this strategy has its limits, since some of the views put forward by Fine are especially compelling. The other strategy is to acknowledge that at least some of the modal-essentialist views invoked by Fine refute the original modal accounts, and to propose a different modal account purported to escape the relevant difficulties. One option along these lines is to modify the conditional account by adding a condition that should be met by a property for it to be essential, and another option to invoke a nonstandard kind of modality.¹ ¹ Both routes have been explored: the first one by Wildman () (but also see Della Rocca (), who does not discuss Fine’s paper), and the second one by Brogaard and Salerno () and myself ().

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

I believe that some variants of this strategy are promising.² Yet I also believe that the Finean overall conception of the relationships between essence and modality has many virtues, and that it is therefore worth developing. I spell out that overall conception in the rest of this section and take it for granted in the rest of this chapter. Fine offers three main positive considerations about essence and its connections to modality. First, he advocates the Aristotelian-sounding view that essence must be understood “on the model of definition,” but nevertheless stresses that this view need not be understood as a reductive one (: –): . . . essence [h]as been conceived on the model of definition. It has been supposed that the notion of definition has application to both words and objects—that just as we may define a word, or say what it means, so we may define an object, or say what it is. The concept of essence has then [been] taken to reside in the “real” or objectual cases of definition, as opposed to the “nominal” or verbal cases. . . . [T]he traditional assimilation of essence to definition is better suited to the task of explaining what essence is. It may not provide us with an analysis of the concept, but it does provide us with a good model of how the concept works.

Elsewhere, he explicitly states that he doubts “whether there exists any explanation of the notion in fundamentally different terms” (a: ). Second, Fine holds that it is metaphysical modality that reduces to essence rather than the other way around. More precisely, the reductive claim concerns metaphysical necessity, and takes the following form (: ): for a proposition p to be metaphysically necessary is for it to be the case that p is true in virtue of the nature of all objects. (Other metaphysical modalities like metaphysical possibility and metaphysical contingency can in turn to be defined in terms of metaphysical necessity in the usual manner.) Third, he extends this reductive strategy to other familiar modal notions, in particular the notions of conceptual and logical necessity: the conceptual necessities are taken to be the propositions which are true in virtue of the nature of all concepts, and the logical necessities the propositions which are true in virtue of the nature of all logical concepts (: –).

 What Does the “in virtue of ” Locution Mean? Thus, Fine advocates the following reductive accounts: • Given any proposition p, for p to be metaphysically necessary is for it to be the case that p is true in virtue of the nature of all objects. ² A view which follows the first route just mentioned and which deserves to be examined in detail holds that for a property to be essential to an object is for it to be the case that as a matter of metaphysical necessity, if the object exists, then it has the property and the property is intrinsic to the object. The account nicely takes care of the Socrates / Eiffel Tower and the Socrates/{Socrates} examples. Notice that I also crucially appeal to the concept of intrinsicality in order to characterize the nonstandard “local” modalities in Correia .

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• Given any proposition p, for p to be conceptually necessary is for it to be the case that p is true in virtue of the nature of all concepts. • Given any proposition p, for p to be logically necessary is for it to be the case that p is true in virtue of the nature of all logical concepts. They have the very same structure, and each involves an instance of the essentialist locution “is true in virtue of the nature of all Fs,” which is itself a special case of the locution: • is true in virtue of the nature of X, where “X” is to be replaced by a singular term (“Socrates,” “the empty set,” . . . ) or a plural term (“all objects,” “all concepts,” “Socrates and the empty set,” “the natural numbers,” . . . ). But how is this locution to be understood in the present context? This is a substantial question, even if it is granted that essence cannot ultimately be explained in nonessentialist terms. For consider the following natural, “conjunctive” view about how the locution is to be understood in the case of plural attributions: to say that a proposition p is true in virtue of the nature of several objects a, b, . . . is to say that p is true in virtue of the nature of each of a, b, . . . , that is, that for every entity x which is one of a, b, . . . , p is true in virtue of the nature of x. As natural as the view may be, however, it is a nonstarter in the context of the Finean reductions, since we do not want to be committed, for instance, to the view that every mathematical truth holds in virtue of the nature of Socrates, or to the view that the proposition that Socrates is human is true in virtue of the nature of the concept of multiplication. Or consider the following other natural, “disjunctive” view: to say that a proposition p is true in virtue of the nature of several objects a, b, . . . is to say that p is true in virtue of the nature of some of a, b, . . . , that is, that for some entity x which is one of a, b, . . . , p is true in virtue of the nature of x. This alternative view is also problematic, although perhaps less obviously so. For consider for example the metaphysically necessary conjunctive proposition that Socrates is human and water is H₂O and the empty set is a subset of any set. In virtue of the nature of which entity (not “entities”) shall we say that this proposition is true? The question of how the essentialist locution at work in the Finean reductions should be understood was the focus of Correia , and it is also the central question in what follows.

