A Metaphysics of Platonic Universals and their Instantiations: Shadow of Universals [1st ed.] 9783030533922, 9783030533939

Table of contents :
Front Matter ....Pages i-x
Introduction (José Tomás Alvarado)....Pages 1-19
Front Matter ....Pages 21-21
Theoretical Roles for Universals (José Tomás Alvarado)....Pages 23-43
The Superiority of Universals Over Resemblance Nominalism (José Tomás Alvarado)....Pages 45-80
The Superiority of Universals Over Classes of Tropes (José Tomás Alvarado)....Pages 81-106
The Superiority of Universals over Theological Nominalism (José Tomás Alvarado)....Pages 107-126
Front Matter ....Pages 127-127
Transcendent Universals and Modal Metaphysics (José Tomás Alvarado)....Pages 129-157
Transcendent Universals and Natural Laws (José Tomás Alvarado)....Pages 159-187
Transcendent Universals and Ontological Priority (José Tomás Alvarado)....Pages 189-200
Objections Against Transcendent Universals (José Tomás Alvarado)....Pages 201-245
Identity Conditions for Transcendent Universals (José Tomás Alvarado)....Pages 247-259
Front Matter ....Pages 261-261
Substrata and Bundles (José Tomás Alvarado)....Pages 263-286
The Nuclear Theory of Trope Bundles (José Tomás Alvarado)....Pages 287-310
The Reformed Nuclear Theory (José Tomás Alvarado)....Pages 311-345
By Way of Conclusion: (Neo) Platonism (José Tomás Alvarado)....Pages 347-351
Back Matter ....Pages 353-362

Citation preview

Synthese Library 428 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

José Tomás Alvarado

A Metaphysics of Platonic Universals and their Instantiations Shadow of Universals

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 428

Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, USA Editorial Board Member Berit Brogaard, University of Miami, Coral Gables, USA Anjan Chakravartty, University of Notre Dame, Notre Dame, USA Steven French, University of Leeds, Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, Amsterdam, The Netherlands Darrell P. Rowbottom, Department of Philosophy, Lingnan University, Tuen Mun, Hong Kong Emma Ruttkamp, Department of Philosophy, University of South Africa, Pretoria, South Africa Kristie Miller, Department of Philosophy, Centre for Time, University of Sydney, Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. Besides monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

More information about this series at http://www.springer.com/series/6607

José Tomás Alvarado

A Metaphysics of Platonic Universals and their Instantiations Shadow of Universals

José Tomás Alvarado Instituto de Filosofía Pontificia Universidad Católica de Chile Macul – Santiago, Chile

ISSN 0166-6991 ISSN 2542-8292 (electronic) Synthese Library ISBN 978-3-030-53392-2 ISBN 978-3-030-53393-9 (eBook) https://doi.org/10.1007/978-3-030-53393-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgments

This work was written during the execution of the research project Fondecyt 1160001 (2016–2019; Conicyt, Chile). It is, however, the fruit of more than 10 years’ work. The ideas presented here have been developed with the support of the Fondecyt research projects 1070339 (2007–2008), 1090002 (2009–2011), and 1120015 (2012–2014). Many more people deserve my gratitude than I am able to include here. In particular, I must mention the Colloquia of Analytic Metaphysics that, since its first version in the Argentine Society of Philosophical Analysis (SADAF) in Buenos Aires in 2008, has continued to meet every 2 years. Many of the theses that appear in this book found their first formulations in those colloquia. I must thank Ezequiel Zerbudis, Gonzalo Rodriguez-Pereyra, Juan Larreta (requiescat in pace) Guido Imaguire, Sebastián Briceño, Carlo Rossi, Robert Garcia, and Horacio Banega for their comments, criticisms, and suggestions on those memorable occasions. In October of 2017, Sebastián Briceño organized a seminar in the Department of Philosophy of the University of Concepción that was dedicated to the discussion of the first version of this book. I am very grateful to Sebastián Briceño and Javier Vidal for their sharp observations that greatly helped to improve this draft. Marcelo Boeri, Juan Manuel Garrido, and Matthew Tugby have kindly read the manuscript and made me see several errors that I have been able to rectify. An anonymous referee for Springer made many useful observations that have improved the book in many parts. Finally, I thank my colleagues from the Institute of Philosophy of the Pontificia Universidad Católica de Valparaíso and from the Institute of Philosophy of the Pontificia Universidad Católica de Chile who have created a stimulating environment for philosophical work; and I thank the generations of students who have been exposed—in one way or another—to the ideas explored here.

v

Contents

1

Introduction [§ 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Properties, Universals, Tropes [§ 2] . . . . . . . . . . . . . . . . . . . . 1.2 Possible Worlds, Grounding, Dependence [§ 3–4] . . . . . . . . . . 1.3 Intrinsic and Extrinsic Properties, Mereology, Concrete and Abstract [§ 5–7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Summary of What Is to Come and some Nomenclature [§ 8] . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

. . .

1 2 8

. . .

11 15 18

. . . . . . . .

23 24 26 27 32 38 40 42

. . .

45 48 54

. . . . . . .

58 61 65 69 72 75 78

Universals

2

Theoretical Roles for Universals [§ 9] . . . . . . . . . . . . . . . . . . . . . . 2.1 One Over Many [§ 10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Many Over One [§ 11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Objective Resemblances [§ 12] . . . . . . . . . . . . . . . . . . . . . . . 2.4 Causality [§ 13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Natural Laws [§ 14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Inductive Practices [§ 15] . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

The Superiority of Universals Over Resemblance Nominalism [§ 16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Resemblance Nominalism [§ 17] . . . . . . . . . . . . . . . . . . . . . . 3.2 Primitive Facts of Resemblance [§ 18] . . . . . . . . . . . . . . . . . . 3.3 How Would We Have Epistemic Access to Such Resemblance Facts? [§ 19] . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Determinate and Determinable [§ 20] . . . . . . . . . . . . . . . . . . . 3.5 Natural Laws and Inductions [§ 21] . . . . . . . . . . . . . . . . . . . . 3.6 Causal Powers [§ 22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 A Vicious Regress [§ 23] . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Modal Consequences [§ 24] . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

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Contents

4

The Superiority of Universals Over Classes of Tropes [§ 25] . . . . . 4.1 Primitive Resemblances [§ 26] . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Imperfect Communities and Companionships [§ 27] . . . . . . . . 4.2.1 Resemblance Classes of Modifier Tropes [§ 28] . . . . . . 4.2.2 Resemblance Classes of Module Tropes [§ 29] . . . . . . 4.3 Modal Consequences [§ 30] . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Natural Classes of Tropes [§ 31] . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 81 . 83 . 91 . 93 . 95 . 98 . 103 . 105

5

The Superiority of Universals over Theological Nominalism [§ 32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Concepts and Mind of God [§ 33] . . . . . . . . . . . . . . . . . . . . . 5.1.1 Concepts [§ 34] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 The Mind of God [§ 35] . . . . . . . . . . . . . . . . . . . . . . . 5.2 A Vicious Regress, Again [§ 36] . . . . . . . . . . . . . . . . . . . . . . 5.3 Divine Aseity [§ 37] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusive Summary of Part I [§ 38] . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

107 114 115 117 120 122 125 126

Part II

Transcendent Universals

6

Transcendent Universals and Modal Metaphysics [§ 39] . . . . . . . . 6.1 Arguments to Accept Transcendent Universals [§ 40] . . . . . . . 6.2 The Function of Universals in Modal Metaphysics [§ 41] . . . . 6.3 Modal Theories Based on Universals [§ 42] . . . . . . . . . . . . . . 6.4 Combinatorial Modal Theories [§ 43] . . . . . . . . . . . . . . . . . . . 6.5 Linguistic Theories of Modality [§ 44] . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

129 132 135 138 145 151 156

7

Transcendent Universals and Natural Laws [§ 45] . . . . . . . . . . . . 7.1 Tooley-Type Cases [§ 46] . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Functional Laws [§ 47] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Powers to the Rescue? [§ 48] . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Generalized Tooley-Type Cases [§ 49] . . . . . . . . . . . . . . . . . . 7.5 The Necessary ‘Nomic Network’ [§ 50] . . . . . . . . . . . . . . . . . 7.5.1 Natural Laws Essential for Universals [§ 51] . . . . . . . . 7.5.2 Necessary Laws [§ 52] . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

159 160 162 169 172 176 176 183 186

8

Transcendent Universals and Ontological Priority [§ 53] . . . . . . . 8.1 The Ontological Priority of Universals [§ 54] . . . . . . . . . . . . . 8.1.1 Priority to Natures [§ 55] . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Priority to Resemblances [§ 56] . . . . . . . . . . . . . . . . . 8.1.3 Priority to Natural Laws and Causality [§ 57] . . . . . . . . 8.2 The Priority Profile of Immanent Universals [§ 58] . . . . . . . . .

. . . . . .

189 189 190 191 191 194

Contents

ix

8.3

Immanent Universals, Similarities, Laws, and Causality [§ 59] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9

10

Objections Against Transcendent Universals [§ 60] . . . . . . . . . . . . . 9.1 The Eleatic Principle [§ 61] . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 What Is a Causal Power? [§ 62] . . . . . . . . . . . . . . . . . . 9.1.2 Causal Powers and Combinatorial Modality [§ 63] . . . . . 9.1.3 Primitive Causal Powers [§ 64] . . . . . . . . . . . . . . . . . . . 9.1.4 Dispositions? [§ 65] . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Do Transcendent Universals Confer Causal Powers? [§ 66] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Lack of Economy [§ 67] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Bradley’s Regress [§ 68] . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Tropes of Instantiation [§ 69] . . . . . . . . . . . . . . . . . . . . 9.2.3 Primitive States of Affairs [§ 70] . . . . . . . . . . . . . . . . . . 9.3 The Epistemological Problem [§ 71] . . . . . . . . . . . . . . . . . . . . . 9.3.1 Perceptual Beliefs [§ 72] . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 A Priori Intuitions [§ 73] . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Universals Epistemologically Transcendent? [§ 74] . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 224 226 229 231 233 235 239 241 243

Identity Conditions for Transcendent Universals [§ 75] . . . . . . . . . 10.1 Structuralism [§ 76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Problem of Regress of the Conditions of Identity [§ 77] . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

247 250 254 259

Part III

201 204 206 213 217 220

Particulars

11

Substrata and Bundles [§ 78] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Substrata or Bundles? [§ 79] . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Difficulties of Substratum Ontologies [§ 80] . . . . . . . . . . . . . 11.3 Difficulties of Bundle Ontologies [§ 81] . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

263 266 270 275 285

12

The Nuclear Theory of Trope Bundles [§ 82] . . . . . . . . . . . . . . . . 12.1 Saturation Structures [§ 83] . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Persistence in Time [§ 84] . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Problem of Substantial Change [§ 85] . . . . . . . . . . . . . . . 12.4 Problems of Ontological Dependence [§ 86] . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

287 290 296 301 304 309

13

The Reformed Nuclear Theory [§ 87] . . . . . . . . . . . . . . . . . . . . . . 13.1 Essences [§ 88] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Unique Nuclear Trope Is a substratum [§ 89] . . . . . . . . . . 13.3 What Qualitative Character Does a Nuclear Trope Possess? [§ 90] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 311 . 314 . 320 . 324

x

Contents

13.4 The Problem of Spatiotemporal Localization [§ 91] . . . . . . . . . 13.5 Analogy of Being [§ 92] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Physical Structures [§ 93] . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Structural Bundles [§ 94] . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

. . . . .

327 329 332 338 344

By Way of Conclusion: (Neo) Platonism [§ 95] . . . . . . . . . . . . . . . . 347 14.1 Everything There Is [§ 96] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 14.2 Open Problems [§ 97] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Chapter 1

Introduction

Abstract This first chapter presents the main line of argumentation that is going to be followed in the book for the defence of an ontology of Platonic universals and their instantiations. Several important concepts and theses for that argumentation are introduced, like ‘property’, ‘universal’, ‘trope’, ‘possible world’, ‘grounding’, ‘dependence’, ‘intrinsic/extrinsic property’, ‘mereological sum’ and ‘abstract/ concrete’. § 1. If we consider everything in the broadest possible sense, we may say that certain entities are grounded on others (cf. § 4), while some entities are fundamental because they are ungrounded. The central thesis of this work is that everything on the fundamental level is either a transcendent universal or a trope. Many other categories of entities can be admitted, such as sets, mereological fusions, states of affairs, events, concrete structures, and—especially—particular objects; but they are ontologically derivative. This thesis has never enjoyed great acceptance, and a long and relatively intricate journey will be undertaken to defend it here. The strategy to justify this central thesis can be summarized in the following sub-theses: (A) (B) (C) (D)

There are universals Every universal is independent of its instantiations There are particular objects Every particular object is a bundle of tropes

Sub-theses (A) and (B) can be taken relatively independently of sub-theses (C) and (D). The defence of (A) and (B) will be made in Parts I and II of this work (Chaps. 2, 3, 4, 5, 6, 7, 8, 9, and 10, §§ 9–77). The defence of (C) and (D) will be made in Part III (Chaps. 11, 12, and 13, §§ 78–94). That is, the thesis to be defended is that there are universal properties (sub-thesis (A)), but that—contrary to what many of its defenders have assumed—these universal properties do not need to be instantiated to exist (sub-thesis (B)). Much of this work will be dedicated to justifying, then, that there are universals that are not instantiated. When it comes to an understanding of the nature of particular objects, on the other hand, I will argue that these should be understood as complexes of the instantiations of transcendent © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_1

1

2

1 Introduction

universals (sub-thesis (D); sub-thesis (C) does not require a justification). These ‘instantiations’ are tropes or particular properties. This ontology of tropes will be explained in Part III. When considering the nature of particular objects it has been usual to argue that there are two mutually exclusive alternatives: they are understood either as being made up of a substratum that has properties, or as a cluster or bundle of properties without a substratum. I will here advance the claim that there is no such opposition and, moreover, will defend an ontology that understands particular objects as tropes, but at the same time as having a substratum (see §89). The central thesis to be defended in this work is that, at the fundamental level, everything that exists is a universal or an instantiation of a universal. The ordinary world of objects and events that happen to these objects is, in some sense, a world of ‘shadows’ of the universals of which the particular properties or tropes are essentially an instantiation. Talk about ‘shadows of universals’ has Platonic resonances, of course (cf. Plato (1982), VII, 514a-521b), but it is not used here except as a suitable metaphor. It is not intended here to argue, for example, that the existence of concrete objects and the particular properties that make them is an ‘illusion’, or that ‘really’ these objects do not exist, or that their existence is of a ‘minor degree’. Rather, the metaphor is used here in the key sense that a shadow projects information about what causes it. The particular objects are instantiations of universals. The reality of the world of particular objects also ‘projects’ information from a different realm: a realm of universals. In this introductory chapter, along with a global presentation of the central defence strategy to be employed, some notions of importance will be presented, such as ‘property’, ‘universal’, ‘trope’, ‘possible world’, or ‘part’.

1.1

Properties, Universals, Tropes

§ 2. We ordinarily attribute characteristics, features, or determinations to the objects with which we have regular contact. We can correctly say of an apple that it is red and of a banana that it is yellow.1 Things are qualified by their form, by their mass, by their location concerning other things, and so on. In this work, we will talk about the ‘character,’ ‘trait’, or ‘nature’ of an object to designate the fact that it is square, or green, or has a height of one metre. A ‘nature,’ ‘trait’, or ‘character’ is not a ‘thing’, therefore, over which to quantify. If it is right to say that x is F then x is F or—which is taken here as merely a paraphrase of this fact—x has the nature of being F. All positions in the metaphysics of properties agree that objects have different traits, characters, or natures. The differences between these positions appear when one 1

Although these are not very good examples, as will be explained below. There is probably no unique real property of ‘being red’ or of ‘being yellow’. The closest thing to such a property would be infinite pluralities of determined colour properties, i.e. of maximally specific colour tones. These pluralities are vague, as there will be certain determined colour tones that will not clearly be red tones or non-red tones. It is doubtful, then, that there are such pluralities.

1.1 Properties, Universals, Tropes

3

considers what it is that grounds such natures—if there is anything on which they are grounded. In the following, a characteristic, feature, or determination of an object that is supposed to be numerically different from the object that possesses it will be called a ‘property’. The connection between a property and the object that owns it will be designated as its “instantiation”, however such a ‘connection’ is finally understood.2 An ontology that postulates properties, therefore, postulates that the objects that are usually presented to us have a specific ontological structure—at least in the sense in which the expression “property” is used here. What is presented to us usually must be a ‘complex’ that must include properties that are numerically different from the object that has them. Not all philosophers use this terminology to designate a ‘property’. David Lewis (cf. 1983, 10) and Gonzalo RodriguezPereyra (cf. 2002, 15–17), for example, use “property” to designate whatever it is that satisfies the theoretical functions usually attributed to universals. The theories that they have proposed, then, are presented as theories about the nature of ‘properties’, even though according to the terminology employed here neither Lewis nor Rodriguez-Pereyra postulates the existence of properties. For simplicity, it is assumed that the properties in question will be monadic, but everything that is said should be considered applicable mutatis mutandis for relations, that is, for dyadic properties or properties with a larger number of arguments (or ‘adicity’). When the distinction between monadic properties and relationships is relevant, the distinction will be made explicitly. A theory that rejects the existence of properties in the indicated sense is a form of nominalism. Here, too, there is considerable terminological variability. In the philosophy of mathematics, any theory that rejects the existence of abstract entities, such as sets or classes (see, for example, Goodman and Quine 1947) has been called “nominalist”. A traditional nominalist strategy to eliminate universals is to propose classes of objects or natural classes of objects that can fulfil their functions. A class is an abstract entity, however, of those that the ‘nominalists’ in the philosophy of mathematics reject. Any theory that rejects the existence of universals has also been called “nominalist” (see, for example, Armstrong 1978a, 138). According to the terminology that is followed here, however, one can defend the existence of properties that are not universal. Properties can be conceived as having a universal or a particular nature. A universal property is a property that can be instantiated in a plurality of exemplifications at the same time. A universal can be shared by many objects. Unlike a universal, a trope is a property of particular and not universal character. The terminology here has also been very variable: they have been called “modes”, “moments”, “abstract particulars”, “perfect particulars”, “concrete properties”, or “instances of property”. From now on, they will be designated as “tropes”. A trope cannot be shared by several objects. For a defender of universals, for example, the mass of an electron is numerically the same property as the mass of another. For a

2 Later on (see § 79), other additional distinctions required for an adequate understanding of particular objects will be introduced.

4

1 Introduction

defender of tropes, on the other hand, the mass of an electron is numerically different from the mass of another electron, if these electrons are numerically different.3 Tropes can be perfectly similar to each other, but this is another matter. There are two ways in which universals can be understood: as immanent or ‘Aristotelian’ universals, or as transcendent or ‘Platonic’ universals. The denominations “Aristotelian” or “Platonic” should not be taken too seriously here. One cannot confidently ascribe to Aristotle the postulation of immanent universals.4 Neither can one confidently attribute to Plato the postulation of transcendent universals (for a general presentation, see Ross 1953), as those universals have been conceived in the recent discussion—and indeed, the acceptance of a theory like the one that will be defended here could not be attributed to Plato. An immanent universal is a universal that exists only if it has instantiations. A transcendent universal is a universal that does not require instantiations to exist. In other words, the difference between one and another conception of universals has to do with the acceptance or rejection of this principle (see Armstrong 1978a, 113): [Principle of Instantiation]

It is necessary for all n-adic universal U that necessarily at least n objects instantiate U.5

The defenders of immanent universals accept this Principle of Instantiation, while the defenders of transcendent universals reject it. Immanent universals are, therefore, ordinarily contingent entities. Except for the case of essential properties for an object, the fact that an object instantiates a property is a contingent fact. Still, however, if it were an essential property for an object, if the object in question were a contingent and not a necessary entity, then the property would also be contingent and not necessary. In effect, as an immanent universal U exists only in the possible worlds6 where it is instantiated, even if it is an essential property of an object, in the worlds where that object does not exist, U will not either—assuming that no other objects in those worlds instantiate U. For a defender of immanent universals, the only cases of necessary universals would be the cases of essential properties of necessary objects that invariably exist in all possible worlds. Necessary universals could be got, perhaps, in the case of essential properties of mathematical objects, if any, and the essential properties of God. The situation is very different when it comes to transcendent universals. It would not be inconsistent to suppose that these universals are contingent entities. All that is

3

An assumption that should always be taken with extreme caution when dealing with entities at the quantum level. Cf. French and Krause 2006, 84–197. 4 Indeed, it has been the subject of discussion whether Aristotle admits universal or particular forms in Metaphysics, especially about Z 13. Cf. for general presentations, Wedin 2000; Lewis 2013. 5 Armstrong’s original formulation (see 1978a, 113) does not include modal operators, but clearly, he does not propose the principle as a happy contingent coincidence affecting all actually existing universals. The principle obtains for the defender of an Aristotelian conception of universals due to the essential nature of universals, i. e. as a matter of metaphysical necessity. The insertion of modal operators in the principle above makes explicit this modal character. 6 On the concept of ‘possible world’, cf. § 3 below.