 The Naïve View A relatively simple answer to the question is provided by what I will call “the naïve view.”³ The naïve view endorses the important distinction between constitutive and consequential essence introduced in Fine a (p. ): We need here to distinguish between what one might call the constitutive and the consequential conceptions of essence. An essential property of an object is a constitutive ³ This corresponds to the “consequentialist account of derivative essentiality” I discuss in Correia .

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

part of the essence of that object if it is not had in virtue of being a consequence of some more basic essential properties of the object; and otherwise it is a consequential part of the essence . . . . The constitutive essence is directly definitive of the object, but the consequential essence is only definitive through its connection with other properties. It is in principle possible that a logical consequence of a constitutive part of the essence of an object should itself be a constitutive part of the essence, but as a general rule this will not be the case. Consider Socrates, for example. His essence will, in part, be constituted by his being a man. But being a man or a mountain will merely be consequential upon, and not constitutive of, his essence.

The basic idea seems clear enough. Each object is associated with two classes of propositions which are true in virtue of its nature: its constitutive essence, whose members are the propositions which are directly definitive of the object, and its consequential essence, whose members are the propositions which logically follow from its constitutive essence. On the understanding of the distinction I shall attribute to the naïve view, “being directly definitive of” is a primitive notion, and consequential essence is to be understood in terms of constitutive essence: for a proposition to belong to the consequential essence of an object is for it to be a logical consequence of the constitutive essence of that object. This distinction can easily be extended to pluralities of objects. A natural way of doing so in the present context goes as follows: • The constitutive essence of several objects a; b; ::: ¼df the class of all propositions p such that p is either directly definitive of some object among a, b, . . . , or directly definitive of several objects among a, b, . . . taken together. • The consequential essence of several objects a; b; ::: ¼df the closure of the constitutive essence of a, b, . . . under logical consequence. Thus, given this extension, the constitutive essence of one or more objects X is the class of all propositions p such that p is directly definitive of some object or of some objects among X, and the consequential essence of X is the closure under logical consequence of the constitutive essence of X. All this being said, the naïve view suggests that we understand the locution “is true in virtue of the nature of X” in the Finean reductions as conforming to the following conditions: • In any instance of the locution, the corresponding instance of “the nature of X” is a referring expression; it refers to a certain class of propositions, namely the constitutive essence of what is referred to by the corresponding instance of “X;” • In the locution, “is true in virtue of” must be read as “is a logical consequence of.” Thus, the naïve view endorses the following general principle: • To say that a proposition p is true in virtue of the nature of one or more objects X is to say that p is a logical consequence of the constitutive essence of X