1.1 Properties, Universals, Tropes

5

required for a defender of transcendent universals is the rejection of the Principle of Instantiation; that is, to suppose that it is possible that there is a non-instantiated universal. A universal can exist non-instantiated in some possible worlds and not exist in others. Positions of this type have been defended on the grounds of economy (see Tooley 1987, 113–120), since postulating contingent universals would be less ‘costly’ ontologically than postulating necessary universals. However, reasons of simplicity and economy militate against this hypothesis. The most sensible thing is to suppose that transcendent universals, if they exist, must be necessary entities, invariant between different possible worlds. Indeed, we might ask, if they were indeed contingent, for what reason would they exist or fail to exist in one possible world rather than in another? Whoever holds such a position ought to give a reason why these variations can occur. It is not possible here to appeal to an instantiation that a universal might have in one possible world and not have in others, because a transcendent universal does not depend ontologically on having an instance. And there do not seem to be any other reasons to suppose such differences. The most economical thing, therefore, is to suppose that there are no such variations between different possible worlds when it comes to transcendent universals. There are also significant differences between immanent and transcendent universals concerning their space-time location. For transcendent universals, it does not seem to make sense to attribute a location to them in space and time. For the immanent universals, however, it does not seem inappropriate to attribute a spatiotemporal location to their instantiations. Unlike a particular object, an immanent universal will be located entirely in each of the regions in which the objects that instantiate it are located. It has to be multi-located. A particular object, on the other hand, is located in a region of space because it has different parts that are located in the different sub-regions of that space. For temporal localization, however, the contrast between universals and particular objects is more difficult to pin down. Traditionally, it has been argued that particular objects persist in time, i.e. are identical in different instants of time; or exist entirely in each of those instants, i.e. are ‘simple’ but extended entities. In recent years, however, a different conception has become popular in which the persistence of an object in time is understood analogously to the location of an object in space, that is, by the possession of different temporal parts in each of the different moments of persistence. If the second mode of temporal persistence is adopted, there is a marked contrast between the form of the temporal location of a universal and the form of the temporal location of a particular object, as happens with spatial localization. If one takes the first mode of persistence, however, individuals are located in time, as are universals: they exist all and entirely at every moment when they exist.7 7 There are other more exotic hypotheses discussed in the literature that will not be discussed here. It has been argued, for example, that a particular object could also be multi-located entirely in different regions of space because an object could travel in time to the past and occupy a different region of space than it occupies at any given time (Ehring 2011, 27–30). It is not intended to argue here that the differences between universals and individuals concerning their spatio-temporal location are the way to analyse the difference between universal and particular.

6

1 Introduction

There has been a debate about how to analyse the distinction between universals and particulars. Many criteria have been proposed, and many of them have been unsatisfactory. One might think, for example, that the difference between a universal and a particular consists in the fact that the universal—in the manner of a Fregean function—is essentially incomplete and required to be saturated by an object. Ramsey, however, showed that one could think analogously of objects as ‘incomplete’ and as ‘requiring saturation’ by one property or another (see Ramsey 1925).8 One might think, too, that the way to analyse the difference between universal and particular is that individuals may enter into connections with an arbitrary number of other objects and universals to form states of affairs. A universal, on the other hand, seems to be able to enter into connection only with a fixed number of objects, corresponding to its adicity, that instantiate it (cf. Armstrong 1997, 168; MacBride 2005, 571–572). It turns out, however, that one cannot exclude a priori the existence of multigrade universals, that is, universals that can be instantiated in arbitrary pluralities of objects. Suppose there were a universal of to surround—a hypothesis not very plausible, but sufficient to clarify this point. It can be given the state of affairs of being Peter, John, and James surrounding x, but also the state of affairs of being Peter, John, James, and Andrew surrounding x, or the state of affairs of being Peter and John surrounding x, etc. The same universal seems to be integrating states of affairs with two, three, four, or n different objects. The fact that there is no way to analyse the difference between universals and particulars, however, should not be taken as a reason to argue that there is no difference between them. Not everything can be analysed. The difference between universal and particular might be of such a fundamental nature that there are no more basic conceptual resources from which it could be made clear. It would not be strange if such a thing should happen. In the following it is assumed that the distinction is simply primitive.9 A distinction has frequently been postulated between sparse and abundant properties, at least since it was presented in these terms by David Lewis (cf. 1983, 11–14). A traditional way of postulating universal properties has been to argue

8

Suppose that the proposition that Socrates is wise corresponds to a state of affairs. The semantic components of the proposition correspond to the ontological components of the state of affairs. In the proposition, we can distinguish a function x is wise in which the ‘x’ indicates the empty place of the variable that requires saturation or to be bounded by a quantifier to generate a complete proposition. Socrates seems to be ‘complete’. This contrast, however, seems excessively determined by the grammatical difference between predicates and names. Ramsey (1925) argued that one could just as well represent Socrates as Socrates is F, where F is a variable that needs to be saturated to form a complete proposition. If the ‘unsaturated’ character implies a certain ontological precariousness and is not merely semantic, the same seems to happen with universals. Universals cannot constitute states of affairs independently of objects—at least in the immanentist conception of universals—nor can objects form states of affairs independently of universals. 9 Which does not rule out, of course, that some appropriate way of doing the analysis should appear. For example, it has been proposed that the difference between universal and particular is given by the fact that it is metaphysically impossible for universals to be perfectly similar to each other or indiscernible among themselves (cf. Ehring 2011, 30–40). On the other hand, it is possible that there are particulars which are indiscernible between themselves, whether they are objects or tropes.

1.1 Properties, Universals, Tropes

7

that they are the semantic value of the predicates of our language or that they are the content of our concepts. Each possible predicate will be correlated with a ‘property’, as well as each concept of a possible thinking subject. For a natural language like English, which allows us to build denumerably infinite many10 different predicates by the recursive application of compositional rules of grammatical construction, there are infinite predicates that seem to select extremely heterogeneous things. For example, there is a predicate like ‘being a cat and being examined before the year 3,000, or being a galaxy and being examined after the year 3,000’. In this conception there is exactly one class of all the things that instantiate the ‘property’ which is the meaning of the predicate in question. This class includes cats and galaxies. Examples like this can be multiplied ad nauseam. There are certain theoretical functions that properties specified in this way have been expected to satisfy. The discussion in this work, however, will concentrate on the so-called ‘sparse properties’. These are ‘sparse’ because in comparison to the ‘abundant’ ones they form a minority. These properties are those that are strictly necessary to ground the objective resemblances and the causal powers of objects. The way in which we can determine if there is a property in this sense is not through reflection on the contents of our thought and our language, but rather through reflection on what are the objective resemblances between the objects and what causal powers they have—according to our best evidence regarding such objects, which will ordinarily be empirical. It is not the task of the philosopher to determine, in general, what properties exist, but the task of the natural scientist. In principle, ‘sparse’ properties should be the minimum basis sufficient to fully characterize the world (see Lewis 1983, 12) or, in linguistic terms, they should be what primitive predicates refer to in a language sufficient for a complete description of everything. In the following, these sparse properties will be referred to as “authentic properties”.11 Yet although it has been usual to maintain that it is natural science that determines a posteriori, by empirical research, what properties exist, it should not be assumed that the postulation of authentic properties is connected with some form of naturalism. Several advocates of theories of authentic properties, such as David M. Armstrong (cf., for example, 1997, 5–10), are naturalists, that is, they hold that everything that exists is the space-time system and the entities it contains that are described by natural science. But there is no systematic connection between naturalism and the postulation of authentic properties. The claim is simply that not every

An infinite plurality is said to be ‘denumerable’ if it can be put in bijection with the set of natural numbers, as with the set of rational numbers or the set of integers. An infinite plurality is said to be ‘indenumerable’ if it cannot be put into bijection with the set of natural numbers, just as happens with the set of real numbers. 11 This is not to deny, of course, that there is an important philosophical problem about what is the content of our thinking, and what semantic values should be given to the expressions of our languages. Nothing prevents, in particular, that operations from universals to universals should allow the generation of such contents—as has been supposed by those who have defended ‘abundant’ properties—from a base constituted only by authentic properties, according to what is indicated here. These, however, are further issues that will not be discussed in this work. 10

8

1 Introduction

term or predicate of our language must be correlated with some authentic property that has to be its ‘meaning’, and that it is more reasonable to think that there are real properties for which we do not have linguistic expressions that designate them. Properties are not the mere correlates of our languages. Moreover, according to everything we know, there could be properties of such a character that they will evade our best efforts to discover or understand them. These properties could still be the kinds of properties with which natural science deals. Something like this is what we should suppose if everything that exists is of a physical nature, but it is not what we should suppose if there are entities that transcend time and space. There could be properties of a non-physical nature. There may be mental properties, for example, that are not reducible to physical properties or not based on physical properties. If so, such mental properties need to be entered into a complete characterization of the world.

1.2

Possible Worlds, Grounding, Dependence

§ 3. The expression “possible world” will be used freely. As is well known, there is controversy about what should be understood by this term (cf. for a general perspective, Divers 2002; Alvarado 2008). Despite the differences in perspective, there is a minimum content that can be assigned to the expression, and which has utility when it comes to considering modal facts—that is, facts about what might be the case, or about what necessarily is the case, or about what is not necessarily the case, etc. By ‘possible world’ we should understand a form in which all the things could be. Anyone who believes that things could be different from how they actually are believes that there are ways in which things could be that is not the way things are (see Lewis 1973, 84). Accepting the existence of possible worlds, therefore, is little more than accepting the existence of possibilities, which seems a matter of common sense. No special assumption is made here about their nature. Modal issues are considered in several sections of the work (see §§ 24, 30, 41–44). It will be seen that when it comes to an understanding of the nature of worlds, an ‘actualist’ conception will be preferred, according to which there is only one concrete world, the actual world. ‘Possible worlds’ are abstract constructions that represent how things could be. For the moment, however, it will not be necessary to assume anything more than that to speak of ‘possible worlds’ is simply to speak of metaphysical possibilities considered globally. § 4. Much of the argumentation in this chapter will have to do with the aptitude or inaptitude of certain postulated entities to ‘explain’ specific facts. The notion of ‘explanation’, however, has been the subject of much controversy throughout the past century. An important tradition has connected it with the notion of ‘causality’. This connection to causality makes its application in ontology difficult, since it is doubtful, for example, that universals ‘cause’ the phenomena of one over many (cf. § 10). An ‘ontological’ explanation must have a different character from what is expected of an explanation in the natural sciences. In a very general sense, one

1.2 Possible Worlds, Grounding, Dependence

9

could argue that universals allow us to answer in a satisfactory way—given certain presuppositions—the question about why oneness is given in what is multiple (see § 10), why there are objective resemblances (see § 12), why objects have causal powers (see § 13), and so on. These ontological ‘explanations’, however, require special clarification. Following a significant number of theorists, we here assume that the ontological ‘explanation’ is given by the highlighting of relations of ‘grounding’ or of ‘dependence’. Grounding will be taken as a primitive, non-analysable, and multigrade relationship that can occur between a plurality of grounding entities and the grounded entity. It is usually assumed that the relata of grounding are ‘facts’, but this restriction will not be imposed here.12 Entities of different categories can be with each other in the grounding relationship. It is a strict order, that is, it is an irreflexive, asymmetric, and transitive relationship.13 If x is grounded on y1, y2, . . ., yn then it is assumed that y1, y2, . . ., yn are constitutively sufficient for the existence of x, given its essence. The entities y1, y2, . . ., yn ‘determine’ the existence of x. For this reason, the following link will be assumed: [Grounding – Implication]

If y1, y2, . . ., yn ground x, then it is necessary that: if y1 exists, y2 exists, . . . and yn exists, then x exists

However, the converse is not valid, because it could be the case that it is necessary that if y1, y2, . . ., yn exist, then x exists, without x intuitively being grounded on y1, y2, . . ., yn. For example, a strict conditional is true if its consequent is necessary, no matter what is in the antecedent. The strict implication [□((I exist) ! (2 + 1 ¼ 3))] is true, then, but it would be extravagant to think that I am the ground of the mathematical truth that 3 is the sum of 2 and 1. Modal covariations do not discriminate which entities are prior—the ground—and which are derivative—grounded (cf. Rosen 2010; Correia and Schnieder 2012; Fine 2012; Alvarado 2013; among many others). Ontological dependence, on the other hand, is a primitive, irreflexive, asymmetric, and transitive relationship, which implies a strict implication, but is not

12

The reason why it has usually been assumed that grounding is a relationship that has as relata ‘facts’ is that many analyses have focused on the uses of the expression ‘–because–’ and the like, that connect complete sentences. For this reason, it has also been proposed that it be treated as a sentential connective (see Fine 2012). If someone has difficulties with the freedom with which the relations of grounding between entities of any category will be treated here, one can replace the expressions ‘x is grounded on y’ with ‘the fact that x exists is grounded in the fact that y exists’. Of course, this is not the place to make a detailed discussion of this topic. For a recent defence of ‘entity’ grounding, see Wilhelm, 2019. 13 With a better understanding of the nature of grounding, other hypotheses have been explored with infinite or cyclic structures of grounding in which there is nothing ‘fundamental’, not based on anything (cf. for example, Bliss 2013, 2014). Although none of these hypotheses is inconsistent, it is far from clear that there are cases in which it is reasonable to postulate such structures. Here we are going to assume that both grounding and dependence are ‘well-founded’ relationships, so there must be something fundamental, not grounded on anything, or independent, not dependent on anything.

10

1 Introduction

analysable as a strict implication. If x depends ontologically on y, then it follows that it is necessary that, if x exists, then y exists. A strict conditional by itself, however, only registers some modal covariation. If it is about the propositions x exists and y exists, the strict conditional will be: it is necessary that if x exists, then y exists. That is, in all possible worlds, either x does not exist—so x exists is false—or y exists— that is, in all possible worlds where x exists, y also does. Of course, this covariation may be based on the fact that it is part of the essence of x that y exists, but it can also be based on other circumstances. The strict conditional may be based on the fact that y is a necessary entity, existing equally in all possible worlds. Trivially, then, in all possible worlds in which x exists, y will exist also. It can also be based on the fact that y necessarily results from the existence of x. For example, given the axioms of set theory, in all possible worlds where x exists, there is also the singleton set {x}. It is evident, however, that x does not depend ontologically on {x}. Ontological dependence goes in the opposite direction. It turns out, therefore, that if x is dependent on y1, y2, . . ., yn, then the entities y1, y2, . . ., yn are included in the essence of x, so that it is necessary that if x exists, then y1, y2, . . ., yn exist. The following link therefore results: [Dependence – Implication]

If x depends on y1, y2, . . ., yn, then it is necessary that: if x exists, then y1 exists, y2 exists, . . ., and yn exists

The relations of grounding and dependence are strict orders, that is, they are irreflexive, asymmetric and transitive relations. Nothing is grounded on itself or is dependent on itself. Some attempts to challenge these formal characteristics of dependence are discussed below (see § 86). We can also define weaker notions of anti-symmetric grounding and dependence: [Weak grounding] [Weak dependence]

x is weakly grounded on y ¼ df x is grounded on y or x ¼ y x depends weakly on y ¼ df x depends on y or x ¼ y

In both cases it is a reflexive, anti-symmetrical, and transitive relation, that is: (i) for all x, x depends weakly on itself and x is weakly grounded on itself; (ii) for all x and y, if x is weakly grounded on y, and y is weakly grounded on x, then x ¼ y; if x depends weakly on y, and y depends weakly on x, then x ¼ y; (iii) for all x, y, z, if x is weakly grounded on y, and y is weakly grounded on z, then x is weakly grounded on z; if x depends weakly on y, and y depends weakly on z, then x depends weakly on z. These weaker notions may have a formal interest, but the notions of ontological importance are those that have to do with relations of ontological priority/posteriority. Moreover, nothing is previous to or derivative on itself, so the notions of strict grounding and strict dependence are relevant to what will be discussed here. It follows from this that ‘ontological priority’ is said in many ways. The ontological ‘priority’ is the converse relation of ontological ‘posteriority’ or ‘derivativeness’. An entity x can be said to be ‘posterior’ to y, either because x is grounded on y, or because x is dependent on y. There are two different forms of priority, then, which should be distinguished, and which can be analysed as follows:

1.3 Intrinsic and Extrinsic Properties, Mereology, Concrete and Abstract

[Grounding Priority]

x is ontologically prior to y ¼ df y is grounded on x

[Dependence Priority]

x is ontologically prior to y ¼ df y depends on x

11

Something can be prior in the first sense without being prior in the second, or vice versa. We will say that an entity is ‘emergent’ if it is ungrounded, but is ontologically dependent on something (cf. Barnes 2012). Analogously, we will say that an entity that is independent, but grounded on x1, x2, . . ., xn, is ‘realized’ in x1, x2, . . ., xn. Something like this would be a ‘multiply realizable’ entity, something that must be grounded ontologically, but such that it need not be grounded on the specific entities on which it depends. For what follows, it will also be useful to consider another notion close to the notion of grounding that has been described: [Partial Grounding]

x is partially grounded on y ¼ df there are z1, z2, . . ., zn such that: z1, z2, . . ., zn, y ground x

Both grounding and dependence are non-monotonic relationships. That is, if y1, y2, . . ., yn ground x, then there is no z, z 6¼ y1, z 6¼ y2, . . ., z 6¼ yn such that: z, y1, y2, . . ., yn ground x. The ground of an entity is that which is constitutively sufficient for the existence of such an entity and nothing else. Otherwise, it would be trivial that everything that fully grounds something, also partially grounds something, because if x is grounded on y, then x could be grounded also on y, z, for an arbitrary z. For dependence, however, there is another notion of ‘generic dependence’ that has proved useful and is opposed to ‘rigid dependence’. If x depends generically on an F, then it is necessary that if x exists, then there is an F. A notion close to that of ‘grounding’ is that of truthmaking or ‘making true’ (cf. Fine 2012, 43–46). The truthmaker of a proposition is that entity whose existence grounds its truth. We will assume, furthermore, that for every true proposition p, it is necessary that there is some entity e such that e grounds the truth of p. From this, it follows that it is necessary that, if e exists, then p is true.

1.3

Intrinsic and Extrinsic Properties, Mereology, Concrete and Abstract

§ 5. A distinction to which we will have several occasions to resort is that between ‘intrinsic properties’ and ‘extrinsic properties’, as well as the distinction between ‘internal’ and ‘external’ relations. It has been notoriously difficult to analyse the difference between intrinsic and extrinsic properties. In some ways of effecting the distinction, the intrinsic properties are those that persist after ‘enlargements’ or ‘reductions’ of the possible worlds in which an object possesses such a property. In other ways of effecting the distinction, a property is intrinsic if it does not imply the existence of other objects different from the object that possesses such a property—assuming forms of ‘relevant’ implication. In other ways of effecting the

12

1 Introduction

distinction, intrinsic properties are those that are not relational. None of these strategies has been entirely satisfactory (see Weatherson and Marshall 2012). Of course, it is not possible to adjudicate this question here. Probably, there is no single precise concept of ‘intrinsic property’. There are certain relatively vague intuitions about properties that determine an object regardless of what other objects exist and what relationships exist with them, but these intuitions are consistent with different and incompatible ways of making them precise. It seems more prudent, for these reasons, to define two different concepts of ‘intrinsic property’ according to two different criteria. The first is the one that corresponds to combinatorial modal intuitions: [Combinatorially Intrinsic]

P is combinatorially intrinsic ¼ df the fact that x instantiates P is independent of whether x is alone or accompanied

Of course, a ‘combinatorially extrinsic’ property is one that is not ‘combinatorially intrinsic’. This formulation comes from Lewis and Langton (1998).14 An object x is alone in the possible world w if and only if there is no other object than x in w. An object x is accompanied in w if and only if it is not alone in w. According to this first way of characterizing what an intrinsic property is, it is one whose instantiations do not vary if the possible world in question includes more or less different objects. If something has 5 gr of mass, for example, its mass does not vary if there is another object or not. On the contrary, the property of being 5 metres from a cube is not intrinsic because nothing will instantiate it in possible worlds in which there is no cube. Adding or removing a cube makes a difference to whether something has the property of being 5 metres from a cube. Note that if there are objects of necessary existence, these will not vary between different possible worlds, because no world lacks them. If the number 3 is a necessary object, then the property of being accompanied by the number 3 will be combinatorially intrinsic to every object, since the fact that an object is accompanied by the number 3 is invariant concerning worlds in which that object is or is not accompanied by others. For this property to be combinatorially extrinsic, there must be possible worlds in which adding or removing an object to a possible world determines that something is no longer accompanied by the number 3 or that it becomes accompanied by the number 3. However, as the number 3 is—ex hypothesi—modally invariant, nothing that is added or removed modally produces a difference.

14

There are sophistications introduced in the analysis of Lewis and Langton that are not important here. These sophistications are motivated by the possibility of disjunctive properties. The proposed analysis ultimately depends on a primitive distinction between ‘natural’ properties and those that are not (see Lewis and Langton 1998, 118–121). A property is said to be basic intrinsic if and only if: (i) the fact that an object instantiates it is independent of whether the object is alone or accompanied (as indicated above), (ii) it is not the disjunction of other natural properties, and (iii) it is not the negation of a natural disjunctive property. Two objects are duplicates if and only if they have the same ‘basic’ intrinsic properties. Now, a property is intrinsic if and only if it cannot differ between duplicates.