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which, given the proposed characterization of consequential essence, is equivalent to: • To say that a proposition p is true in virtue of the nature of one or more objects X is to say that p belongs to the consequential essence of X.⁴ Given how constitutive essence for pluralities has been characterized, the following monotonicity principle holds: (P) If a proposition belongs to the constitutive essence of one or more objects, then it belongs the constitutive essence of any plurality comprising this or these objects.⁵ An immediate consequence of (P) is: (Q) A proposition belongs to the constitutive essence of a, b, . . . if it belongs either to the constitutive essence of a, or to the constitutive essence of b, or . . . . It is in principle open to a proponent of the naïve view to endorse the converse principle, namely: (R) A proposition belongs to the constitutive essence of a, b, . . . only if it belongs either to the constitutive essence of a, or to the constitutive essence of b, or . . . . However, it is should be clear that this “disjunctive” principle is problematic in the present context. For consider the metaphysically necessary proposition that Socrates is distinct from the Eiffel Tower. By the Finean reduction in combination with the naïve view, the proposition would have to be a logical consequence of the constitutive essence of some given object or objects. And given (R), it would accordingly also have to be the case that either the proposition is a logical consequence of the constitutive essence of some given object, or there are objects a, b, . . . such that the proposition is a logical consequence of the class-theoretic union of the constitutive essence of a, the constitutive essence of b, . . . . Yet it is hard to see which object or objects could satisfy this condition. For on one hand, we cannot hold (at least if we follow the Finean intuitions) that the proposition is directly definitive of Socrates or that it is directly definitive of the tower. And on the other hand, the proposition does not appear to be a logical consequence of a class of propositions each of whose members is directly definitive either of Socrates or of the tower. The naïve view is thus best construed as rejecting (R), and this is how I will understand it in what follows. On that account, the naïve view holds that there are “non-disjunctive” plural constitutive essences. On the view so construed, for instance, it may be held that the proposition that Socrates is distinct from the Eiffel Tower belongs to the constitutive essence of Socrates and the tower taken together, ⁴ Notice here that opting for the alternative view obtained by replacing “consequential” by “constitutive” would be hopeless. For consider again the proposition that Socrates is human and water is H₂O and the empty set is a subset of any set. It is metaphysically necessary. But it is not directly definitive of Socrates, or of water, or of the empty set. Nor is it directly definitive of all three taken together, or even of all three plus the concept of conjunction. More generally, it does not appear to be directly definitive of any object, or of any objects taken together. ⁵ Given (P), consequential essence is also subject to monotonicity. (I am here assuming that logical consequence is itself subject to monotonicity.)

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but neither to the constitutive essence of Socrates nor to the constitutive essence of the tower.

 Alternative Views The naïve view is a fairly natural view about how the essentialist locution in the Finean reductive accounts should be understood. In this section, I wish to present other views which might be taken as serious alternatives. Some of them are based on suggestions made by Fine himself or inspired by material one can find in his writings. Yet it is not altogether clear to me what his “official” view on our main question— which is, again: how should the locution “is true in virtue of the nature of” be understood in the reductive accounts—really is, and accordingly none of the views to be presented here will be attributed to him. The naïve view understands the essentialist expression “is true in virtue of the nature of ” in the Finean reductions as equivalent to “belongs to the consequential essence of,” and is in addition committed to two principles concerning consequential essence: (i) it is to be understood in terms of constitutive essence, and (ii) it is closed under logical consequence, that is: • If a proposition p is a logical consequence of a class C of propositions, and each member of C belongs to the consequential essence of one or several objects X, then p belongs to the consequential essence of X. Could the basic understanding of the essentialist expression be kept while giving up on (i), or (ii), or both? Some parts of Fine  and Fine a point towards a positive answer. In the first paper, Fine claims that he is inclined to reject (i) in favor of the view that it is constitutive essence that should be understood in terms of consequential essence (p. ). The account he suggests is framed in terms of the notion of grounding, and runs as follows in the case of constitutive/consequential essences of single objects: • For a proposition p to belong to the constitutive essence of an object x is for it to be the case that (a) p belongs to the consequential essence of x, and (b) there is no proposition q in the consequential essence of x such that q helps ground p.⁶ And in the second paper, he puts forward a view according to which (ii) fails (pp. –). He distinguishes between two conceptions of consequential essence, the unconstrained and the constrained conceptions. On the unconstrained conception, consequential essence is subject to logical closure (as on the naïve view). On the constrained conception, in contrast, consequential essence is not subject to logical closure, but it nevertheless conforms to a restricted form of the principle. In the case of consequential essences of single objects, the restricted principle runs as follows: ⁶ A proposition is grounded in other propositions when the first holds in virtue of the fact that the latter hold, where “in virtue of” is understood as expressing a metaphysical explanatory tie, and a proposition helps ground another proposition when the latter proposition is grounded in the former, or in the former together with other propositions. The relevant notion of ground is thoroughly discussed in Fine  (see also Correia and Schnieder ).

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• If p is a logical consequence of a class C of propositions, and each member of C belongs to the consequential essence of x, then provided that x depends on all the objects involved in p not involved in any member of C, p belongs to the consequential essence of x, where the relevant concept of dependence may be characterized as follows: • x depends on y just in case for some ϕ, the proposition that y ϕ’s belongs to the (constrained) consequential essence of x. Three views emerge from these considerations. On the most radical of these views, the essentialist locution at work in the Finean reductions should be understood as expressing a certain constrained notion of consequential essence, not definable in terms of constitutive essence. The less radical views accept only half of this. One of them holds that the essentialist locution should be understood as expressing some constrained notion of consequential essence, definable in terms of constitutive essence; and the other less radical view holds that the essentialist locution should be understood as expressing a certain unconstrained notion of consequential essence, not definable in terms of constitutive essence.⁷ On the naïve view and the three alternative views just characterized, thus, the essentialist locution at work in the Finean reductions is subject to logical closure, at least in restricted form. I will reserve the label “consequentialist” to qualify the views of that sort. A fourth alternative view I wish to mention invokes the concept of grounding where the naïve view uses that of logical consequence. More precisely, on that view, to say that a proposition p is true in virtue of the nature of one or more objects X is to say that p is  in the constitutive essence of X, where constitutive essence may be characterized as on the naïve view, and “p is  in class of propositions C” is understood as “either p is in C, or p is grounded in some propositions which belong to C.” The view so characterized is not consequentialist. For take p in the constitutive nature of x, and q a pure proposition. Then p is  in the constitutive nature of x, and p∧ðq∨:qÞ is a logical consequence of p, but we cannot generally infer that p∧ðq∨:qÞ is  in the constitutive nature of x.

 Against the Proposed Views None of the views introduced so far is satisfactory. I here briefly run through a number of objections before introducing a view I take to be much better in the next section. In Correia , I argued against the naïve view on the grounds that it operates with a concept of essence which is closed under logical consequence. The point can ⁷ In his two papers on the logic of essence (Fine b, ), Fine’s primitive essentialist notion is consequential and falls under the constrained conception. Yet it is not clear that the notion at work there is adequate for the Finean reduction of all three modal notions of interest to us: although metaphysical necessity is characterized as truth in virtue of the nature of all objects, conceptual and logical necessity are taken to provide possible interpretations of the notion of truth in virtue of no object.