1.3 Intrinsic and Extrinsic Properties, Mereology, Concrete and Abstract

13

Another way of analysing the notion of ‘intrinsic’ property is that proposed by Rosen (2010, 112) and depends on the concept of ‘grounding’ already explained. F is an intrinsic property if and only if, necessarily for all x: (i) if x is F in virtue of a fact involving y, then y is an improper part of x; and (ii) if x is not F in virtue of a fact involving y, then y is an improper part of x. The following alternative analysis can be given: [Grounding Intrinsic]

P is grounding intrinsic ¼ df the fact that an object x does or does not instantiate P must be grounded on an improper part of x if it is grounded

Of course, a property is ‘grounding extrinsic’ if and only if it is not grounding intrinsic. This concept is substantially stronger than that of being ‘combinatorially intrinsic’. According to this alternative analysis, it is not enough that the instantiation of the property in question by an object is invariant in possible worlds in which there are or not other objects The instantiation—or the non-instantiation—must be grounded on the object that instantiates the property or on its parts. In this way, although the property of being accompanied by the number 5 turns out to be combinatorially intrinsic—assuming that the number 5 is a necessary object—it is not grounding intrinsic, since the existence of the number 5 is not grounded on the existence of any other concrete object. In the following, it will be assumed as a general rule that the ‘intrinsic’ properties are grounding intrinsic, and the ‘extrinsic’ properties are grounding extrinsic. In some cases, the alternative notions of being ‘combinatorially intrinsic’ or ‘combinatorially extrinsic’ may be relevant, but when this is so it will be expressly indicated. Given the concept of ‘intrinsic property’, the notion of an ‘internal relation’ can be analysed. A relation R is internal if and only if the fact that x1, x2, . . ., xn are in the relation R is grounded on the intrinsic natures of x1, of x2, . . . and of xn. The intrinsic nature of x is the totality of the intrinsic properties instantiated by x. For example, if x1 is 1 metre and x2 is 2 metres, then it follows that x1 is in the relation of being smaller than with respect to x2. The relationship in question is not an ‘ontological addition’ over the intrinsic properties possessed by the relata. A relationship is said to be ‘external’ if and only if it is not internal.15 Of course, if one conceives intrinsic properties as combinatorially intrinsic, internal relationships will be ‘combinatorially internal’. If one conceives of intrinsic properties as grounding intrinsic, internal relationships will be ‘grounding internal’. As it is generally assumed that intrinsic properties are grounding intrinsic, as explained above, internal relationships will be understood—correlatively—as grounding internal, unless there is an explicit prevention.16 Although cf. Lewis 1983, 26, note 16; Lewis calls ‘external’ the dyadic relations that are supervening on the pair of relata considered together, although they are not supervening on the intrinsic natures of these relata. 16 The expression ‘internal relationship’ has also been used to designate a relationship that is essential to their relata. That is, if R is an internal relation for x1, x2, . . ., xn, then none of those objects would exist if it were not in the relation R with the rest. The designation ‘internal’ has also 15

14

1 Introduction

§ 6. It was mentioned in § 5 above that an object can be part of another. The notion of ‘part’ will be understood according to standard extensional mereology. Mereology is a theory that formalizes the relationships between wholes and parts. It can be characterized by the following axioms: (a) if x is a proper part of y, and y is a proper part of z, then x is a proper part of z [Transitivity of the relation of ‘being a part of’] (b) if x is a proper part of y, then y is not a proper part of x [Asymmetry of the relation of ‘being a part of’] (c) if x is a proper part of y, then there is a z such that: z is a proper part of y and z is disjoint from x [Principle of supplementation] (d) if there is any F, then there is a unique object x such that: something overlaps with x if and only if it is overlapped with an F [Principle of unrestricted sums] The notion of ‘being a part of’ has been used here as a primitive. It corresponds to our ordinary intuition about a part that has to be ‘smaller’ than the whole of which it is a part. A part that may be ‘smaller’ or possibly identical to the whole of which it is a part is designated an ‘improper part’. The relation of ‘being an improper part of’ is therefore reflexive and antisymmetric. Two objects are ‘overlapping’ if and only if they have some (improper) part in common. Two objects are ‘disjoint’ if and only if they do not overlap each other (cf. Simons 1987, 1–45). In standard extensional mereology, it is automatic that for any objects—no matter how far apart they may be spatially or temporally, or how heterogeneous they are—there is the mereological sum of precisely those objects. It is also characteristic of standard extensional mereology that the only thing that is relevant for the conditions of identity of a mereological sum is what its parts are. It does not matter what kind of ‘structure’ such parts have or do not have. § 7. We will assume that entities can be ‘concrete’ or ‘abstract’. However, as in so many other areas, this distinction has been mired in controversy (for a general presentation, see Cowling 2017, 69–105). In the present work, it will be assumed that universals are paradigmatic cases of abstract entities, whereas particular objects are paradigmatic cases of concrete entities. It will not be necessary to consider whether one or another analysis is the most pertinent to specify the distinction.17 It

been used for a relationship that has a plurality of objects that are not entirely independent of each other. These notions will not be considered in this work. 17 The most important difficulties that have arisen for these proposals of analysis have to do with the status that should be granted to entities that have as ‘constituent’ something that is clearly concrete. Attempts have been made to distinguish between the concrete and the abstract through, for example, the distinction between what has spatio-temporal location and what does not, what can enter into causal interactions and what cannot, what is necessary and what is contingent, and also in relation to features of categories of entities. Thus, Hoffman and Rosenkrantz (1994, 182–187) argue that something is concrete if and only if it instantiates a category C such that C possibly has at least one instance with spatial or temporal parts. Assume the particular object b, which is a clear case of a concrete entity. It is, in effect, localized spatio-temporarily, it can enter into causal interactions, it is contingent, and it belongs to a general category of entities that possibly possess instances with spatial or temporal parts. Consider now the set {b}. Is it located where b is located? Does it

1.4 Summary of What Is to Come and some Nomenclature

15

will also be assumed that the distinction satisfies certain theoretical restrictions (cf. Cowling 2017, 70–71): (i) something is concrete if and only if it is not abstract, and is abstract if and only if it is not concrete (although cf. Williamson 2013, 7); (ii) everything is either concrete or abstract; (iii) if something is concrete, then it is necessarily concrete, and if something is abstract, then it is necessarily abstract; (iv) a mereological sum, one of whose parts is concrete, is a concrete entity.

1.4

Summary of What Is to Come and some Nomenclature

§ 8 The central thesis defended in this work is that, at the fundamental level, all that there is are universals and their instantiations. To defend this thesis, it is necessary to justify, in the first place, the claim that there are universals. This is done in Part I (Chaps. 2, 3, 4, and 5, §§ 9–38). As is well known, the philosophical discussion around universals has undergone a profound transformation in the last fifty years, especially since the work of Armstrong (1978a, b, 1983, 1989, 1997). It had been traditional to approach the question about the existence of universals as a problem about the ontological commitments to which our everyday communicative practices compel us, in which it seems that we quantify over universals, or reference is made to universals (see on this approach, Jackson 1977). After Armstrong’s contributions, however, the postulation of universals has come to be seen as having more to do with the roles that must be fulfilled for the explanation of objective resemblances—as these are described in our best scientific theories—as well as causal connections and natural laws. This is the approach followed here. If the reasons for the postulation of universals as part of our ontology have to do with specific theoretical roles that cannot be satisfied by other entities, then it will be essential to pay some attention to the nature of such ‘roles’. It will also be essential to carefully consider whether universals systematically fulfil these roles better than their alternatives. Chap. 2 (§§ 9–15) explains the theoretical roles usually attributed to universal properties or their systematic alternatives. In Chaps. 3, 4, and 5 (§§ 16–38), a comparative evaluation is made between universals and those alternatives. As will be explained later, this relative weighting will not be carried out with respect to all the conceptually possible alternatives but only with respect to the conceptions that seem more relevant, at least for me. These are: resemblance nominalism (Chap. 3, §§ 16–24), the theories of tropes (Chap. 4, §§ 25–31), and theological nominalism (Chap. 5, §§ 32–38). The question that will be considered in Part I, then, is whether there are universals rather than resemblance classes of objects, resemblance classes of tropes, or concepts in the mind of God.

intervene in the causal relationships in which b intervenes? Since, in general, a set depends on its elements, the singleton set {b} is contingent, since it does not exist if b does not. It is not obvious if {b} should count as ‘concrete’ after all. Similar considerations could be made regarding the proposition b exists, and of the property of being identical to b.

16

1 Introduction

The justification of the claim that there are universals, however, leaves open whether universals are immanent or transcendent. This is discussed in Part II (Chaps. 6, 7, 8, 9, and 10, §§ 39–77). Traditionally, those who have postulated the existence of transcendent universals have basically done so through considering the functions that universals would have as the content of thought and language. Since here it is assumed that the properties are those that are required to fulfil the theoretical roles indicated in Chap. 2, it cannot be assumed that for each concept that is the content of our judgments there must be a correlative property. The way in which the existence of transcendent universals will be justified—or, more precisely, the conditional according to which, if something is a universal, it must be a transcendent universal—will be through considering the theoretical functions of universals in three areas: modal metaphysics (Chap. 6, §§ 39–44), the metaphysics of natural laws (Chap. 7, §§ 45–52), and the profile of ontological priority that a universal must have (Chap. 8, §§ 53–59). In each of these cases, it will be shown that the only alternative coherent with the functions attributed to a universal is to suppose that it is a universal whose existence does not depend on having any instantiation. For many, however, these arguments will not be persuasive if the difficulties usually directed against them are not addressed. In Chap. 9 (§§ 60–74), then, such difficulties are considered, and I show that none of them is a reason to reject transcendent universals. Attention will be focused on three principal objections: the assumption that a transcendent universal would have no causal power, nor would make any difference in the causal powers of something; the criticism that the postulation of transcendent universals would be less economical ontologically than the postulation of immanent universals; and the epistemological objection that transcendent universals would not be cognizable by our ordinary cognitive capacities. Part II includes a chapter devoted to considering the question of what are the conditions of identity for transcendent universals (Chap. 10, §§ 75–77). It goes on to show that, contrary to what might have been assumed, transcendent universals make up a unitary structure of necessary existence. The universals are the nodes of such a structure. The postulation of transcendent universals leaves unaddressed the question about how particular objects should be understood. This is dealt with in Part III (Chaps. 11, 12, and 13, §§ 78–94). Given what has been argued, especially in §§ 67–70, the ontology of particular objects must include tropes that are essentially the particular instantiation of a transcendent universal. In the abstract, it would be consistent with the postulation of tropes with these characteristics to understand particular objects with a substratum that is characterized by tropes but also to understand particular objects as trope bundles. In Chap. 11 (§§ 78–81), these theoretical alternatives will be presented, along with the traditional difficulties directed against ontologies of substrata and ontologies of bundles. One of the alternatives seems especially appropriate for solving the problems discussed in Chap. 11: the so-called ‘nuclear theory of bundles of tropes’ (Chap. 12, §§ 82–86). This is a form of a theory of trope bundles, but with two important features that distinguish it from theories of traditional trope bundles. In this conception, the unity of an object is grounded on ontological dependence relations, and there is a distinction between essential and accidental properties for an object. This is achieved with the distinction between two

1.4 Summary of What Is to Come and some Nomenclature

17

different ontological strata in a particular object, called the “core” or “nucleus” and the “periphery”. Despite the advantages offered by the traditional theory of nuclear trope bundles, it has serious problems that have to do crucially with how the relation of ontological dependence must be understood (especially § 86). This makes it convenient to propose a ‘reformed’ nuclear theory (Chap. 13, §§ 87–94) that evades these difficulties by postulating a nucleus consisting of a single trope. The reformed nuclear theory retains the advantages of the traditional nuclear theory, but also achieves the unification of the ontologies of properties and substrata, customarily considered as incompatible with each other. The only nuclear trope is, in effect, a substratum (see § 89). The reformed nuclear theory is a theory that conceives particular objects, at the same time, as bundles of tropes and as constituted by a substratum. We also consider how the reformed nuclear theory can make intelligible sense of structures formed from beginning to end only by relations, without relata (cf. §§ 93–94). Ontologies of this type have been of interest to conceive a multitude of different types of physical entities, both at the quantum level and for the grand scales of the universe. The reformed nuclear theory has enough flexibility to give an adequate treatment of these structures. I have tried to offer in this book a comprehensive defence of a Platonistic ontology of universals that presents the advantages and discusses its counterintuitive consequences. In effect, for an adequate assessment of the feasibility of such an ontology it is necessary to have a synoptic perspective of those theoretical advantages and disadvantages. I do not intend, then, to convince the reader just with some short argument, but with a much more arduous revision of most of the reasons that have been put forward in decades of philosophical discussion –in some cases, centuries of philosophical discussion. So, it has been convenient to include here the exposition of many ideas that for most specialists are hardly new, to gain a broader perspective appropriate for a correct theoretical assessment. Besides, a more comprehensive presentation of the range of questions that involve universals has the additional advantage that the book results more accessible to non-specialist readers that want to be introduced to the ontological debates that is both informative and ‘opinionated’. Throughout this work, reference will be made to predicates (that is, sequences of phonemes or graphemes of some language), concepts (that is, the content of mental states in which we judge or consider something), and universal properties. To avoid confusion, predicates will be presented in double quotes, as usual. Concepts, on the other hand, will be presented in single quotes. A universal property, however, will be presented in italics. In this way, “__ is a cube” is the predicate that is truthfully attributed to something that is a cube; ‘being a cube’ is the concept that a rational subject truly judges of something that is a cube; being a cube, on the other hand, is the universal property that all objects that are cubes instantiate. Analogously, the propositions—that is, the content of what is stated by whoever states a complete

18

1 Introduction

sentence—will also be presented in cursive letters. The proposition that is stated when it is stated that there is a cat is there is a cat.18

References Alvarado, J. T. (2008). ¿Qué son los mundos posibles? Intus legere, 2(1), 1–23. Alvarado, J. T. (2013). Fundación y reducción. Aporía. Revista internacional de investigaciones filosóficas, 6, 59–74. Armstrong, D. M. (1978a). Universals and scientific realism, Volume I, Nominalism and realism. Cambridge: Cambridge University Press. Armstrong, D. M. (1978b). Universals and scientific realism, Volume II, A theory of universals. Cambridge: Cambridge University Press. Armstrong, D. M. (1983). What is a law of nature? Cambridge: Cambridge University Press. Armstrong, D. M. (1989). Universals. An opinionated introduction. Boulder: Westview. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Barnes, E. (2012). Emergence and fundamentality. Mind, 121, 873–901. Bliss, R. (2013). Viciousness and the structures of reality. Philosophical Studies, 166, 399–418. Bliss, R. (2014). Viciousness and circles of ground. Metaphilosophy, 42, 245–256. Correia, F., & Schnieder, B. (2012). Grounding: An opinionated introduction. In Fabrice Correia & Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality. Cambridge: Cambridge University Press, pp. 1–36. Cowling, S. (2017). Abstract entities. London: Routledge. Divers, J. (2002). Possible worlds. London: Routledge. Ehring, D. (2011). Tropes: Properties, objects, and mental causation. Oxford: Oxford University Press. Fine, K. (2012). Guide to ground. In Fabrice Correia & Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality. Cambridge: Cambridge University Press, pp. 37–80. French, S., & Krause, D. (2006). Identity in physics: A historical, philosophical, and formal analysis. Oxford: Clarendon Press. Goodman, N., & Quine, W. V. O. (1947). Steps towards a constructive nominalism. The Journal of Symbolic Logic, 12, 105–122. Hoffman, J., & Rosenkrantz, G. (1994). Substance among other categories. Cambridge: Cambridge University Press. Jackson, F. (1977). Statements about universals. Mind, 86, 427–429. Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Lewis, F. A. (2013). How Aristotle gets by in Metaphysics zeta. Oxford: Oxford University Press. Lewis, D. K., & Langton, R. (1998). Defining ‘intrinsic’. Philosophy and Phenomenological Research, 58, 333–345. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 116–132.

18

In general, I have tried not to include any formulations in the usual symbolism of formal logic. When this has seemed useful for greater precision, expressions in logical symbolism have been included in footnotes. Quantified first-order modal logic is used as a framework for these formulations, without Barcan’s Formula or the Converse Barcan’s Formula.

References

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MacBride, F. (2005). The particular-universal distinction: A dogma of metaphysics? Mind, 114, 565–614. Plato, (1982). República. Traducción, notas e introducción de Conrado Eggers Lan. Madrid: Gredos. Ramsey, F. P. (1925). Universals. Mind, 34, 401–417. Reprinted in R. B. Braithwaite (Ed.), The foundations of mathematics and other logical essays, London: Routledge & Kegan Paul Ltd., 1931, 112–134. Rodriguez-Pereyra, G. (2002). Resemblance nominalism. A solution to the problem of universals. Oxford: Clarendon Press. Rosen, G. (2010). Metaphysical dependence: Grounding and reduction. In B. Hale & A. Hoffman (Eds.), Modality. Metaphysics, logic, and epistemology (pp. 109–135). Oxford: Oxford University Press. Ross, D. (1953). Plato’s theory of ideas. Oxford: Clarendon Press. Simons, P. (1987). Parts. A study in ontology. Oxford: Clarendon Press. Tooley, M. (1987). Causation. A realist approach. Oxford: Clarendon Press. Weatherson, B., & Marshall, D. (2012). Intrinsic vs. extrinsic properties. In E. Zalta (Ed.), Stanford encyclopedia of philosophy. http://plato.stanford.edu/entries/intrinsic-extrinsic/. Accessed on 13.03.15. Wedin, M. V. (2000). Aristotle’s theory of substance. The categories and metaphysics zeta. Oxford: Oxford University Press. Wilhelm, I. (2019). An argument for entity grounding. Analysis. https://doi.org/10.1093/analys/ anz065. Williamson, T. (2013). Modal logic as metaphysics. Oxford: Oxford University Press.

Part I

Universals

Chapter 2

Theoretical Roles for Universals

Abstract Universals have been traditionally postulated because they can satisfy certain theoretical functions. Universals supposedly explain why different objects have the ‘same’ nature (the one over many problem), why the same object has different natures (the many over one problem), why different objects are objectively similar, why objects have the causal powers they have, what are natural laws, and why our epistemic practices of induction seem reliable. This chapter presents these theoretical roles for universals. Of course, those who reject universals argue that other entities can satisfy the same functions. If those alternatives are acceptable, though, depends on their aptitude to play the explanatory role of universals. § 9. There are several reasons why it seems necessary to postulate the existence of universal properties. Of course, there is also a motive of economy that militates against its acceptance, because—apparently—they are not entities with which we have a contact in common experience (although, see §§ 71–74). It seems more reasonable, in principle, to accept only the existence of particular objects. If one is going to postulate something additional to these particular objects, there should be good reasons for doing so. The reasons that are going to be presented in this section are reasons to postulate universals since these are specific theoretical roles that universals can satisfy. The universals adequately explain why the facts in question obtain because universals if they exist—would ground such facts. These are not, of course, all the explanatory roles that the universals or what can fulfill their functions have been expected to satisfy. As indicated, this work will be concentrated on the ‘authentic’ or ‘sparse’ properties that are relevant to explain objective resemblances, causal powers, and natural laws, among others. The theoretical roles that have been attributed to ‘abundant’ properties, such as being the meaning of common names and predicates, the content of thought and, very notably, the nature of mathematical entities, are not discussed here (see for these areas, Orilia and Swoyer 2016, §§ 3.3, 4.1, 4.2; Oliver 1996, 14–20). For the existence of universals to be justified, however, it is not enough for universals to be adequate to fulfill the proposed theoretical roles. The theoretical alternatives to universals also aim to satisfy those same functions, but with a lower © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_2

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ontological ‘cost’. The postulation of universals cannot be considered justified, therefore, as long as an evaluation of the comparative virtues of the alternatives is not made. This comparative examination will be done, however, in the following chapters (see Chaps. 3, 4, and 5, §§ 16–38).

2.1

One Over Many

§ 10. The one over many has traditionally been considered as the problem of universals, that is, the main reason for their acceptance. Suppose that two objects, a1 and a2. Both a1 and a2 are F. So: (1) (2)

a1 is F a2 is F

The verb “to be” is sometimes used to designate identity. If so, from (1) and (2) it would follow that a1 ¼ a2, which is absurd. There is some ‘form’ of identity in this case, but this should not imply the identification of the objects in question. This is why this has been called “the problem of oneness in the multiple”: the multiplicity of different objects must be respected in some way, but also to ensure, in some way, their ‘unification’. From (1) and (2) it follows that: (3)

a1 is F and a2 is F

And this proposition (3) admits the existential generalization: (4)

There is an X such that: a1 is X and a2 is X

More colloquially, from the fact that a1 is F and a2 is F it seems to follow that there is something that both a1 and a2 are. We are admitting, then, the existence of the same entity in both a1 and a2. What is such an entity? A direct answer to that a question is to argue that there is something numerically the same in both cases, that is, a universal. It would seem that admitting the passage from (3) to (4), that is, the quantification on the variable that replaces F in (3), would be accepting a universal at the outset, because it is already admitting ‘something’ in our ontology that must be in the quantification range in (4). Higher order quantification, however, does not prejudge the issue in any way. “Something” that is shared by a1 and a2 could also be interpreted as a class of resemblance to which a1, and a2 belong, or as a class of resemblance of tropes to which belong tropes of F of objects a1 and a2.1 The point is that if one pretends to dispense from universals, one must offer some form of ‘unifying’ different objects. The way of effecting such unification can be varied

1 The entities over which higher-order quantifiers range have also been interpreted as pluralities or sets. An expression of higher-order logic, like [∃X Xx], accordingly, has been interpreted as “there is a plurality X such that x is one of the Xs” or as “there is a set X such that x 2 X”. It is not mandatory that it be interpreted as “there is a property X such that x instantiates X”.