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be made as follows using the vocabulary introduced previously. A consequence of the (unrestricted) closure principle mentioned above is that: • If p is a logical validity (i.e. a logical consequence of the empty class), then for any object x, p belongs to the consequential essence of x. Given the naïve view’s understanding of the essentialist locution “is true in virtue of the nature of,” it follows that: • If p is a logical validity, then for any object x, p is true in virtue of the nature of x. But this goes against the Finean conception of essential truth, according to which some logical necessities (e.g. the proposition that there are ravens if there are ravens) hold in virtue of certain concepts (in the example, the concept expressed by “if . . . , then . . . ”) as opposed to others. Fine (a: ) makes the very same point. In response to this objection, it might be replied that the unique goal of the naïve view is to provide an interpretation of the essentialist locution that makes the Finean reductive accounts correct, and that succeeding in doing so is compatible with failing to come up with a concept of essential truth that fits extra Finean desiderata. Be it as it may, it is clear that the naïve view even falls short of achieving this goal. For consider again the proposition that there are ravens if there are ravens. On the naïve view, it is true in virtue of the nature of all logical concepts. But none of the logical concepts plays any special role here, for, still on the naïve view, the proposition is true in virtue of the nature of any plurality of objects whatsoever. Therefore, taking the naïve view for granted, it would be pointless to insist that the proposition’s being logically necessary nevertheless consists in its being true in virtue of the nature of the logical concepts. The three alternative consequentialist views presented above are subject to the same difficulties. This is immediate in the case of the view which invokes an unconstrained notion of consequential essence. For the other two cases, it is enough to notice that since the constrained notion is subject the restricted closure principle mentioned above, the following principle holds: • If p is a pure logical validity (i.e. a logically valid proposition which involves no object), then for any object x, p belongs to the consequential essence of x, and that the proposition invoked in the previous objections is pure. There are also problems for those of the alternative views which hold that the notion of consequential essence they invoke is not to be understood in terms of constitutive essence. For one thing, it is not at all clear to me that we can grasp the idea of consequential essence, unless we take it to be defined in terms of constitutive essence in something like the way suggested in the naïve view. A further problem concerns the alternative views in question which would endorse Fine’s particular account of constitutive essence in terms of consequential essence and grounding, namely: • For a proposition p to belong to the constitutive essence of an object x is for it to be the case that (a) p belongs to the consequential essence of x, and (b) there is no proposition q in the consequential essence of x such that q helps ground p. For take x = Socrates and p = the proposition there are ravens if there are ravens. As we saw, p belongs to the consequential essence of any object (whether consequential

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essence is understood as unconstrained or constrained), and so condition (a) is satisfied. Condition (b) is also certainly satisfied: even though p is perhaps grounded (say, in the fact that there are ravens), there is certainly no proposition in the consequential essence of Socrates which helps ground p. Yet we do not want to say that p belongs to the constitutive essence of Socrates.⁸ Let me finally turn to the fourth alternative view. It is not a consequentialist view, and it is clear that it is not subject to the triviality difficulty discussed above. And it also escapes the other difficulties, since it does not make the relevant problematic claims about consequential essence. But it is nevertheless flawed. For let p be the contingent proposition that there are ravens. Then p∨:p is metaphysically necessary, and hence by the Finean account it should be true in virtue of the nature of all things. By the view under focus, it must then be the case either that (i) p∨:p belongs to the constitutive nature of all things, or that (ii) p∨:p is grounded in some propositions q, r, . . . which belong to the constitutive nature of all things. But (i) means that p∨:p is directly definitive of some object, or of some objects taken together, which is implausible. And it is hard to believe that (ii) is true: for which propositions q, r, . . . could we possibly invoke?⁹

 The Rule-Based Account The rule-based account of the Finean reductive theses put forward in Correia  escapes all the previous difficulties. It stems from a brief suggestion made by Fine (a: ), namely that the constitutive essence of the logical concepts is given by rules of inference rather than by propositions: . . . what properly belongs to the nature of disjunction is the inference from p to (p or q) rather than the fact that p implies (p or q). (Thus this is a case in which one might want to think of the nature of something as being nonpropositional in character.) That Socrates is a man or a mountain will then follow from the nature of Socrates and of disjunction in the way in which a proposition is said to follow from certain propositions by means of certain rules. The concept of consequence is not presupposed but is already built into the rules.

I present the account in two parts, departing from the original presentation only in minor respects. The first part presents an account of the essentialist locution “is true in virtue of the nature of,” and the second part presents reductive accounts of metaphysical, conceptual, and logical necessity in line with Fine’s original view.

. The rule-based theory of essence We assume for the sake of illustration that the logical concepts are those of negation, conjunction, disjunction, material implication, universal quantification, and existential quantification,¹⁰ and that each is associated with an introduction and an elimination rule as in a standard natural deduction system for classical logic. ⁸ If it is assumed that there are pure logical validities which are ungrounded, one can change the example and take p to be such a validity. ⁹ Which original proposition is taken as example is important for this argument. For suppose that we take for p the metaphysically necessary proposition that Socrates is human if he exists. Then it is plausible to hold that p is directly definitive of Socrates, and that it grounds p∨:p. ¹⁰ Here and below, by “logical concept” I mean basic logical concept.

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The notion of “being directly definitive of ” is taken to be primitive, as on the naïve view. We take it that what is directly definitive of a logical concept comprises the rules associated with it (this is in line with Fine’s suggestion), but not only. For instance, the proposition that the concept of conjunction is an abstract object is presumably directly definitive of the concept, and the proposition that the concepts of conjunction and of disjunction are numerically distinct may be taken to be directly definitive of these two concepts taken together. Notice that since “being directly definitive of ” is taken to be primitive, the view is immune from my previous objection against the view that it should be understood in terms of consequential essence and grounding. Constitutive essences are characterized as on the naïve view. Notice that constitutive essences so characterized contain only propositions, and accordingly that the rules which are directly definitive of the logical concepts do not belong to any constitutive essence. At the heart of the ruled-based account is a notion of relativized logical consequence, which is defined in the following way. Given any proposition p, class of propositions Δ, and class of logical concepts S, we say that p is a logical consequence of Δ relative to S—in symbols: Δ ‘S p—iff there is a proof of p from Δ, such that given any introduction or elimination rule appearing in that proof, the concept associated with that rule is a member of S. We take for granted that unrelativized logical consequence (which we denote by “‘,” without subscript) is logical consequence relative to some class of logical concepts (or equivalently: relative to the class of all logical concepts). Notice that a proposition p is a logical consequence of a class of propositions Δ relative to the empty class iff p belongs to Δ. Where X is one or more objects, let C(X) be the constitutive nature of X, and log(X) the class of all logical concepts among X. According to the naïve view: • To say that a proposition p is true in virtue of the nature of one or more objects X is to say that p is a logical consequence of the constitutive essence of X—i.e. that CðXÞ ‘ p. According to the rule-based account, we should rather opt for the following construal of the essentialist locution: • To say that a proposition p is true in virtue of the nature of one or more objects X is to say that p is a logical consequence of the constitutive essence of X relative to the logical concepts among X—i.e. that CðXÞ‘log ðXÞ p. In order to get a good grasp of how the account works, it may be useful to distinguish between the following types of cases. (A) Cases where log(X) is empty. In such cases, p is true in virtue of the nature of X iff p belongs to the constitutive essence of X. Thus, on the proposed account, what is true in virtue of the nature of Socrates is just what is directly definitive of him, and what is true in virtue of the nature of Socrates and the Eiffel Tower comprises exactly what is directly definitive of Socrates, what is directly definitive of the tower, and what is directly Socrates and the tower taken together. On this account, thus, the notion of essence expressed by the

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essentialist locution fails to satisfy the closure principles I invoked against the naïve view and the alternative consequentialist views. (B) Cases where log(X) is the class of all logical concepts. In such cases, logical consequence relativized to log(X) and logical consequence tout court coincide, and the proposed view is extensionally equivalent to the naïve view. (C) Cases where log(X) is neither empty nor the class of all logical concepts. It is in such cases that relativized logical consequence does some specific work. Here are two illustrations: — There is a proof of the proposition that there are ravens if there are ravens from the empty class that involves, among the introduction and elimination rules, only the introduction rule for material implication. Therefore, there is a proof of that proposition from the constitutive essence of material implication that involves, among the introduction and elimination rules, only the said rule. Therefore, on the rule-based account, that proposition is true in virtue of the nature of material implication. In contrast, the account predicts that the proposition in question is not true in virtue of the nature of conjunction, or in virtue of the nature of Socrates and the Eiffel Tower— contrary to what the naïve view and the alternative consequentialist views entail. Here we see that the rule-based account escapes the very first objection I made against the naïve view (which also carried over the other consequentialist views). — Consider a person b and its brain a. Then it is plausible to say that it is directly definitive of b that it has a as a proper part. It is also plausible to say that it is directly definitive of proper parthood that proper parts are distinct from the corresponding wholes. Taking all that for granted, the proposition that a is a proper part of b ða