2.1 One Over Many

25

according to the form of nominalism in question, of course, but it would not be appropriate to reject the existence of universals without putting anything that fulfills its functions. The problem of the one over many, as it is formulated here, seems to be applicable to any attribution made to a plurality of objects. If two objects are perfectly spherical, it should be explained how they can share that spherical character. However, if two objects are green and are examined before the year 3000 or are blue and are examined after the year 3000, it also seems that there is ‘something’ that these objects share and that requires an ontological explanation. The first case does not offer too many problems for those who defend authentic properties, since having a perfectly spherical shape seems to be a ‘natural’ property. The situation is very different, however, concerning the second case, since the defender of authentic properties will expressly maintain that there is no such thing as the property of being green and being examined before the year 3000 or being blue and being examined after the year 3000. It would be a disjunctive property that groups very heterogeneous objects. The problem of the one over many, in what has to do with the postulation of authentic properties, is not a problem about our linguistic expressions or our concepts (see Alvarado 2010). When one is presented with propositions like (1) and (2) above, one is inclined to think of linguistic statements in which something is predicated. The variable ‘F’ in ‘a1 is F’ looks like any predicate. Seen in this way, what has been presented as the problem of the one over many seems to be the problem of assigning semantic value to our predicates or content to our concepts. These are important philosophical problems, of course, but it is not the problem that motivates the postulation of authentic universals or what can replace them. For this, it is more accurate to say that the problem of the one over many is a problem about how different objects can have the same nature. That is, if one considers a characteristic or determination of an object, of those that ground objective resemblances between objects and their causal powers, as our best empirical theories seem to show, then it seems evident that many objects could have that same characteristic or determination. The problem of the one over many is the problem whether that plurality can be unified. It is important to make two observations here. In the first place, it can be appreciated that this is not a problem that has to do primarily with the semantic content of our statements or with truthmakers. One way in which the problem of the one over many has been presented, in effect, is as a problem about which is the truthmaker of propositions such as (1) and (2) (see Rodriguez-Pereyra 2002, 14–30; but, Alvarado 2010). If it is true that a1 is F, then there must be something that grounds that a1 is F is true. What works as truthmaker can include a universal, or a class of resemblance, or a concept, and so on. It seems, however, that the problem has a much more general character. It is about an ontological explanation about why a plurality of objects can have the same nature, whether that fact is described or not by some proposition, or some statement of our natural languages. Universals, resemblance classes of objects, resemblance classes of tropes or concepts in the mind of God are solutions to this explanatory problem because they are entities that

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would ground the fact that a plurality of objects have the same nature—understanding in a different way, of course, the character of such ‘common nature’. As the truthmaking relationship between a proposition and the entity that makes it true has been seen as an instance of the grounding relationship (see Correia and Schnieder 2012, 25–28; Fine 2012, 43–46), there is no difficulty in accepting that the problem of the one over many is sometimes a problem about finding the truthmaker for propositions such as (1) or (2). The point is that no proposition is required for the problem to arise. It is important to note, secondly, that the problem of the one over many has a modal dimension. Only in exceptional cases are we faced with different objects that have exactly the same nature. We can find two objects expressly manufactured as of the same type that may be indiscernible to our ordinary perceptive abilities, but if one examines them carefully enough and with the appropriate measuring devices, differences in shape, size or mass will appear. Although, for example, two bottles of wine may seem ‘equal’ to us, it is inevitable that they will not have the same shape, the same mass, or the same size. The differences will be, possibly, imperceptible, but they will exist. Observed with a sufficient level of detail the surfaces of the bottles will no longer appear as ‘smooth’. An atom more or less makes a difference as to what is the exact mass of an object. It would be a cosmic coincidence that these bottles were composed of exactly the same number of atoms and of the same type. Those bottles can count as remarkably similar to each other, therefore, but not with the ‘same nature’. Moreover, unless an immensely improbable chance has occurred, there is no other object in the universe that has exactly the same shape as the computer I am writing in now. How, then, can the problem of the one in the multiple arise regarding the shape of my computer? It appears because, although there is no other object actually with the same shape that my computer has, there could be a plurality of objects with exactly that same shape. For the great majority of the characteristics, the problem of the one over many is not a problem that arises concerning how many objects actually have the same nature, but concerning how many objects could possess the same nature. An exception to this is the case of the fundamental physical particles in the Standard Model. Every electron has exactly a mass of 0.511 MeV/c2, a charge of 1 and a spin of 1/2. Every quark of the type ‘above’ has exactly a mass of 171.2 GeV/c2, a charge of 2/3 and a spin of 1/2. Even if there were no particle with exactly the same natures, however, the mere possibility of a plurality of objects with the same nature is sufficient to generate the explanatory demands of the one over many.

2.2

Many Over One

§ 11. A problem of certain similarity with that of the one over many is ‘the many over one’. It was formulated much more recently than the first by Gonzalo Rodriguez-Pereyra (see 2002, 46–48). It is a problem that seems to press for the

2.3 Objective Resemblances

27

introduction of properties as numerically different entities from the objects that instantiate them. Assume an object a and characteristics F1 and F2 such that: (5) (6)

a is F1 a is F2

As in the case of the problem of the one over many, if one here interpreted the verb “to be” in (5) and (6) as identity, it would result in the absurdity that F1 ¼ F2. The generalization of this result would be the unification of all the characteristics of an object into a single characteristic. If one wants to respect the difference between F1 and F2, it must be possible, in some way, to discriminate a structuring or complexion in particular objects between an ‘individual’ component and another component that contribute the characteristics of the object. This is an important challenge for nominalist theories, since these positions claim, in effect, to be dispensed with properties numerically different from the objects that putatively possess them. It is not, however, a difficulty in principle insurmountable.2 Resemblance nominalism, for example, posits classes of resemblance between objects that should satisfy the theoretical roles assigned to universals. If an object is a perfect sphere and measures 1 m high, this fact will be grounded on the resemblance of that object to the other members of the resemblance class assigned to spheres and by the resemblance of that object to the other members of the resemblance class assigned to 1 m high. The difference between these classes of resemblance is sufficient to explain why the fact that it is spherical is different from the fact that it is 1 m high. As in the case of the problem of the one over many, this is not a problem about linguistic attributions to the same object. What should be understood by the variables ‘F1’ and ‘F2’ in (5) and (6) is a characteristic of those that should be grounded on authentic properties. It is not a specifically semantic problem but an ontological one.

2.3

Objective Resemblances

§ 12. One of the theoretical functions that have been assigned to universals is to ground objective similarities between objects. Resemblance, as it is ordinarily understood, is an internal relationship, that is, it is a relationship grounded on the intrinsic natures of the relata. Since the relata possess specific intrinsic properties or a precise intrinsic nature, it automatically follows that they fall under the internal relationship in question. If a measures 1 m and b measures two meters, then nothing else is required ontologically to make b taller than a. Being taller than is a typical With the exception, perhaps, of the so-called “ostrich nominalism” or “priority nominalism” in which it is a primitive ontological fact that, for example, a is F1 and a is F2, without the introduction of the existence of nothing that has to ground such facts. Even in this case, however, the difference between F1 and F2 is, by itself, an ontologically primitive fact. One could argue that this would be to postulate too many primitives or primitives that are too obscure or too implausible (see Lewis 1983, 23), but there does not seem to be a problem of coherence in this position. 2

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internal relationship that automatically arises once there are facts about how high the different objects are. Similarity and dissimilarity also seem to be internal relations that arise automatically once the facts about the intrinsic nature of the objects to be related are established. If two objects have a perfectly spherical shape, it follows that they are similar to each other—at least regarding their shape. Universal properties ground facts of similarity, for if two objects instantiate the same universal, it follows automatically that they will be perfectly similar to each other in respect of this universal. In the different forms of resemblance nominalism, this order in the direction of grounding is inverted. It is not the properties of objects that ground the similarities or dissimilarities between them, but rather it is the facts of similarity of some objects with others that ground which characteristics can be attributed to them as intrinsic features of such objects (see Armstrong 1978a, 50–51, 1989, 43–47; RodriguezPereyra 2001, 2002, 87–90; § 17, 2004). Although a nominalist must generally maintain that every dyadic relation must be reduced to a class of resemblance of ordered pairs, resemblance relations are postulated as primitive facts that are not subject to the same restrictions as the other relations. These facts of similarity are not grounded on anything.3 In any metaphysical theory, some or other fundamental ontological facts must be postulated, as well as certain primitive not analysable notions. For the nominalist, the facts of primitive resemblance are what ground the rest, so clearly resemblance cannot be an internal relationship. On the contrary, what we can later describe as the intrinsic nature of an object must be grounded on what similarities or dissimilarities this object has with others. Something analogous happens with the theories of tropes. Usually, it has been argued by its proponents that universals can be replaced by resemblance classes of tropes. In more traditional theories, however, it had been argued that tropes possess an intrinsic qualitative nature. The similarities and dissimilarities between tropes would then be internal relations between those tropes grounded on such natures (see, for example, Campbell 1990, 37, 38, 59–60; Williams 1953a, b). This conception, however, has encountered a multitude of difficulties (see Moreland 2001, 53–71; Ehring 2011, 175–187). These difficulties will be discussed later (see §§ 27–29). Due to these difficulties, some friends of tropes have come to postulate resemblance classes of the same character as the resemblance classes proposed by the nominalists. The relation of resemblance is here also primitive and external. It might seem, then, that there is not an explanatory theoretical role to be satisfied concerning objective similarities and dissimilarities between the different alternatives in metaphysics of properties. If one postulates universals, then facts about similarity and dissimilarity between objects will be grounded on such universals and their instantiations. If one postulates a form of resemblance nominalism or 3

Rodriguez-Pereyra has said that the truthmaker of a statement of resemblance of different objects a and b is “just a and b together” (Rodriguez-Pereyra 2001, 403; see also 2002, 110–121). This is not to say that facts of resemblance are grounded in the ‘nature’ of a and the ‘nature’ of b. Objects a and b ‘together’ resemble each other and are the ground of the properties that could be attributed to them.

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resemblance classes of tropes, then the facts of similarity are the ground of what fulfills the functions of a universal property. Thus, the question seems to be whether to consider the facts of similarity as something grounded on properties or as the ground of the natures shared by objects. There would not, therefore, really be the same theoretical function that must be satisfied both by universals and by their alternatives. But this is a mistake. Facts of objective similarity have a peculiar structure that is inscribed in the systematic relations between determinate and determinable properties. These facts impose much stronger explanatory demands for all alternatives in the debate in metaphysics of properties (see Alvarado 2014). The relation of ‘determination’ which can obtain between two properties is like the relation between the property of having 10 kgr and the property of having mass or as the relation between the property of being perfectly spherical and the property of having a form. A determined property is a ‘specification’ of a determinable property. The same property can be determinable and determinate at the same time concerning different properties. For example, being red is a determination of the property of having a color, but is determinable concerning the property of having exactly one specific shade of scarlet red. Some properties are not determinable concerning any other. They have been called “super-determinate”. There are also properties that are not a determination of any property. They have been called “super-determinable”. The relation of determination among properties imposes important restrictions that seem completely objective (see Armstrong 1978b, 111–113, 1997, 47–63; Funkhouser 2006, 548–549, 2014, 16–54). Let P1, P2, . . ., Pn be determinable properties of the same level with respect to the determinable properties Q1, Q2, . . ., Qn. So, the following principles seem to be valid: [Upward Necessitation]

[Downward Necessitation]

If something instantiates the determinate property P, then it instantiates the determinable properties Q1, Q2, . . ., Qn of which P is a determination. If something instantiates the determinable property Q, then it instantiates one and only one of the determinate properties P1, P2, . . ., Pn under Q.

Note that Downward Necessitation not only imposes that whatever instantiates the determinable property Q must instantiate at least one of the determinate properties P1, P2, . . ., Pn, but it must also be instantiated at most one of them. For this reason, what is presented here as the principle of Downward Necessitation has often been characterized as a couple of principles of ‘downward necessitation’ and of ‘exclusion’. In effect, determinate properties of the same level under a determinable seem to metaphysically ‘repel’ each other. If something has exactly 10 gr of mass, then it cannot have 11 gr of mass. If a whole surface is blue, then it cannot be red. The most relevant feature for the facts of objective similarity, however, does not have to do with these necessary connections, but with the so-called ‘dimensions of determination’ linked to a determinable property (see Funkhouser 2006, 551–556). A determinable property has—at least, usually—a fixed number of ‘dimensions’ in which it

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can be determined.4 In the simplest cases, these are properties with a single dimension. Each of these dimensions can be represented by a set. For example, the infinite determinate properties of mass can be put into bijection with the set of real numbers. Each real number can represent a super-determinate mass property. In other cases, a dimension of determination can be represented by the set of natural numbers. For color, it has usually been assumed that dimensions of determination of tone, brightness, and saturation are relevant. For sounds, it has been assumed that dimensions of determination of tone, volume, and timbre are relevant. A determinable property of being a triangle, on the other hand, has as dimensions of determination the lengths of each of its three sides. It is especially interesting here that each determinable property can then be represented as a region in an n-dimensional space, according to how many are its dimensions of determination. The determinable property of being a triangle, for example, can be represented as a three-dimensional space. A region in this space represents a determinable property of being a triangle—such as an equilateral triangle—but not yet a super-determinate property. Each point in this space represents a super-determinate property, with maximally specific lengths for each of its three sides. Something analogous can be done for each determinable property and its determinations. Natural science has exploited those structures of determination for the formulation of natural laws. Consider this famous principle of mechanics: (7)

f¼ma

Here the variables ‘f’ and ‘a’ have as range vectors of force and acceleration, respectively, while the variable ‘m’ has as range masses. What this natural law does is not to correlate directly determinate properties of mass, acceleration, and force. What is directly correlated by it are determinable properties of mass, acceleration, and force (see § 47). It does so by exploiting the possibility of representing such determinable properties in certain spaces in such a way that a real number can be assigned to each given mass and each acceleration can be assigned a vector with a scalar component and a direction. The determinable property of having mass is represented by a one-dimensional space, and the determinable property of having acceleration is represented by a vector space. The product of that real number and that vector has another vector as a value that, due again to the possibility of representation of the forces in a vector space, represents exactly a force vector with a scalar component and a direction component. The operations on real numbers and vectors—abstract entities—have importance to discover determinate properties because a bijection between such determinate properties and such abstract objects can be established. 4 Not in all cases a determinable property has a finite number of determination dimensions. This happens in properties such as, for example, having a form. Are determinations of this property being a triangle, being a square, . . ., being a chiliagon, and so on. Each side of a polygon generates a dimension of determination. The property of being a square has four dimensions of determination, but the property of being a chiliagon has a thousand dimensions of determination. All of these properties are, however, determinations of having a form.

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The existence of these isomorphisms between mathematical structures—such as that of real numbers or vector spaces—with structures for different determinate properties determined under a determinable, also allows us to specify objective facts of similarity between different objects. Typically, the ‘degree’ of similarity between objects that instantiate determinate properties under the same determinable is given by the ‘distance’ between those properties in the space generated by the determinable property. For example, an object that has a mass of 10 gr is more similar to an object that has a mass of 11 gr than to an object that has a mass of 1 kg. These ‘degrees’ of similarity seem to be perfectly objective and are representable by the absolute values |11–10| and |1000–10|, respectively. These degrees of similarity are apparent when it comes to determinate properties under a uni-dimensional determinable, as is the case with the determinable mass. The degrees of similarity are less noticeable when it comes to determinables with several dimensions of determination. Even in these cases, nevertheless, the comparability between different determinate properties has to do with their ‘distance’ in the various dimensions of determination. It has been frequent that the philosophical discussion in metaphysics of properties has been centered on the cases of ‘perfect resemblance’ which is that which occurs when two objects have exactly the ‘same’ nature—however such ‘sameness’ is understood. It is the kind of similarity that exists between two objects when, for example, both have the ‘same’ perfectly spherical shape. When attention is concentrated only in cases of this kind, it seems indeed, that the difference between theories of universals, nominalist theories, and trope theories consists only of whether the similarity will be fundamental or derivative. ‘Perfect’ similarity, nonetheless, is a limiting case of a much more complex phenomenon. The question of objective similarity does not seem to be just a problem with an adequate selection of primitives in the theory. Objective similarities—plurally—seem to be connected with the structures of determination between determinate and determinable properties. The problem in the metaphysics of properties, then, is to explain those structures of determination ontologically. And this is a problem that must be addressed by any of the positions in dispute. It is not enough, especially, for the nominalist to maintain that the facts of similarity are primitive, ungrounded external relations. Whatever the nature of the primitive facts of resemblance they must explain how to have 10 gr of mass and to have 20 gr of mass are determinations of the same determinable and how is that there is a precise degree of similarity between the objects that have those masses that is isomorphic to the ratio 10:20. There is, then, a theoretical function that both the universals and the different alternatives proposed by nominalists and defenders of tropes must be able to resolve concerning objective similarities and dissimilarities.

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Causality

§ 13. It seems intuitively obvious that the properties of an object affect the causal connections in which that object may be involved (see Armstrong 1978a, 22–24, 42–43, 56–57, 75). It has been common to argue that causality is a relationship between events and not between objects. For example, it would seem that if the explosion of a bomb causes the destruction of a car, it would not be accurate to say that “the bomb destroyed the car”. A bomb is an object that integrates an event—a blast—that is what causes another event—the destruction of the car. The car is also merely a component of this destruction event. A fairly standard way of understanding the nature of an event is as the instantiation of a property in an object at a time, or of a relationship between several objects at a time (see Kim 1976; Alvarado 2013). If causality is a relationship between events and events are instantiations of properties, then it seems evident that the causal relationship requires the existence of properties that is what may or may not be instantiated for the configuration of an event. At least it seems to need something that can fulfill the functions attributed to a property. This general intuition, however, requires many precisions and qualifications. Different conceptions about the causal relationship have been defended and not in all those theories, the demands regarding metaphysics of properties are equal. In a general way, it can be affirmed that the interaction between metaphysics of properties and metaphysics of causality is a factor of great relevance for the adjudication of the debates regarding both the nature of properties and the nature of causality. Of course, one cannot do here a review of all the theories about causality that have been proposed, but only a very general examination of some of the most important of them. These different conceptions impose different restrictions and problems for positions in the metaphysics of properties. The views regarding causality in which the attention will be concentrated here are the theories of regularity, the counterfactual theories, and the non-reductivist theories.5 In regularity theories, causal facts are reduced to facts about the regularity of types of events (see Psillos 2002, 19–79, 2009). The core of the theory is a conception like this: the event c (of type C) causes the event e (of type E) if and only if: (i) c is spatial-temporarily contiguous to e, (ii) c is temporally prior to e, and (iii) all events of type C are regularly followed by events of type E. Although the regularity theory usually includes a clause such as (i) requiring the ‘local’ character of the causal relation, causal facts are not really ‘local’ facts concerning what happens with two events, i. e., concerning what happens in the spacetime region which those events are occupying. There could be no causal relationship between events c and e if such events were not instances of certain types and if there were no

5

Therefore, conceptions as important as the probabilistic, the interventionist, and those that try to reduce the causality to processes of transmission of conserved quantities are not going to be considered. I hope that the discussion restricted to the positions indicated above will be sufficiently illustrative of the problems and theoretical functions that are expected of the universals and their alternatives.

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general fact concerning the regular succession of events of one type with the events of the other. For example, according to Newtonian theory, any two objects with mass attract each other—the measure of this attraction is a force that is directly proportional to these masses and inversely proportional to the square of the distances between those bodies. Our intuition is that the masses of two objects located at a distance cause these objects to attract each other. For a defender of the theory of regularity, such a causal event reduces to the totality of the behavior of all other objects with a mass in the world. Suppose a possible world w in which the Earth and the Moon exist as they exist in the actual world. The Moon orbits around the Earth just as it does in the actual world. The phenomena of the Earth are just like the phenomena of the actual world. Objects fall, and the measure of their acceleration is just like what they have in the actual world—approximately 9.8 ms/sec2. The phenomena on the Moon are just as in the actual world as well. It happens, however, that in this world all the rest of the universe behaves very differently. For example, the rest of the objects with a mass of w repel with force directly proportional to the masses and inversely proportional to the square of the distances between them. It can be assumed that the Earth and the Moon are sufficiently far away from the other bodies so that their interaction with them is negligible. For a defender of the theory of regularity, although in w what happens in the Earth and in the Moon is ‘equal’ to what happens in the actual world, it would not be the case that the masses of the Earth and the Moon and their distance would cause the Moon to orbit the way it does. Nor would it be the case that the Earth’s gravity would cause objects to fall from height. In w it is not a regularity what actually is a regularity, then, there are not in w the same causal relations that exist in the actual world although—at least concerning a region of that world—events at that world and the actual world are indiscernible. It would seem, then, that in a conception of this type universal properties or what fulfills their functions are indispensable since one event can only cause another due to the type of event in question. But these appearances are deceptive. The theory of regularity does not impose substantive restrictions on metaphysics of properties. The theory of regularity requires that events cause and effect to be of a certain type, but there is no restriction on the nature of this type. Suppose that in a world w the following regularity obtains: every event of there being a cat and there being a dog is followed by an event of a meow of a cat or a bark of a dog. One can suppose that the temporal posteriority requirements of the event effect and spacetime contiguity are satisfied here. An event of there being a cat and a dog is an event that will occupy a precise spacetime region. It may be a spatially disconnected region—it will include the region occupied by the cat and the region occupied by the dog—but we do not have a priori reasons to prohibit disconnected events of this type. The event that happens if there is a cat meowing or a dog barking will also occupy a spacetime region. It can be assumed that it will happen in an instant immediately following the time of the event cause and in a contiguous spacetime region—because it is contiguous to the region occupied by the cat or because it is contiguous with the region occupied by the dog. In this world it is a regularity that the existence of cats and dogs causes the barking of dogs, but not because of dogs’ special interest in cats,

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but simply because a dog bark is trivially a cat meow or a dog bark, and it is a regularity that the existence of a cat and a dog is followed by a cat’s meow or a dog’s bark. One can here protest that the property of meowing a cat or barking a dog is not an authentic property. It is a disjunctive property that groups very heterogeneous events together. It has been usual for defenders of authentic properties to reject the existence of disjunctive properties (see Armstrong 1978b, 19–29, 1989, 82–84), since there does not seem to be any objective similarity between what instantiate such properties. It seems gratuitous to say that a cat’s meow resembles a dog’s bark because of both falling under the ‘property’ of meowing a cat or barking a dog. The point here is whether this should have relevance to the regularity theory of causality. And it seems that it is not relevant. For a defender of the theory of regularity, the causal facts are facts about regularities—with the other usual qualifications. It does not matter that the respects or characteristics relevant to these regularities are ‘natural’. We are inclined to think that it is “natural” properties that must enter the causal connections because we are used to thinking that such properties actually confer some power to cause something else. But from a regularist perspective, such an inclination is prejudiced. A ‘causal power’ is nothing more than the metaphysical possibility of entering into a regularity, which seems guaranteed for any predicate in which we can think. No matter what ‘being F’ is, it seems that—except for unusual cases—there are worlds where everything that is F in that world is followed by a G, no matter what ‘being G’ is. It turns out, then, that in a theory of regularity the assignment of ‘causal powers’ is extremely cheap ontologically. We are also inclined to think that what gives a causal power to an object must be something, in some way, located in that object or intrinsic to that object. But this, again, is for the defender of the theory of regularity a prejudice. For him, causal facts are not ‘local’ facts about what happens strictly between two events, but facts about complete possible worlds in which a regularity is given or not. When the problems of the one over many, the many over one and the question of objective resemblances have been dealt with, it has been found that those who try to propose something alternative to universals face important explanatory requirements. They must postulate some ontological structure that, for example, effectively allows to ‘unify’ a multitude of objects numerically different from each other, or must offer some structuring of facts of similarity that provides for the differentiation between determinable and determinate properties, with their mutual relations. When it comes to explaining why something causes something and one is guided by the theory of regularity, on the other hand, there are no such demands. One could postulate a putative property for each class of objects, or each predicate of our languages, and this would not make any difference. Any way to generate a ‘type’ of events is as good as any other. If there is regularity between these types of events— with the other usual restrictions—there will be a causal connection. It is not

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necessary for the nominalist or the defender of tropes to generate something that should satisfy theoretical roles ordinarily attributed to an authentic universal.6 A second tradition to understand the nature of the causal relationship is the family of counterfactual theories of causality. It is an approach inaugurated by David Lewis (1973b, variations in 2004) that has enjoyed great popularity in the last 40 years (see for a general overview, Collins et al. 2004; Paul 2009). The core of the counterfactual theory is a conception of this type: the event e depends causally on the event c if and only if (i) if c did not exist, then e could not exist; and (ii) if c existed, then e could exist. Causation is the ancestral relationship of causal dependence.7 The same year that Lewis published this theory of causality, he also published his theory of counterfactual conditionals (see 1973a) which is the theoretical tool that is here operating. In the semantics of Lewis, a counterfactual conditional if p were the case, then q could be the case is true in w if and only if in all the closest possible worlds to w, not p obtains or q obtains (see Lewis 1973a, 1–43). A possible world is more or less ‘close’ to another according to their mutual global similarity. It is a presupposition of Lewis’s theory that a ‘metric’ of global similarity can be established between possible worlds by which relative distances between possible worlds can be built according to such a metric. How similar two worlds are to each other is something that suffers from a certain vagueness, because different respects of similarity must be weighed—even if these similarities are objective. This vagueness is reflected in the truth conditions of the counterfactual conditionals that will also have oscillating truth values according to which worlds seem most relevant for the evaluation of a conditional. This is not the time to go into more detail about this semantics. According to the counterfactual theory, causal facts are reducible to facts about counterfactual dependencies. And these facts about counterfactual dependencies are reducible to facts about the similarities and dissimilarities between possible worlds. Causal facts are reducible, then, to facts about the similarity and dissimilarity between possible worlds. Indeed, from the perspective of the counterfactual theory, the fact that the event c causes the event e is reduced to the fact that: (i) if c occurs then e could occur, and (ii) if c does not occur, then e could not occur. One is inclined to accept this counterfactual dependence but grounded on a previous causal relationship. In the 6 Of course, one could argue that the fact that the regularist theory of causality does not require ‘natural’ properties as that which enters into regularities makes the theory much less plausible since it seems to leave room for spurious causal connections. It is not our business here, however, to develop a more credible regularity theory of causality and consider what other requirements, apart from the usual ones, it should have. 7 These complications are due to certain peculiarities of the semantics of the counterfactual conditionals, at least according to Lewis’s semantics (see 1973a, 1–43). Lewis expresses the counterfactual if p were to occur, then q could occur as [p⃞ ! q]. This conditional, unlike the material implication and the strict implication, is not transitive. Causality, on the other hand, is. One cannot, then, try to reduce the causal facts directly to facts about counterfactual dependencies. It is said that the events e1 and e2 are connected by the ancestral of causal dependence if e1 causally depends on e2 or e1 causally depends on e3, and e3 causally depends on e2, or e1 causally depends on e3, e3 causally depends on . . . that causally depends on e2, etc.

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counterfactual theory, this order of grounding is inverted. There is a causal connection because conditionals (i) and (ii) are true. According to Lewisian semantics, (i) and (ii) are true in a possible world w because, if you consider all the closest worlds to w it will be found that in all those worlds in which the event c occurs, the event e also occurs, and in all the worlds where the event c does not occur, neither does the event e. It is not, again, that the existence of the causal relation determines facts about the similarity or dissimilarity between possible worlds. It does not determine the ‘closeness’ or ‘remoteness’ of possible worlds in the similarity metric. Counterfactual dependence must already presuppose previous facts of similarity and dissimilarity between possible worlds and, with it, a metric of similarity between them. A causal fact appears as a result of specific correlations and invariances in different worlds in certain ‘spheres of possibility’—that is, if possible worlds are supposed to form a modal space ordered by the similarity metric; that is, having a given world as its center, there is a sphere of worlds closer to that world. Facts of similarity or dissimilarity between possible worlds are crucial for this perspective about causality since they are what ultimately ground the causal facts. Since objective similarity is a matter of such importance for the different alternatives in metaphysics of properties, it is not surprising that, then, the nature of properties is crucial for the counterfactual perspective of causality. It is very relevant also for these facts of similarity or dissimilarity between possible worlds which nature is postulated for such worlds, but this is a question that cannot be dealt with at this point.8 In principle, possible worlds are similar or dissimilar to each other considered globally by the distribution of intrinsic properties in their objects and the external relations between these objects. For properties and relationships to be comparable, it is essential that there are such properties and relations. For a defender of universals, there is no difficulty here. Possible worlds are entirely determined, at least with respect to their qualitative aspects by which they can be compared with other possible worlds, by the pattern of instantiations of universals, whether monadic or polyadic. That same function should be satisfied by resemblance classes of objects, or tropes, or by concepts in the mind of God. A nominalist cannot merely reject the existence of universals without proposing anything that fulfills its functions. From the perspective of the counterfactual theory of causality, such a thing would imply throwing away causal facts along with universals. For the theories of regularity, there is no great problem, as explained above, because there are regularities between very unnatural types of events, not acceptable for defenders of authentic properties. The

Different ways of understanding the nature of possible worlds will be discussed below (see §§ 42–44). There are significant differences between them because, for example, in most of them, the same object exists in different possible worlds. Possible worlds, then, can be compared not only with respect to what purely qualitative characteristics they possess but also about what objects exist in them. Two possible worlds could differ, for example, solely with respect to what individuals exist in them and no qualitative respect. (In the following, a ‘purely qualitative’ property will designate a property that does not involve an individual. A non-purely qualitative property is, for example, to be identical with Socrates, which includes Socrates and, therefore, ontologically depends on Socrates. This property does not exist in possible worlds in which Socrates does not exist.) 8

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situation here, on the other hand, is very different. Whether resemblance classes of objects, resemblance classes of tropes, or concepts in the mind of God are postulated, these entities must be able to ground facts of objective similarity between possible worlds by determining the global qualitative character of each world. A third family of theories about causality is that of non-reductivist theories. These are theories that do not intend to reduce or ground causal facts on other more basic facts, such as regularities of events or counterfactual dependencies (see Anscombe 1971; Tooley 1987, 173–314). Causal facts are primitive, not grounded on anything. In the conceptions of this type, causal facts ground regularities or counterfactual dependencies. It is because there is a causal relation between events of type C with events of type E that it always happens that an event of type E always follows an event of type C. In the same way, it is because there is a causal relationship between the events c and e, that the event e counterfactually depends on the event c. It has been traditional since Hume to assume that a causal relationship between two events can only obtain if there is a general law that connects events of the first type with events of the second type. This assumption has, of course, been linked to the regularist views of causality,9 but it is not exclusive of such theories. If causal facts are not reduced or grounded on regularities, then it is not necessary for the existence of a causal relation that there is a general law that connects events of the type of the cause with events of the type of the effect. Nor, however, is the non-reductivist conception of causality incompatible with the existence of natural laws of a general nature under which causal relations occur. For what matters here, if one adopts a non-reductivist conception of causality, the intuition that properties confer causal powers to the objects that possess them imposes substantive restrictions on metaphysics of properties. In a non-reductivist conception of causality, in effect, causal connections between two events must be local facts that involve the objects that enter into such connections. If an object endowed with mass, for example, is attracted to another at a distance, this causal relation must be dependent on the properties possessed by the objects in question. And the fact that those objects have such properties must be something that intrinsically involves such objects.10 It is difficult to reconcile these requirements with theories in which the nature of an object is grounded on ‘extrinsic’ facts. In resemblance nominalism, for example, the fact that x is F is grounded on the fact that x resembles such and such other objects which form the respective class of resemblance that fulfills the functions of the universal F. In the nominalism of predicates, for example, the fact that x is F is grounded on the fact that it can be

9

Counterfactual theories, on the other hand, have always left room for singular causal connections not linked to general laws. Of course, the fact that two possible worlds have the same natural laws— however, such laws are understood—is important for the similarity between such worlds (see Lewis 1979). Thus, if according to a general law, in w an event c should have happened followed by an event e, it is reasonable to think that in all the closest possible worlds to w, if c occurs e occurs, and if c does not occur, neither does e. Possible worlds in which the law does not obtain will be farther away. 10 Note here what has been explained above concerning the notion of ‘intrinsic property’. See § 5.

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truly asserted that “x is F”. In both cases, the fact that x be F is not grounded on what happens with x in the region of spacetime occupied by x in which it is F. The fact that x is F under predicate nominalism also depends on what happens with other objects occupying other regions of spacetime, or what happens with a cultural entity such as a language. In any of these forms of nominalism, that x is F does not count as a fact ‘intrinsic’ to F. It is then difficult to understand how it is that the character of being F should have an impact on the causal connections in which the object x may be involved. In the case of other theories, such as those that posit classes of tropes similar to each other or concepts in the mind of God, this problem acquires different forms that will not be addressed now. In any case, any theory in metaphysics of properties must explain how a property—or what is put instead—has an impact on the causal powers of an object. As you can see, then, there is not a problem of causality in metaphysics of properties, but various problems of varying importance according to the type of conception about the causal relationship. If one defends a regularist theory of causality, there is no special requirement that properties should meet. If one supports a counterfactual theory of causality, all that is demanded is that there are properties that ground objective similarities. If one defends, finally, a non-reductivist theory of causality, it is necessary to explain how the properties ground the causal powers of objects as something ‘intrinsic’ to the object. The non-reductivist theories are, then, by far the most demanding.

2.5

Natural Laws

§ 14. Another theoretical function assigned to universal properties has to do with the ontology of natural laws. Traditionally, natural laws have been conceived as regularities of events—the same regularities to which causal relationships should be reduced. This family of conceptions, however, has been subjected to intense criticism for some 40 years. These criticisms have focused on the intuitive difference between an authentic natural law and a mere accidental regularity (see Armstrong 1983, 11–73; Tooley 1987, 43–66; Carroll 1994, 28–85). It could happen, for example, that no human being has a height higher than 1.70 ms. In a possible world where this happens, it would be a regularity that no human being measures more than 1.70 ms. It seems unlikely, however, to think that it is a natural law in that world that every human being should measure less than 1.70 ms because some human being could perfectly measure more than 1.70 ms. Our intuition is that an authentic natural law is not only an enunciation of what actually happens, but it is something with a specific modal force, capable of grounding counterfactual conditionals and that “regulates” not only what happens but also what could or could not happen—at least for some space of possibilities. These difficulties have led to the postulation of theories of natural laws in which these are universal or constructions of universals. A well-known theory defended simultaneously by Fred Dretske (1977), Michael Tooley (1977, 1987, 67–141) and David Armstrong (1983,

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75–173), holds that natural laws are relations of a higher order among universals. It is not that it happens that all events of one type are followed by events of another type. It is that the universal that is instantiated when an event of the first type occurs is ‘nomologically’ connected with the instantiation of another universal, or ‘necessitates’ the instantiation of another universal that is the one that is instantiated when an event of the second type occurs. Of course, given this connection between universals, a regularity obtains between the events of their instantiations. This regularity is, however, grounded ontologically on something more robust. In this conception, natural laws effectively “regulate” events and ground counterfactual conditionals. These theories have been criticized for several reasons. Dretske, Tooley, and Armstrong have argued that natural laws, even when though they generate a space of possibilities in which laws have a regulating function, are contingent. A natural law is constituted by universals—be they monadic or relational—but what constitutes the nomological ‘connection’ or the ‘necessity’ of the instantiation of a universal is a relational universal of a higher order. This relationship may not connect those universals. The connection between the universals—the nomological or necessitating relation—is, to put it in some way, ‘extrinsic’ to the connected universals. Because of this, natural laws are contingent. For many, these characteristics of natural laws according to the theory of Dretske, Tooley, and Armstrong are unreasonable. This will be discussed with detention below (see § 51). Due to these difficulties, other theories have been proposed in which natural laws are reduced to universals, but not by a nomological relation of extrinsic necessity between the connected universals. In these theories, natural laws are eliminated by primitive causal powers, or are reduced to primitive causal powers (see Mumford 2004, 127–205; Bird 2007, 43–65, 189–203). Natural laws are not here relations of a higher order between universals, but merely the causal power or disposition conferred by a property. The law is nothing ‘extrinsic’ to a property, but the unfolding of the intrinsic nature of the property in question. It is not necessary here to discuss these alternatives in detail. It is obvious that any of them requires an ontology of universals. If there are no universals, then obviously there are no natural laws as necessitating relations between universals or as primitive causal powers conferred by universals. These theories that depend on universals are not the only non-Humean theories (see, for example, Lange 2009), but they seem the most likely.11 The point is that if one does not want to postulate universals, it is essential to offer a reasonable theory of natural laws that is capable of solving all the difficulties that have been appreciated in the old theory of regularity. There seem to be two significant strategies here that those who reject universals could adopt. In the first place, one could try to develop an ontology of natural laws that, following the 11 Lange (2009), in particular, has argued that natural laws are sets of primitive counterfactual facts. Usually, it is supposed that the existence of a natural law—as a relation of higher-order of necessitation between universals, or as a primitive causal power—grounds counterfactual facts. Lange reverses this order of ontological grounding. It is also a non-Humean theory in which natural laws have a true regulative function. Lange’s position has the difficulty, nevertheless, that primitive counterfactual facts seem extremely unlikely.

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central lines of the theories mentioned above, substitute the universal properties for other entities that are more acceptable. A natural law should be a class of similarity of relational tropes of ‘necessitation’ between tropes of certain kinds, or it should be a sort of similarity of ordered pairs of objects belonging to specific resemblance classes (see § 21); or it should be a concept in the mind of God correctly attributed to other concepts in the mind of God. It is not apparent that these alternatives are adequate to replicate natural laws according to a non-Humean conception. Second, one might try to develop a Humean theory of natural laws in which laws are, in effect, regularities of events, perhaps with specific special qualifications. This is what Lewis has tried to do (see 1973a, 73) by arguing that natural laws are the axioms of those theories that describe everything, and that achieve the best balance of simplicity and informational content. It turns out, then, that a defender of universals has no difficulty in adopting any of the non-Humean theories indicated above. Whoever does not want to accept universals must replicate the explanatory advantages of universals for the ontology of natural laws, or else it must show the viability of old Humean conceptions or some sophistication of them.

2.6

Inductive Practices

§ 15. Another theoretical function that has been assigned to universals is to explain the reliability of our inductive practices. Since the eighteenth century the so-called “problem of induction” has been a central epistemological issue. Under a perspective that can be characterized in a general way as “Humean”, it is a challenging problem to solve. Why having observed a finite number of particular cases in which an F is also a G, can one be warranted epistemologically to believe that every F is a G? The thesis that everything supervenes on local facts about the distribution of intrinsic properties at different points of spacetime and external relations of spatiotemporal distance between such points has been termed as “Humean supervenience” (see Lewis 1986b, ix–xvi). Reality is a vast mosaic of local facts. The ‘global’ features of a world, such as what natural laws exist in that world, or what causal relationships exist in that world, are grounded on local facts. In Lewis’s metaphysics, moreover, the modal traits that can be attributed to a world are based on a greater mosaic of local facts in different possible worlds (see Lewis 1986a, 5–20). There are no necessary connections between different existents. There is nothing in a region of spacetime that can ‘make it necessary’ things to be different in another region of spacetime. From a perspective of Humean supervenience, how come we get to know that all F is G? Nothing we can observe in a localized spacetime region can be a guarantee of what happens in other regions. Nothing we have seen in the past may make it necessary or more likely that things in the future will be in some way more than another. And while we do not know what happens in the future, we simply do not have how to know if a regularity in the world is fulfilled or not. From a Humean

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supervenience perspective, for example, to suppose that nature behaves uniformly would be completely unmotivated. For instance, before the sun rises over the horizon tomorrow, it is rational to assume that it is as probable that it will come out as it is that it does not. Earth is as likely to keep rotating as it does on its axis as it does not. Moreover, the objective probability that the Earth will continue to rotate tomorrow on its axis in the way it has done up to now, that is, making the Sun show the precise trajectory that we expect it to show tomorrow according to our best astronomical calculations, is infinitesimally small, tending to zero. That trajectory is as probable as any of the infinite other possible trajectories. From the perspective of Humean supervenience, the success we have had so far in predicting that the Sun rises and sets every day is the product of a coincidence of a cosmic scale. For a Humean, there are infinite other possible worlds in which everything that has happened so far is just like in the actual world, but tomorrow the Sun shows any of the other infinite different trajectories. Precisely, that our world is one of the very few in which the Sun behaves in a temporarily uniform manner is something that should surprise us. It has been highlighted, then, that, if the fact that all F is also a G is a coincidence, it seems inevitable the skepticism about our inductive practices (see Armstrong 1983, 52–59; Foster 1982/3; BonJour 1998, 187–216). The postulation of universal properties, however, offers a straightforward explanation of the success of our inductions. Now they are not coincidences. A universal property is numerically the same in all its instances. The causal powers conferred by a universal property are the same in all its instances, at least, if one supposes that natural laws are relations between universals or are based on universals. When one projects a regularity given what has been observed in a sufficient number of particular cases, what is being done is to explain what has been seen in all these cases by the presence of the same universal property. Inductions are here cases of inferences to the best explanation.12 If one supposes that there are universals and one assumes that such universals determine definite causal powers, then the most sensible thing is to think that there will be a regularity. This is an important advantage of theories of universals. Any alternative theory in metaphysics of properties must deal with the problem of induction. For doing this, one might prefer as a strategy to show how resemblance classes of objects, or resemblance classes of tropes, or concepts in the mind of God can replicate the epistemological virtues of universals. One could also choose as a strategy to directly confront the epistemological problem of induction, showing that this is not as serious as it has been presented, so the postulation of universals is not offering so many theoretical advantages at this point. Be that as it may, it is something that no serious theory in metaphysics of properties can elude.

12

An inference to the best explanation is a form of inference in which a fact H is justification for a theory T1 if: (i) T1 explains H, and (ii) T1 explains H better than any of the alternative explanations of H, T2, T3, . . ., Tn. For a general presentation, see Lipton 2004.

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References Alvarado, J. T. (2010). El problema de los universales. Filosofia unisinos, 11(2), 112–129. Alvarado, J. T. (2013). Estados de cosas en el tiempo. Revista de humanidades de Valparaíso, 1(2), 83–104. Alvarado, J. T. (2014). Propiedades determinadas, propiedades determinables y semejanza. Discusiones filosóficas, 15(24), 129–162. Anscombe, G. E. M. (1971). Causality and determination. Cambridge: Cambridge University Press. Reprinted in Ernest Sosa & Michael Tooley (eds.). Causation. Oxford: Oxford University Press, 1993 pp. 88–104. Armstrong, D. M. (1978a). Universals and scientific realism, Volume I, Nominalism and realism. Cambridge: Cambridge University Press. Armstrong, D. M. (1978b). Universals and scientific realism, Volume II, A theory of universals. Cambridge: Cambridge University Press. Armstrong, D. M. (1983). What is a law of nature? Cambridge: Cambridge University Press. Armstrong, D. M. (1989). Universals. An opinionated introduction. Boulder: Westview. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Bird, A. (2007). Nature’s metaphysics. Laws and properties. Oxford: Clarendon Press. BonJour, L. (1998). In defense of pure reason. A rationalist justification of A Priori justification. Cambridge: Cambridge University Press. Campbell, K. (1990). Abstract particulars. Oxford: Blackwell. Carroll, J. W. (1994). Laws of nature. Cambridge: Cambridge University Press. Collins, J., Hall, N., & Paul, L. A. (2004). Counterfactuals and causation: History, problems, and prospects. In John Collins, Ned Hall & Laurie A. Paul (eds.), Causation and Counterfactuals. Cambridge, Mass.: MIT Press, pp. 1–57. Correia, F., & Schnieder, B. (2012). Grounding: An opinionated introduction. In Fabrice Correia & Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality. Cambridge: Cambridge University Press, pp. 1–36. Dretske, F. (1977). Laws of nature. Philosophy of Science, 44, 248–268. Ehring, D. (2011). Tropes: Properties, objects, and mental causation. Oxford: Oxford University Press. Fine, K. (2012). Guide to ground. In Fabrice Correia & Benjamin Schnieder (eds.), Metaphysical Grounding. Understanding the Structure of Reality. Cambridge: Cambridge University Press, pp. 37–80. Foster, J. (1982/1983). Induction, explanation, and natural necessity. In Proceedings of the Aristotelian Society 83, pp. 87–101. Reprinted in Carroll, J. (Ed.). (2004). Readings on laws of nature (pp. 98–111). Pittsburgh: University of Pittsburgh Press. Funkhouser, E. (2006). The determinable-determinate relation. Noûs, 40, 548–569. Funkhouser, E. (2014). The logical structure of kinds. Oxford: Oxford University Press. Kim, J. (1976). Events as property exemplifications. In Brand, M., & Walton, D. (Eds.), Action theory (pp. 159–177). Dordrecht: Reidel. Reprinted in Supervenience and mind. Selected philosophical essays. Cambridge: Cambridge University Press, 1993, pp. 33–52. Lange, M. (2009). Laws and lawmakers. Science, metaphysics, and the laws of nature. Oxford: Oxford University Press. Lewis, D. K. (1973a). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1973b). Causation. The Journal of Philosophy, 70, 556–567. Reprinted with postscripts in David Lewis, Philosophical Papers. Volume II. New York: Oxford University Press, 1986, pp. 159–213. Lewis, D. K. (1979). Counterfactual dependence and time’s arrow. Noûs, 13, 455–476. Reprinted with Postscripts in Lewis (1986b), pp. 32–66. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55.

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Lewis, D. K. (1986a). On the plurality of worlds. Oxford: Blackwell. Lewis, D. K. (1986b). Philosophical papers, volume II. New York: Oxford University Press. Lewis, D. K. (1999). Papers in metaphysics and epistemology. Cambridge: Cambridge University Press. Lipton, P. (2004). Inference to the best explanation (Second ed.). London: Routledge. Moreland, J. P. (2001). Universals. Montreal: McGill-Queen’s University Press. Mumford, S. (2004). Laws in nature. London: Routledge. Oliver, A. (1996). The metaphysics of properties. Mind, 105, 1–80. Orilia, F., & Swoyer, C. (2016). Properties. In E. Zalta (Ed.), Stanford encyclopedia of philosophy. https://plato.stanford.edu/entries/properties/. Accessed on 28 Dec 2018. Paul, L. A. (2009). Counterfactual theories. In H. Beebee, C. Hitchcock, & P. Menzies (Eds.), The Oxford handbook of causation (pp. 158–184). Oxford: Oxford University Press. Psillos, S. (2002). Causation and explanation. Montreal: McGill-Queen’s University Press. Psillos, S. (2009). Regularity theories. In H. Beebee, C. Hitchcock, & P. Menzies (Eds.), The Oxford handbook of causation (pp. 131–157). Oxford: Oxford University Press. Rodriguez-Pereyra, G. (2001). Resemblance nominalism and Russell’s regress. Australasian Journal of Philosophy, 79, 395–408. Rodriguez-Pereyra, G. (2002). Resemblance nominalism. A solution to the problem of universals. Oxford: Clarendon Press. Rodriguez-Pereyra, G. (2004). Paradigms and Russell’s regress. Australasian Journal of Philosophy, 82, 644–651. Tooley, M. (1977). The nature of laws. Canadian Journal of Philosophy, 7, 667–698. Tooley, M. (1987). Causation. A realist approach. Oxford: Clarendon Press. Williams, D. C. (1953a). On the elements of being: I. The Review of Metaphysics, 7, 3–18. Williams, D. C. (1953b). On the elements of being: II. The Review of Metaphysics, 7, 71–92.

Chapter 3

The Superiority of Universals Over Resemblance Nominalism

Abstract Three alternatives to universals are discussed in this work: resemblance nominalism, trope theories and theological nominalism. This chapter discusses resemblance nominalism, especially in its strongest formulation by Gonzalo Rodriguez-Pereyra. It is argued that there are important difficulties for resemblance nominalism because it requires to postulate primitive and complicated facts of resemblance, it requires an inversion of the direction of ontological priority, it cannot explain structures of determination of properties, it cannot explain adequately causal powers, natural laws, or the reliability of our inductive practices, and it requires a possibilist modal metaphysics. Besides, resemblance nominalism under the specific formulation of Rodriguez-Pereyra suffers from a vicious regress. § 16. A series of functions or theoretical roles that should be fulfilled by universals or any other alternative in metaphysics of properties that are supposedly to do their work have been explained. These are reasons to accept the existence of universals, but only if a comparative examination shows that they satisfy these theoretical functions better than their alternatives. This comparison is what is intended to be shown in this and the next two chapters. It should be considered that there are a priori reasons of economy that seem to favor some nominalist alternative or some theory of tropes. For common sense, it is strange to find an entity that is located entirely in different disconnected regions of space or an abstract entity for which the attribution of spatiotemporal location does not make sense. These intuitions of ‘common sense’ against universals are motivated more than by the detailed examination of all the reasons for or against their existence, only by the fact that ordinarily sensible people explicitly consider objects in his ordinary dealings with the world and not the properties of those objects.1 In any case, these reasons of economy exist. Theories that can satisfy the explanatory requirements without having to postulate categories of ‘strange’ entities should be preferred. Thus, if there were no obvious 1 My experience, moreover, with common people, without previous philosophical training, is that once the pros and cons of the different positions in metaphysics of properties are explained to them, they are usually inclined to accept the existence of universals.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_3

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advantages of the universals concerning their alternatives regarding the satisfaction of the theoretical roles indicated in the previous Chap. 2, it would be more reasonable not to accept them. What will be shown in what follows, however, is that there are such advantages and that, therefore, the most reasonable thing to do is to postulate universals in our ontology. Three alternatives to universals will be considered: (i) resemblance nominalism without paradigms, (ii) theories of tropes with resemblance classes of tropes, and (iii) theological nominalism according to which the concepts in the mind of God can replace universals. Many more alternatives have been discussed throughout history, but it is not intended here to review all these positions or all the positions logically or conceptually possible about these issues. Such a thing would be entirely out of the possibilities of this work. This selection of alternatives, however, seems well justified given the merits of the positions selected. They look the best articulated positions in metaphysics of properties, at least to me. Probably the least familiar of those positions is theological nominalism. It will not seem a feasible option to a philosopher for which the postulation of God is not feasible. The problem is that, in the contrary, if one thinks that God is a theoretically reasonable entity, conceiving properties as ‘concepts in the mind of God’ is an extremely strong form of nominalism. It is so strong that it seemed to me important to dedicate an entire chapter to its discussion. If you attend to the usual expositions, I am leaving aside several positions. For example, the so-called “predicate nominalism” according to which an object x is F because you can truthfully attribute the predicate “is F” to x is omitted. It is also omitted the so-called “concept nominalism”, according to which x is F because one can judge with truth of x that it falls under the concept of ‘being F’. These positions are subjectivist. In effect, they make the fact that an object has some nature to be grounded on the contingent vicissitudes about what languages have we developed or what concepts we have or have not come to possess (for more detailed criticism, see Armstrong 1978a, 17–27; Edwards 2014, 86–94). Class nominalism and mereological nominalism are also being set aside. According to class nominalism, an object x is F because x belongs to a class of objects that can be characterized informally as the class of Fs. Of course, the class must be specified independently of the characteristic of being F. According to mereological nominalism, x is F because x is part of the mereological fusion of all and only the Fs. There are classes or mereological fusions that select every and only objects that intuitively seem to share the same nature. The problem is that there are also classes or mereological fusions that group completely heterogeneous objects in a completely indiscriminate way (for more detailed criticisms, see Armstrong 1978a, 28–43, 1989a, 21–38; Edwards 2014, 94–104; Allen 2016, 67–91). The so-called “ostrich nominalism” is also set aside. This form of nominalism has been called pejoratively as “ostrich”, because it is a position in which, although one admits that, for example, x is F and many other objects could have that same nature, one refuses to offer an ontological explanation of what can ground such facts, in a way similar to how an ostrich puts his head in the sand to evade danger. It seems merely to refuse the need to face the problem of the one over many, the many over

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one, objective resemblances, and so on (see for such criticism, Armstrong 1978a, 16, 1980; Rodriguez-Pereyra 2002, 43–46). Nevertheless, several interesting and carefully argued works have appeared lately that develop this theoretical alternative, making it now a live option for nominalists (see Melia 2005; Imaguire 2014, and especially 2018; Schulte 2019). Under the more sophisticated formulations, this type of nominalism makes a choice of primitive facts different from the selections done by other alternatives in metaphysics of properties. Defenders of universals maintain that primitive facts of ‘instantiation’ of universals are what ground the character of objects. Defenders of trope theories maintain that the existence of tropes with a primitive qualitative character and the fact that those tropes happen to be part of a bundle are the ground of the character of objects. Defenders of resemblance nominalism, on their turn, propose primitive facts of resemblance as ground. So, if every alternative requires one or other kind of primitive fact, why don’t just assume that it is a primitive fact that any object has the character it has. Instead of searching for a ground of the fact that, for example, x is F in terms of universals, tropes, resemblance or whatever, one can simply postulate that the primitive ungrounded fact is that x is F. This stance has been called, then, “priority nominalism” and also “grounding nominalism”, because it supposes that the fact that things are as they are is ontologically prior and need no grounding in something else. As it will be discussed below, almost all positions in metaphysics of properties should face a problem of regress (see § 18), but ‘priority nominalism’ don’t. I have my own view about how to handle the regress problem (see §§ 67–70) without any ‘primitives’ for universals, but under the usual treatments, the strategy of the priority nominalist appears at least as acceptable as any other. Nevertheless, I don’t think that priority nominalism even under these more sophisticated formulations is one of the strongest forms of nominalism. The main problem of the position is not that it refuses to give an explanation –as it was alleged by Armstrong and all those that followed him– but that it is an extremely uneconomical position. Priority nominalism is not more uneconomical that trope theory –or the type of Platonism that is going to be defended in this work– in the quantitative sense. But economy or its lack of is also qualitative. Priority nominalism postulates infinite different types of primitive facts. In effect, for the priority nominalist it is a primitive fact that, for example, x1 is F; it is a different type of primitive fact, though, that x2 is G, and it is also a different type of primitive fact that x3 is H, etc. The price of the explanatory economy of the priority nominalist is the multiplication ad infinitum of different types of primitive facts, each for any character an object may have. Forms of ‘aristocratic’ similarity nominalism are also being set aside. In these forms of nominalism, what grounds that an object x is F is the resemblance to a ‘paradigm’ or a group of objects that function as ‘paradigm’. The form of resemblance nominalism that will be treated here is called, by opposition, as “egalitarian”, since there are no objects to which the function of paradigm is assigned. The nominalism of ‘aristocratic’ resemblance has seemed much less reasonable than the ‘egalitarian’ one. An object can be similar to the paradigm in more than one respect. A red object of spherical shape can be taken as a paradigm of red, but something can be similar to the paradigm because it is spherical. If the resemblance

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to the paradigm is what makes something red, then the property of being red seems to be identified with the property of being spherical. For example, if it is a necessary fact that everything that is F is also G, there do not seem to be reasonable ways of selecting something that could serve only as a paradigm of being F (see for more detailed criticisms, Armstrong 1978a, 44–57; Rodriguez-Pereyra 2002, 124–141). In what follows, therefore, attention will be focused on the theories that seem the best alternatives to universals. If these alternatives present significant comparative faults, we will have very justified reasons to accept the existence of universals. The universals will be shown as those entities that offer the best explanation about how facts about the one over many, the many over one, objective resemblances, causal powers, natural laws, and our inductive practices can be grounded.

3.1

Resemblance Nominalism

§ 17. The form of resemblance nominalism that is going to be considered here is the so-called “egalitarian” one where there is no ‘paradigm’ for the constitution of the resemblance classes. The central idea of this type of conception was initially proposed by Carnap (1928, 113). The fact that an object x is F must be grounded on the fact that x resembles the objects y1, y2, . . ., yn that make up a class. The facts of similarity, then, are those that ground the natures of the objects. A ‘resemblance class’, be α, is specified as satisfying the following two requirements: (A) (B)

Any two objects belonging to α resemble each other. Nothing that does not belong to α, resembles all the objects that belong to α.

Condition (A) ensures that all objects belonging to class α satisfy the requirement of being similar to each other. It is assumed here, of course, that the similarity is a dyadic, reflexive and symmetrical relationship.2 Condition (B) ensures that only the objects belonging to α satisfy the requirement that they are all similar to each other. The class of similarity α is, then, the largest class of objects that satisfy the condition of being all similar to each other. It is assumed that, if a plurality of objects seems to share the same property, then they must all belong to the same class of similarity. If a plurality of objects belongs to the same class of similarity, on the other hand, they should share the same ‘nature’. The details of the forms of resemblance nominalism that have been defended more recently have been motivated by several notorious difficulties for a conception of this type. In the first place, it has been suggested that resemblance nominalism would force us to identify what appear to be different properties. A classic example is that of

2 The relation of resemblance is not generally transitive. This here presents no problem, however, since for objects belonging to a resemblance class according to requirements (A) and (B), there are no cases in which, for x, y, z: x is similar to y, y is similar to z, but x is not similar to z. By construction, if x were not similar to z, either x or z should not belong to the resemblance class.

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animals with hearts and animals with kidneys. Assume that all and only animals with hearts also have kidneys.3 Intuitively to have heart and to have kidneys seem different properties. However, if the resemblance class corresponding to animals with hearts is the same resemblance class corresponding to animals with kidney, then a resemblance nominalist should maintain that—contrary to our intuitions about the issue—they are the same ‘nature’. To address this problem, David Lewis proposed that the resemblance classes that can fulfill the functions of universals are classes not only of actual objects but also of merely possible objects (see Lewis 1983, 10). It is followed in this by Gonzalo Rodriguez-Pereyra (see 2002, 99–100). If resemblance classes have as elements possibilia, then the eventual cases of actually co-extensive classes should not concern the nominalist. Although actually all and only animals with hearts are also animals with kidneys, there will be possible worlds with animals with heart and without kidneys, or with kidneys and without hearts. This is enough to differentiate the resemblance classes assigned to each of these natures. In cases where the classes of similarity assigned to what appear to be two different properties have exactly the same elements in all possible worlds, there would be good reasons to think that they are the same nature. The most notorious problems for ‘egalitarian’ resemblance nominalism are the problems of the ‘imperfect community’ and the problem of ‘companionship’. These problems were already formulated by Nelson Goodman (1966, 160–164; but see Rodriguez-Pereyra 2002, 142–155).4 The problem of the imperfect community is a difficulty that affects the sufficiency of the requirement (A) indicated above to select the objects that, intuitively, seem to share the same nature. Suppose a class of all objects—actual and merely possible—that are either perfectly spherical and have 10 kg of mass, or have 10 kg of mass and are 10 cm tall, or are 10 cm tall and are perfectly spherical. This class will have as elements infinite objects. Any two objects of this class will resemble each other. In effect, any pair of objects of the class will be similar either because both have a perfectly spherical shape, or because both have 10 kg of mass, or because both have a height of 10 cm. Condition (A), then, is satisfied, but it does not seem reasonable to think that this class selects objects that share the same nature. These are extremely heterogeneous objects in which there is not a single ‘nature’ shared by all of them. Of course, if one admitted disjunctive properties one could, perhaps, argue that all those objects possess the same property of being perfectly spherical or having 10 kg of mass or having 10 cm of height. But in no way would anything like that be acceptable as an authentic property.

3 Which seems false, indeed. There are human beings without kidneys—which must be dialyzed regularly—but with a heart. There are human beings without hearts—with an artificial device that circulates their blood—but with kidneys. A resemblance nominalist could also differentiate between the resemblance class assigned to having a heart and the resemblance class assigned to having kidneys. See Rodriguez-Pereyra 2002, 97–98. The classic example will be maintained, however, to simplify the discussion. 4 The problem of the company has been confused many times with the problem of co-extensive classes. A precise formulation of this problem has only been made in Rodriguez-Pereyra 2002, 149–155.

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The companionship difficulty, on the other hand, affects the necessity of condition (B), since it is a type of case in which there is a class of objects that seem to share the same nature, but where requirement (B) is violated. Suppose a class of—actual and merely possible—objects that are all F. Assume also that it is a necessary fact that all that is F is G, although not everything that is G is F. If there is a class of similarity assigned to the nature of being G, this class will be clearly different from the class of the Fs, but the class of the Fs will be included as a sub-class of the class of the Gs.5 In a case of this kind, it happens that the objects that belong to the class of similarity of the Gs and that do not belong to the class of similarity of the Fs will be similar to all the objects of the resemblance class of the Fs. In effect, since all F is G, every object that is G will be similar to every object that is F since these objects are also G. Then, we have the case of a class of similarity that satisfies requirement (A) and that seems to include objects that share the same nature. However, the condition (B) is not satisfied, since there are objects that do not belong to such class of similarity that are similar to all the objects of the class. David Lewis has tried to overcome these difficulties by postulating a quite drastic modification in the basic characteristics of the relationship of similarity (see Lewis 1983, 14–15). Instead of assuming a dyadic relation of similarity—the notion of similarity that is familiar to us—he postulates a relation of multigrade and ‘contrastive’ similarity that would be formulated in the following way: x1 , x2 , . . . , xn are similar
to each other and are not similar
to y1 , y2 , . . . , yn : Since the classes of similarity posited by Lewis are classes of possibilia, the variables in this predicate must be infinitely many. A concept like this would solve the problem of the imperfect community because for a class of resemblance of n individuals the multigrade relation of resemblance* would be applied ‘at once’ to the n individuals. It would not be the case that each pair of these individuals were in a relation of dyadic similarity to each other. The resemblance* would not be analysable as the conjunction of facts of dyadic similarity between pairs of objects of the class.6 This notion of resemblance* should be, then, conceptually primitive. It could not happen that a few individuals of the class are similar to each other for being both F while another pair of individuals are similar to each other for being both G.

5 The examples that come to mind when a resemblance class has a ‘companion’ are cases like the property of having a perfectly spherical shape and the determinable property of having a form. If one wants to restrict the examples to super-determinate properties, however, it is more difficult to find examples. There would be cases of this kind in particle physics if it were accepted that natural laws are necessary. Every quark of the type ‘down’ has a mass of 4.8 MeV/c2 and an electromagnetic charge of 1/3, but not every particle with a charge of 1/3 has such a mass. 6 One could define, in effect, the similarity* of n objects to each other as the conjunctive fact that: x1 is similar to x2, and x2 is similar to x3, and . . . and xn-1 is similar to xn. A notion of similarity* defined in this way in terms of the dyadic notion of resemblance, however, could not evade the problem of the imperfect community, since objects that are similar to each other without sharing the same nature would count as similar* to each other.

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There would also be no reason to suppose, then, that the similarity* between n objects should imply the similarity of each pair of them. The facts of similarity* are disconnected from the facts of similarity. This disconnection has been considered an important disadvantage of this approach compared to that of Rodriguez-Pereyra (see 2002, 81). The notion of similarity* has another ‘contrastive’ part, because it is attributed not only to n objects that are all similar to each other ‘at once’, but at the same time it ‘separates’ those objects as non-similar from all the others. It does, therefore, a ‘contrast’ between the objects for which the resemblance* obtains from those for which it does not. In this way, the problem of the companionship can be solved, because an authentic class of similarity can always be differentiated from its companions. All the objects of the companion class will automatically be excluded in the contrasting part of the basic fact of similarity* as not similar* with the objects that share the relevant nature. Note that, according to this theory, all facts of similarity* must involve all possible objects, without exception. Either these objects are included in the resemblance class, or they remain in the contrast part. In any case, each fact of similarity* must include them all. This imposes additional difficulties on this form of resemblance nominalism—as Lewis himself acknowledges (see 1983, 14)—for ‘similarity*’ is an extremely complex and artificial primitive notion.7 What is gained by avoiding the problems of the imperfect community and the companionship is lost correlatively by drastically distancing the relationship of similarity* from our ordinary intuitions about dyadic resemblance. No natural language could even express a predicate that has indenumerable infinite arguments. It is very doubtful that we can understand the notion of similarity* in these terms. We would never be in a position to judge that something falls under it correctly. All the cases in which the notion would be applied correctly are cases in which it should be applied to an indenumerable infinite multitude of objects that, naturally, escape our finite cognitive capacities—and they would surpass the cognitive capabilities of angels, for the same reasons. Many times, we have come to know properties or, at least, we have come to have well-justified beliefs concerning properties or what fulfill their functions. If in this form of resemblance nominalism, however, the primitive facts of resemblance* are facts that irreducibly involve all possibilia, how could we ever come to have knowledge of a ‘nature’ shared by many objects? The knowledge we claim to possess by doing natural science about connections between different ‘natures’—if those natures are anything other than primitive resemblance facts— would be inexplicable. Every position in metaphysics of properties must be able to explain objective similarities. In principle, in resemblance nominalism, these objective similarities are primitive and not grounded on anything. It happens, however, that in this form of resemblance nominalism that which grounds the natures of 7

Something similar happens with the variant of resemblance nominalism proposed by Paseau (2015, 110–115). He postulates a relation of similarity of comparative character that has pluralities as arguments. The form of the basic resemblance facts is the following: the X1s are more similar to the X2s than the X3s are similar to the X4s. Here the variables ‘X1’ to ‘X4’ range over pluralities. The details of this alternative will not be discussed here.

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objects are not facts of similarity, but facts of similarity*. Contrary to what one might have supposed, then, it would be necessary here to deliver an explanation of objective similarities. But it is already evident that dyadic resemblances are disconnected from the multigrade and contrasting relation of similarity*. Therefore, Lewis’s resemblance nominalism seems much less feasible than the version defended by Gonzalo Rodriguez-Pereyra (2002). In this version, a relationship of primitive similarity is used, as it is obvious, but it is the familiar dyadic similarity. The problem of the imperfect community is solved here by appealing to the ‘hereditary’ character of the similarity in ‘perfect’ communities. A ‘perfect’ community is a class of similarity that effectively selects objects that share the same nature. For a class of similarity of objects, be α, you can define a ‘hereditary pair’ of objects of α, as a pair of objects of α, or a pair of pairs of objects of α, or a pair of pairs of pairs of objects of α, and so on. Let a class of similarity of level 0 be a class of objects that satisfy the condition (A) indicated above. Let a class of level 1 be a class of ordered pairs of level-0 objects. A class of level-2 is a class of level-1 pairs, and so on. The class ‘αn’ is the class of the nth level of objects that, at their level-0, must be objects that satisfy requirement (A), that is, they are all objects that are similar to each other. The central idea of Rodriguez-Pereyra is that in a perfect community there is primitive and ungrounded similarity for all entities x, y of a class αn of any level n. These entities can be objects or any of the infinite hereditary pairs. That is, there is similarity of all the objects of α to each other, as required in (A), but there is also similarity between all the pairs of objects of α, and there is similarity between all pairs of pairs of objects of α, and so on. When it comes, on the other hand, to an imperfect community, even if there is similarity between all the objects among themselves, there will be some level in which hereditary pairs of such objects will not to be similar to each other. To make precise what the similarity between pairs of a certain level in the hierarchy of hereditary pairs consists, Rodriguez-Pereyra (see 2002, 163–165) defines a function f(x) that assigns to x its ‘properties’—in the terminology used here, it assigns it a ‘nature’. If x is a particular, then f will assign its first order properties. If x ¼ then the value of f(x) will be determined by the value of f( y) and f(z). In particular, the value of f(x) is determined by the value of ( f( y) \ f(z)). If y and z share the first order property F0, then f will assign the second-order property F1 to x. If there is a pair of pairs that share property F1, then f will assign it F2, etc. The similarity of a pair of pairs of objects is then fixed as that primitive relation that occurs between those pairs when their constituent objects share the same nature. More precisely (see Rodriguez-Pereyra 2002, 164):  f ðxÞ ¼ X 0 1 , X 0 2 , . . . , X 0 m if and only if x is a particular object and the  elements of X 0 1 , X 0 2 , . . . , X 0 m are all the sparse properties of x;

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 f ðxÞ ¼ X nþ1 1 , X nþ1 2 , . . . , X nþ1 m if and only if x ¼< y, z > and ð f ðyÞ \ f ðzÞÞ ¼ fX n 1 , X n 2 , . . . , X n m g; f ðxÞ ¼ ∅ in any other case: The ‘properties’ in question, in the terminology used by Rodriguez-Pereyra, must be grounded on similarities. The function f specifies how a hereditary pair may or may not be similar to another hereditary pair and how this similarity grounds ‘properties’ of such pairs. The properties of a hereditary pair will always be ‘inherited’ from the properties of the objects of the pair. The definition of a perfect community is as follows (see Rodriguez-Pereyra 2002, 170): α0 is a perfect community ¼df for all n, for all x, for all y: if x and y belong to αn, then x is similar to y. Rodriguez-Pereyra designates as “R*” the similarity relation for any hereditary pair. It is a dyadic relation of similarity, but it must apply to objects or hereditary pairs of objects. This makes it different from the ordinary dyadic similarity between objects, but it is a minor difference compared to the peculiarities of Lewis’s relation of resemblance*. When it comes to the problem of the companionship, Rodriguez-Pereyra proposes to solve it with what he calls “maximal perfect communities”. A ‘degree of similarity’ is assigned to each class. The degree of similarity of a pair of objects is the number of properties shared by such objects. In the terminology followed here, it is the number of ‘natures’ shared by such objects. A class has a degree of similarity n if and only if the objects belonging to that class share n properties. This assumes that objects have a finite number of authentic properties—which is a reasonable assumption. Given how the function f has been defined, if the elements of a similarity class α0 share d properties, then the elements of αn share at least d properties. This allows a degree of similarity to be assigned to a class of any level in the following way (see Rodriguez-Pereyra 2002, 181): α0 is a perfect community of degree d ¼df α0 is a perfect community and d is the lowest degree in which two elements of αn stand to each other in the resemblance relationship. Once the degree of similarity is fixed for each class of objects, a maximal perfect community can be defined as (see Rodriguez-Pereyra 2002, 182): α0 is a maximal perfect community of degree d ¼df α0 is a perfect community of degree d, and there is no perfect community of degree d of which α0 is a sub-class. Maximal perfect communities are not affected by the problem of the companionship. Indeed, if all F is G, but not all G is F, then whatever the degree of similarity that has the class of similarity assigned to F, that degree of similarity must be higher than the degree of similarity assigned to G. The objects that are F are also G, so there is an

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additional nature that all those objects share with each other that all the objects that are G do not share. In the theory of Rodriguez-Pereyra, therefore, it turns out that the resemblance classes that fulfill the functions of a universal are maximal perfect communities. This requires replacing the requirements (A) and (B) indicated above. (A) must be replaced by the condition required for something to be a perfect community. (B) must be replaced by the condition required for something to be a maximal perfect community.8 A series of difficulties against this form of resemblance nominalism, as defended by Rodriguez-Pereyra, will now be considered. It is the strongest form of nominalism and, therefore, if there are severe difficulties with it, there will be sufficient grounds to suppose that it is better to abandon resemblance nominalism.

3.2

Primitive Facts of Resemblance

§ 18. No theory should be criticized for postulating primitive ontological facts, not grounded on anything. Every theory must propose one or other type of facts that should be taken as basic according to the theory. Grounding chains of entities must finish at some point. Resemblance nominalism cannot be criticized, then, for postulating basic facts of similarity. A theory, however, can be criticized for proposing primitive facts that are too obscure, or too strange, or too complicated, or for postulating too many primitive facts. The first difficulty with resemblance nominalism in all its forms is that it contradicts some of our intuitions about the relation of similarity. The relationship of similarity seems a relationship and an internal relationship. It is doubtful that anything that does not meet these conditions can be appropriately qualified as ‘resemblance’. In the first place, a nominalist asks us to accept that all the characteristics of objects and all relations between objects are grounded on similarities. Similarities between objects, or between n-tuples of objects are of fundamental ontological importance. It is on those facts of similarity that almost everything else is grounded. If the nominalist explained such similarities in the same way in which he explains any other relationship, a vicious infinite regress would be generated (see Armstrong 1978a, 18–21, 41–42, 53–56, 1989a, 53–57), so that it has come to be accepted by many philosophers that ‘resemblance’ must be a ‘primitive’ that cannot be understood as having the same nature as any other relational fact. Two factors seem to have made reasonable the postulation of a ‘primitive’ fact of similarity. First, when it is said that similarity is a ‘primitive predicate of the theory’ the nominalist maneuver is presented as if it was a simple matter of choice of vocabulary, without higher ontological weight than the choice one could make between using implication and

8

In Rodriguez-Pereyra’s theory, an additional requirement is added to distinguish the classes of similarity that fulfill the functions of universals from mere intersections of classes of similarity (see Rodriguez-Pereyra 2002, 186–198). These sophistications will not be considered here.

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negation as primitive connectives rather than disjunction and negation in propositional logic. The theories that result from this selection of primitives are equivalent, so it does not seem that anything is at stake here other than an aesthetic question. It would be—as sometimes it is formulated—a matter of ‘ideology’ and not of ‘ontology’. But this way of presenting the question is most confusing. Perhaps the term “primitive” used here so often to qualify ontological questions should be avoided. This is not a question about the selection of predicates to be attributed to a fixed domain of objects. It is not at all a question about ‘predicates’. Although there were no natural languages with terms such as the English expression “similarity”, there would still be similarities for the nominalist, as it would still be a fact that the electrons have a mass of 0.511 MeV/c2, because the fact that the electrons have this mass is grounded on similarities. The postulation of ‘primitives’ is the indication of what is fundamental in the ontology that is being proposed. For a nominalist, what is fundamental in his ontology is the existence of objects and similarities between these objects.9 It seems evident to me that resemblance is a relationship. Although one might want to call it otherwise to differentiate it from the other relations that are nothing more than similarities between n-tuples, I do not see that they can be considered in any other way. The ontology of the resemblance nominalist is decidedly an ontology that postulates in its base objects and relational facts between these objects. Once the existence of relationships has been admitted—since we accept similarities that are relations—it is obscure why the nominalist refuses to accept any other. If there are similarities, why there are no causal connections, for example? A second reason why it has seemed to many an acceptable answer to maintain that similarity is simply a ‘primitive’, has been the existence of analogous problems of a vicious regress for almost all positions in metaphysics of properties. This is especially clear for the case of universals where this problem is famous and has been called the “Bradley’s regress” (see Bradley 1897, 18; § 68). It is not necessary for a universal to be instantiated in one particular object rather than in another. Even when the universal is essential to the object—so that in every possible world in which the object exists, it instantiates that property—the object might not exist. Except for essential properties, an object does not need to instantiate a universal rather than other. The mere existence of universal and particular does not guarantee, then, the existence of the state of affairs of that object possessing that universal. For example, suppose a possible world w1 in which the object a1 instantiates the universal U1, and the object a2 instantiates the universal U2. There are in this world the entities a1, a2, U1 and U2. Suppose now the possible world w2 in which a1 instantiates U2 and a2 instantiates U1. In w2, there are the same entities a1, a2, U1, and U2 that in w1, but it does not have the same states of affairs. There is a substantive difference, hence, 9 A nominalist will probably protest against this formulation. It is not the case, he will say, that similarities between objects are postulated, but simply that the objects are (or are not) similar to each other. But it is an acceptable paraphrase of “a is similar to b” to say that “there is a similarity between a and b”. The nominalist will agree that, for example, (i) it is a fact that a is similar to b, and (ii) that this fact is not the existence of an internal relationship between a and b. This is all that is required for what is held here.

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between w1 and w2, but it has nothing to do with its constituents—or with its basic constituents if you prefer. One might be inclined to say that the difference between w1 and w2 is explained because in w1 the universal U1 is instantiated in a1, whereas in w2 it is instantiated in a2, as well as the universal U2 is instantiated in a2 in w1 and is instantiated in a1 in w2. The instantiation relationship, then, is what would explain the differences. The problem that arises here is that ‘instantiation’ seems to be, indeed, a relationship. If one defends the existence of universals, then one should assume that instantiation is a universal relational property like any other. It is not necessary that the instantiation relationship is instantiated in certain universal and objects rather than others. In w1 the instantiation relation is instantiated in the ordered pair and not in the ordered pair . In w2 it is instantiated in rather than . The mere existence of the relationship of instantiation and is not sufficient to explain why in one case it is instantiated and not in others. If the instantiation relationship is adduced at this point again, the same problem will be generated.10 A universal relationship of instantiation might not be instantiated in the ordered pair in which it is instantiated, etcetera. Armstrong, who was well aware of the problem, said that instantiation is a “fundamental tie or nexus” (see Armstrong 1978a, 108–113, 1989a, 108–110, 1997, 127, 2004, 46–48). Strawson said before him that it was a “non-relational tie” (see Strawson 1959, 167–168). Bergmann said that it was a fundamental “nexus” (see Bergmann 1967, 9–12). Conspicuous advocates of universals have resisted the idea that ‘instantiation’ is a relationship like any other because doing so would generate a known vicious regress. ‘Instantiation’ is taken as a ‘primitive’. But if this is legitimate for the defender of universals, why should not it also be for the nominalist? Why cannot one replace the primitive ‘instantiates’ with the primitive ‘resembles’? There are regresses similar to Bradley’s that affect almost all forms of nominalism with the exception of priority nominalism (see Armstrong 1978a, 18–21, 41–42, 53–56, 1989a, 53–57). Why not allow these different forms of nominalism a judicious choice of primitives to evade these regresses? I think, however, that the appeal to a ‘primitive fact’ in these cases is unacceptable both for the nominalist and

10

Let inst be the universal relation of instantiation. The explanation of why there is a state of affairs of a1 having universal U1 is given by inst (a1, U1). But inst is a universal relationship, so it should be explained why it is instantiated in the pair rather than in any other ordered pair. One could appeal here to the same universal relation of instantiation inst or a relation of instantiation of a higher logical type. In the first case, it is generated inst (inst, (a1, U1)), which then requires inst (inst, (inst, (a1, U1))), and so on. In the second case, a relation inst1 would be required assuming that inst is of the lowest type, be inst0. What explains inst0 (a1, U1) is inst1 (inst0, (a1, U1)). And what explains inst1 (inst0, (a1, U1)) is then inst2 (inst1, (inst0, (a1, U1))) and so on. The first alternative has the drawback that there would be a universal of being an X such that: Øinst (X, X). That is, the universal of not be instantiated in itself, that it is inconsistent, because if it were instantiated in itself, it would not be instantiated in itself, and if it were not instantiated in itself, then it would be instantiated in itself. The second alternative avoids this incoherent universal, but it is much less economical since it is necessary to postulate an infinite sequence of different universal relations of instantiation for each logical type. In either case, it is a vicious regress because there is an infinite chain of grounding relationships without a first element.

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for the defender of universals. Defenders of universals have felt inclined to accept the nominalist maneuver because they have thought—wrongly—that it is the only way to avoid similar problems that affect universals. But this is not the case, as will be shown (see §§ 69–70). There are ways of conceiving the instantiation of universals in a way that does not require postulating any primitive fact and does not require exceptions to the categories of entities already accepted from the beginning. The postulation of a primitive of ‘instantiation’ must be criticized, just as the postulation of a primitive of ‘resemblance’ must be criticized. In the case of ‘instantiation’, it is obscure to postulate a relationship that is not a relationship, as it is obscure to reject the existence of properties and then make an exception concerning similarities. If nominalism is the program of developing an ontology without properties as the basic constituents of the world, the acceptance of relationships—as are similarities—is simply the failure of the program. Contrary to what the nominalist may say about resemblances, then, these are relations. It is also an allegation of the nominalist that resemblances should be taken as external relations, that is, as not grounded on the intrinsic natures of the relata. Admitting that it is an internal relationship would be disastrous for his position. In effect, the nominalist wants to ground the fact that, for example, x is F on the similarity of x with other objects that are elements of the relevant similarity class. If similarity were internal, then what would ground that, for example, x is similar to y would be the intrinsic natures of both x and y. Part of this intrinsic nature should include that x is F. So, the fact that x is F would be grounded on the fact that x resembles y, which in turn would be at least partly grounded on the fact that x is F. Since grounding is a transitive relationship, it would follow that the fact that x is F would be grounded on itself, which is absurd. Worse still, however, for the nominalist, resemblances would be dispensable here because they would not fulfill any explanatory function. For the nominalist, then, accepting internal relations of similarity would be a fatal step. Resemblances must be primitive external relationships, not grounded on any previous nature of the relata. It is very doubtful, however, that a relationship of this kind can even count as an authentic relationship of ‘resemblance’. Resemblance nominalism acquires credibility to the extent that it leans on our familiar practices of detecting similarities or dissimilarities. From these familiar practices, it would be explained why we should not see anything strange in our assignment of ‘properties’ to objects, even if such ‘properties’ are nothing more than shadows of resemblances. The plausibility of this form of nominalism rests on our ordinary intuitions about similarity so that “similarity” in the nominalist’s mouth must mean the same thing as “similarity” in ours. The problem here is that it seems part of our ordinary notion of similarity that similarities are internal relations, grounded on the nature of the relata. Similarity seems different than the relation of spatial distance. Anyone who maintains that there is a similarity between two objects can be asked to show what grounds it on the natures of the relata. This does not happen in external relations such as distance. Distance is not grounded on the nature of the related objects, and no one should be required to explain how a distance would be grounded on such natures. To require such a ground in the relations of distance would seem nonsense. If one wants to

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formulate this in Wittgensteinian terms, one could say that learning the language game of attributing similarities is at the same time learning the language game of knowing how to explain ontologically such similarities. Learning the language game of attributing distances, on the other hand, does not imply it. Of course, these observations about our ordinary intuitions about similarity must be taken with extreme caution and cannot, by themselves, be considered a definitive argument against nominalism. Many times, “common sense”—i. e., what sensible people without specialized philosophical training is willing to accept—includes theses utterly incoherent with each other. Many times, those intuitions have been shown as errors. What ‘common sense’ is willing to accept is not untouchable, nor incorrigible. Good theoretical reasons can be a sufficient justification to abandon those intuitions. I suppose that the nominalist will be inclined to hold something like that in this case. The problem is that much more work is required than what has been done up to this moment to justify that a primitive external relation of similarity is intelligible to us. It is not enough to argue that it is a relationship that is “just like the usual, internal resemblance, but external”. An external relationship of ‘resemblance’ cannot be ‘just like’ the usual internal relationship. It is something of a very different character. The understanding of such a concept seems to require—in Wittgensteinian terms—to learn a language game that is not the one we usually learn to understand the use of “similarity”. Which exactly? It is the nominalist’s job to explain it. As long as such an explanation is not delivered, we cannot assume that it is sufficiently clear what primitive similarities are.

3.3

How Would We Have Epistemic Access to Such Resemblance Facts?

§ 19. Authentic properties are characteristics—in general—sufficiently relevant to enter into natural laws. Of course, a nominalist will not maintain that natural laws are relations between universals, since there are no such things in his ontology, but—if he defends a regularist theory of laws—he must assume that resemblance classes should be mentioned in the descriptions of the regularities that achieve the best balance of simplicity and informativeness (see Lewis 1973, 41–43). If we have succeeded in discovering some natural laws, then we must assume that we have succeeded in finding authentic properties. An acceptable theory in metaphysics of properties should not make impossible or implausible the knowledge of authentic properties that we claim to possess, for example, by doing natural science. There is an integration challenge to fulfill. The ontology must be in harmony with the best epistemology. In these matters, as in so many others, perfect harmony cannot be expected. A sufficiently reasonable ontology can impose restrictions on our epistemology. In the same way, the demands of our best conception of how we acquire justified beliefs can impose some reforms in our ontology. There are, however, degrees of mismatch. An ontological theory about the Fs that has the consequence

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that we almost know nothing about the Fs, against our ordinary epistemic practices, is a theory that must be strongly reformed or dismissed. I am afraid that in the case of resemblance nominalism, we find ourselves in a situation of this kind. It has already been noted that the external resemblances to which the nominalist appeals are obscure. The nominalist, however, is not merely adducing facts of dyadic similarity between objects. Lewis postulated a primitive relationship of multigrade and contrastive similarity—which adds complexity to darkness. Rodriguez-Pereyra has proposed, on the other hand, similarities that should generate ‘maximal perfect communities’. In the case of Rodriguez-Pereyra, each of the similarities is dyadic, which makes his position quite more reasonable than Lewis’s, but there are infinite similarities that must be applied to pairs of objects and hereditary pairs of objects of any level. How could we ordinarily have epistemic access to such kinds of similarity? How could one have cognitive contact with an authentic nature? It is instructive here to contrast with what happens with universals. If one knows of, for example, an object of 10 kg of mass, then one has a cognitive contact with the universal of having 10 kg of mass. It is that same mass with which one has had such contact that could be instantiated in multiple objects. That which is known—or what we come to have familiarity with—is the same numerically as would be instantiated in other cases. For this reason, knowledge of universals allows—if there is such a type of entity—a projection of what happens with states of affairs with which we have no perceptual contact and with which, perhaps, we cannot have perceptual contact. If we know that it is a natural law that every instance of the universal U1 is followed by an instantiation of the universal U2, then we have eo ipso knowledge about regions of spacetime with which we are separated by distances from which no causal signal can reach us. In every such a region an instantiation of a U1 is going to be followed by an instantiation of a U2. We cannot have perceptive contact with what happens in such regions, but our knowledge of universals allows, nevertheless, to know many things about them. In the same way, a biologist with knowledge of the universals instantiated in certain enzymes and the way they work in cellular metabolism acquires a vast knowledge of states of affairs with which he has never had a perceptive contact. You do not need to make observations in each case to know what kind of states of affairs exist in a metabolic process, because it is enough to know the universals or the types of universals that are there instantiated. Of course, there are multiple epistemological questions that arise regarding our knowledge of universals. It is not obvious how we have perceptive knowledge of them—if we have it. These questions will be discussed later (see §§ 71–74). For the nominalist, no nature is something ‘local’. The state of affairs that, for example, x is F is not a fact located in the region of spacetime that x occupies. No matter what intuitions we have about the intrinsic character of the nature of being F, we must deal with a fact concerning the vast disconnected region of different spacetimes of different possible worlds in which all the possible objects belonging to the resemblance class of the Fs are located. If there are universals when one is confronted with a particular state of affairs of being instantiated the universal U in a particular object x one is already confronted with the ‘entire’ universal. From the point of view of resemblance nominalism, when one is confronted with the state of

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affairs of the object x being an F one has not faced but a tiny ‘piece’ of the class of similarity in which to be F consists—an infinitesimal ‘piece’, since every class of similarity must have infinite elements. If there are universals, it is not important that one is not familiar with all its actual instantiations and it is much less important that one is not familiar with all its possible instantiations. It is accidental to a universal what instantiations it has.11 When it comes to a resemblance class, it is not accidental what elements it has. Each of them is essential for the class. If that specific class of similarity is the nature of, say, being F, then each of the infinite objects that belong to the class of similarity is essential for the nature of being F to be what it is and not another. So, it seems that if one is going to know the nature of being an F one can only do it if one has some familiarity with each one of the infinite objects that are elements of the resemblance class assigned to being F. But this would be metaphysically impossible for us and even for an angel. We could not have familiarity with merely possible objects. Then it seems that resemblance nominalism would make impossible the knowledge of the natures of the objects that we claim to possess when, for example, we do natural science. Perhaps a resemblance nominalist would be inclined to answer this objection by making a distinction between ontological and epistemological priorities between natures and resemblance classes. The defense that the nominalist could attempt is to maintain that, although from an ontological point of view, facts of similarity—which ground the class of similarity which is the nature in question—have ontological priority, since they are the ground of, for example, the fact that x is F, from an epistemological viewpoint what has priority is the fact that x is F. This fact only concerns x and is something with which we can have perceptual contact. Thus, one can perfectly verify, for example, that something is a sphere. What one cannot verify is that the nature of being a sphere is grounded on the primitive external similarities of this object with infinite others in different possible worlds. One may have familiarity with the nature of an object, then, but not with that on which such a nature is grounded. Moreover, a nominalist can come to know that the fact that something is a sphere is grounded on primitive similarities, but without having to know exactly what other objects something must be similar to be spherical. Someone who defends a position like that of Rodriguez-Pereyra may come to know that something is spherical because it belongs to a maximal perfect community, without having to know exactly that maximal perfect community, which would require knowledge of what its elements are. An answer of this type, however, generates an immediate difficulty. It makes sense to argue that one can come to know the nature of being an F that possesses the object x and not to know that in which such a nature is grounded—primitive external similarities—if the nature and which grounds it are numerically different entities. But such a thing would be anathema to the nominalist. If there is a numerical difference between the natures of the objects and the similarities on which they are

Although, of course, if it is an ‘immanent’ universal, it must have some instance, although not one or another in particular. If it is a transcendent universal, it is merely accidental to have instances. 11

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grounded, then the nominalist should have the natures in the range of their quantifiers. But then, the nominalist would be postulating entities numerically different from the objects that possess them and that are ‘characteristics’, ‘respects’ or ‘determinations’ of such objects. Such an entity would be a property with another name. The nominalist wants to solve the problem of the many over one postulating a plurality of resemblance classes to which the same object can belong. Being x an F is different from being x an G because the resemblance class assigned to F is different from the resemblance class assigned to G. If one wants to differentiate between an object and its nature—as it is required to solve the problems of the many over one and the one over many—then one must hold that the nature in question should be identical to the resemblance class. If the nominalist argued, on the other hand, that the resemblance class is numerically different from the nature it grounds and that nature is numerically identical to the object that possesses it, then he would not be delivering an answer to the problem of the many over one, because there would not really be a distinction between being F and being G of the same object x. But, if the natures of the objects are for the nominalist identical to resemblance classes, it cannot be that one comes to know a nature without knowing a resemblance class. This leaves us again at the starting point. How could one know that, for example, something is spherical without having to know infinite objects of infinite possible worlds?

3.4

Determinate and Determinable

§ 20. It has been said above that one of the theoretical functions expected to be fulfilled by universals or what work as universals is to explain objective resemblances between objects that are grounded on determinate properties under the same determinable property. It is an objective fact that, if x has 10 kg of mass, y has 11 kg of mass and z has 20 kg of mass, then x is more similar to y than it is to z. These facts of objective similarity admit to being compared since the determinate properties of mass can be put into bijection with real numbers. Thus, the ‘distance’ or degree of similarity between the properties of having 10 kg of mass and having 11 kg of mass can be represented by the difference |11–10|, while the ‘distance’ or degree of similarity between the properties of having 10 kg of mass and having 20 kg of mass can be represented by the difference |20–10|. There is no single way to explain the relations between determinate and determinable properties by universals. It is not necessary to enter here in the detail of these alternatives. David Armstrong, for example, has argued that the unity of all determinate properties under the same determinable comes from certain ‘partial identity’ of such universals (see Armstrong 1978b, 116–131, 1997, 47–63). The central idea of Armstrong is that what makes the universal to have exactly 10 kg of mass is a determinate property of the same determinable than the property of having exactly 20 kg of mass is that—literally—everything that instantiates the universal to have 20 kg of mass also includes the universal to have 10 kg of mass. Indeed, everything

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that instantiates the universal to have 20 kg of mass has a proper part that instantiates the universal to have 10 kg of mass.12 Then, all the determinate properties of mass must share at least one ‘part’ in common. ‘Partial identity’ is what explains the unity of the determinate properties under a determinable, although, in Armstrong’s approach, there are no determinable properties. Other defenders of universals have proposed different solutions. John Bigelow and Robert Pargetter have argued that all determinate properties of the same level of being F are unified by the possession of the same ‘second-grade’ property of being F properties. Objects that have determinate properties of some type F also have a ‘second order’ property of owning some property of type F. The second-order properties are properties of objects that make it necessary for those objects to possess properties of first order of a certain type. Second-degree properties are properties of properties (see Bigelow and Pargetter 1990, 51–54). These different theoretical alternatives are not going to be discussed here.13 It is well known that in all of them, however, the unity of the determinable is given by the unity of the same universal, whether it is a “minimum” universal that has to be “part” of any other universal, or whether it is a universal of second degree. In any case, objective resemblances of those determinate properties under a determinable have to do with universals and with their aptitude to unify a plurality. In the different forms of nominalism, there is very little development regarding these issues. One might be inclined to think that any position that is the most theoretically appropriate in metaphysics of properties should be able to accommodate the relations between determinable and determinate properties. It has been suggested, for example, that all that the resemblance nominalist requires is to offer suitable resemblance classes to fulfill the functions of determinate properties. The determinable properties will result solely from the disjunction of these determinate properties (see Rodriguez-Pereyra 2002, 48–50). But clearly, this is insufficient to solve the problems involved in the relationship of determination between properties. The relation of determination is not reducible to the relation that may exist between a disjunctive property and the properties that constitute this disjunction, even though formally the relations of Upward Necessitation and Downward Necessitation may occur between, for example, the property of being F or being G and the properties of being F and being G taken in isolation.14 Anything that is F should be F or G. Anything F or G must be either F or be G. The problem here is that disjunctive

12

This presupposes that every physical object has arbitrary proper parts that occupy exactly one arbitrary spatial region that is a sub-region of the spatial region that the object occupies. Such an assumption has been called into question. Armstrong’s position would also require that there are ‘minimal’ properties that must be instantiated in the atomic parts of an object. If there are no mereological atoms, it is also doubtful that there are ‘minimal’ properties and, with that, that there is partial unity in the terms contemplated by Armstrong for all those determinate properties under a determinable. 13 Another proposal is that of Evan Fales in which a determinable property is fixed by classes of causal powers (see Fales 1990, 166–178). Neither will the details of this proposal be discussed. 14 Something similar happens with the relationship that may exist between the conjunctive property being F and G and the property of being F.

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properties are entirely indiscriminate. There is no requirement of a connection between the properties that may be part of the disjunction. This lack of connection is why disjunctive properties as authentic properties are inadmissible. But determinate properties under a determinable are specifying all the ways of possessing a determinable property, such that each of these forms is exclusive concerning the others. If one disjunctively connects properties F and G there is no requirement that F and G be mutually exclusive. Nor is there any requirement that disjunctively connected properties should be put into bijection with mathematical structures that will then help us to represent facts of objective similarity. The determination relationship requires much more structure than is captured in disjunctive properties. From the perspective of resemblance nominalism, relations between determinate and determinable should be explained by characteristics of primitive and external similarities (see Alvarado 2014). Natures of objects must be systematically connected in the way required by the relation of determination between properties. Each of the determinate properties can be identified by the nominalist with a resemblance class. They will be assumed to be resemblance classes as conceived by Rodriguez-Pereyra, that is, as ‘maximal perfect communities’, in the sense in which he understands these expressions. The unity of all determinate properties under the same determinable—i. e., the fact that all determinate properties are determinations of the same determinable—, the exhaustive character of all the determinate properties of the same level—i. e., the fact that they are all the ways in which something can have the determinable—and the exclusive character of those determinates among themselves—i. e., the fact that what has a determinate property cannot have any other determinate property of the same level—must be grounded on resemblances between objects. There must be, in particular, primitive external similarities between all the objects that make up a determinable, as there must be maximal perfect communities as determinate properties. Just as it is necessary to postulate primitive external similarities between all the objects and all the hereditary pairs of objects that make up a class of similarity associated with a given property, also primitive external similarities are required between all the objects and the hereditary pairs of objects that make up a class of similarity associated with determinable properties. Otherwise, it would not be possible to solve the problem of the imperfect community. In the same way, just as each of the perfect communities associated with the determinate properties must be maximal, the perfect community associated with the determinable property must also be maximal. Otherwise, it would not be possible to answer the companionship problem. It is notorious, moreover, that each determinable property is a companion of each of the determinate properties under it. In Rodriguez-Pereyra’s scheme, this does not offer any particular difficulty, because the degree of similarity that can be assigned to a determinate property must always be higher than the degree of similarity that can be associated with its determinable property. The degree of similarity that can be assigned to the determinable property, for its part, should always be higher than the degree of its companion. How one should try to account for the difference between determinable and determinate is through relationships of ‘superimposed’ similarities between objects

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and their hereditary pairs. There will be more strict similarities for determinate properties and more lax similarities for determinable properties. What differentiates a perfect community from one that is not, is that there are primitive similarities between all the objects of that community and between all the hereditary pairs of objects of that community. A perfect community, no matter if you have a finite or infinite number of elements, requires infinite facts of similarity for every level of hereditary pairs of objects. Both the class of similarity associated with a determinate property and the class of similarity associated with a determinable property must be perfect communities in this sense. Suppose a maximal perfect community of all objects that have exactly 10 kg of mass. All objects and all hereditary pairs of objects of the kind are similar to each other in respect to their mass. They are not similar to the objects of other maximal perfect communities with other determinate masses, such as, for example, the maximal perfect community of all objects that have exactly 10,0001 kg of mass. These same objects of different maximal perfect communities must be, however, similar to each other concerning their mass, for all of them must be elements of the same maximal perfect community associated with the determinable property of having mass. How can they be similar and dissimilar to each other in the same respect? In traditional cases of the companionship problem, it is usual to consider situations—real or hypothetical—in which, for example, all objects with the same mass have the same electromagnetic charge, but not all objects with that electromagnetic charge have the same mass. It is clear, however, that there are superimposed similarities here concerning mass and concerning charge, which are different respects. What happens when considering similarity classes of determinable and determined properties is that they require overlapping similarities concerning the ‘same’. Objects belonging to perfect maximal communities associated with different determinate properties must be unlike each other—otherwise, they would form the same maximal perfect community—and similar to each other— for they must build the same maximal perfect community concerning, for example, mass. Surely, a nominalist will argue that there is no such thing as a ‘respect of similarity’ between different objects. Everything there is—at least at the fundamental level—are simple primitive external similarities, with no surnames. What we call a “respect” is ontologically derivative from such fundamental similarities. The same object is F for being similar to these objects, and it is G for being similar to these others objects. Any object always has a multitude of similarities and dissimilarities superimposed by the simple fact that it is something multiply characterized. The overlapping similarities and dissimilarities regarding determinable and determinate should not generate any surprise, therefore. But this is what makes the situation so mysterious. An object is similar to certain other objects with which it forms a maximal perfect community and is unlike the rest. These similarities are what ground, for example, that something has a mass of 10 kg. That same object is similar to other objects with which it forms another maximal perfect community and unlike the rest. These similarities are what ground it, for example, to have a spherical shape. But the determination relation between having 10 kg of mass and having mass has substantial differences concerning the relationship that could be established between

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having 10 kg of mass and having a spherical shape. There is a connection in the first case that does not exist in the second. Whether or not an object has a mass of 10 kg does not seem to make it necessary or more likely to have a spherical shape. The fact that an object, on the other hand, has a mass of 10 kg makes it necessary for it to have a mass, and the fact that an object has a mass makes it necessary for it to have a certain mass and only one mass. Nor would it serve here for the nominalist to maintain that the connection between determinate and determinable is fixed by the fact that the maximal perfect community assigned to the determinable is a companion of the maximal perfect community assigned to the determinate. This maximal perfect community of having a charge of q can also be a companion of having n MeV/c2 of mass, but this does not make electromagnetic charge a determinable of having mass. This brings the question back to the point of departure. Why does it happen that the same objects—and the respective hereditary pairs—are similar and dissimilar to each other in the same respect? To appeal here to primitive facts of implication between similarities is to multiply the darkness of the primitive facts of external similarity and, more than an explanation, it is a label for the absence of it. The multiplication of primitive facts is not a good symptom of philosophical health for a theory. Resemblance nominalism makes the relations of determination, so important for the way we understand natural laws, a mystery.

3.5

Natural Laws and Inductions

§ 21. It has been indicated above (§§ 14–15) that universals make it possible to explain why natural laws have a regulatory role of what actually happens, why they ground counterfactual conditionals and why they imply regularities but are not identified with them. The postulation of universals also allows us to explain why our inductive practices are epistemologically reliable—at least, ordinarily. Can resemblance nominalism replicate these explanatory advantages? The primary motivation to sustain that natural laws are universals or are constituted by universals is that laws understood in this way allow to make an ontological difference between the regularities that happen accidentally and authentic natural laws.15 Of course, regularist theories renounce making this distinction. Many of the

15

Although it has been common to maintain that natural laws must, at least, imply regularities, this is not the case—a point that Daniel von Wachter has insisted on for years. Armstrong distinguishes between ‘iron’ laws without exceptions and ‘oaken’ laws that admit being overridden (see Armstrong 1983, 147–150). But natural laws, as they are ordinarily understood, do not imply regularities. If it is a natural law, for example, that opposite charges are attracted, and there are objects with charges of +q and –q, respectively, they will not necessarily come closer together. Some additional force could be exerted to prevent the deployment of that force of attraction. What the law predicts is the existence of a tendency that may or may not be reflected in the phenomena, according to what other forces are operating in the circumstances. In any real physical system, there is a multitude of

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objections that have been directed against the regularist theories can be answered if a judicious choice is made of the regularities that deserve the name of ‘laws’, and that differentiate them from the rest.16 However, the central question remains. If it seems reasonable to think that laws are really ‘regulating’ and ‘determining’ what happens, they should be more than regularities. The mere fact that one type of event is succeeded by another type of event does not bring any connection between them. It is interesting to consider here whether there would be any way to develop a theory of natural laws that preserves its special modal character within the framework of resemblance nominalism. Could resemblance nominalism replicate the functions of universals? Both the nominalist and the friend of universals can always appeal to some regularist conception of laws, but the defender of universals can also claim that laws are relations of a higher order between universals or simply universals. Is there any similar option for the nominalist, or should he content himself with a regularist view? As explained above (§ 14), there are two ways in which the ontology of natural laws has incorporated universals. One of these forms is defended by Armstrong (1983), Tooley (1977, 1987) and Dretske (1977) in which natural laws are relations of ‘necessitation’ or ‘nomological relations’ between universals. The other is holding that natural laws are identified with the causal powers that universal properties confer essentially to the objects that instantiate them (see Mumford 2004; Bird 2007). If the classes of similarity of the nominalist, then, are going to try to replicate the theoretical functions of universals, it should be considered how there could be analogous ‘nomological relations’ between resemblance classes, or how it is that a class of similarity or the facts of primitive external similarity that constitute it could confer causal powers. The connections between resemblance nominalism and causation will be dealt with in the next section (see § 22), so the question about nominalist causal powers will be deferred by then. According to the usual presentations, the nominalist only postulates particular objects, so one would be inclined to think that any ‘nomic’ connection would have to occur between such objects directly. This makes it difficult to think that, for example, there is a second order relationship between the nature of being F and the nature of being G in virtue of which an F is followed by a G. The nature of being F should not be something above or outside objects that are F. Things are not so simple in resemblance nominalism, however, as it has been shown. The nominalist postulates not only objects but also primitive external similarities. The ‘natures’ for

forces operating. Since many of these forces can be neglected, our mathematical models that ignore them have sufficient predictive value. Strictly, however, none of the laws implies a regularity. 16 This is what happens with the Lewis-Ramsey theory in which natural laws are those statements that achieve the axiomatization of everything that happens in a possible world that best satisfies the requirements of simplicity and informativeness (see Lewis 1973, 73). Note that here, the laws are ‘statements’ in a theory that describes how things are in a possible world. They do not have any regulatory function. Laws are those statements that fulfill a systematic function in such a description. In the theories of natural laws that give to these a regulatory role, laws are not statements, but entities that can be described in many different ways.

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the nominalist are the classes of similarity grounded on primitive resemblances. The question here, then, is whether one could think of natural laws as nomological connections between the ‘natures’ understood in this way. Since ‘natures’ are grounded on primitive similarities, if there is something that should enter into natural laws, they are these same similarities. For a natural law N(F, G), the general idea should be that the similarities that make something to be a F would be, by its very nature, that which makes it necessary that later there are similarities that establish that something is a G, in a way analogous as the way by which, for the defender of universals, the universal F, by its very nature, makes it necessary the instantiation of the universal G. The second order ‘nomological’ relation N of ‘necessitation’, on the other hand, should be understood as a class of similarity of ordered pairs of resemblance classes of objects. It is of a higher order since it does not group objects that are similar to each other but rather ordered pairs of resemblance classes of objects. Ordered pairs of resemblance classes are similar between them because they collect resemblance classes of objects that—given its ‘nature’—are followed by objects of other resemblance class. According to everything I know, nobody has wanted to explore a theory in these lines. Most nominalist philosophers have been reassured by some form of regularity theory. And this caution seems prudent because a theory in these lines would have several significant problems. In effect, the knowledge of a natural law allows us to ‘unify’ reality. What initially may seem to us as scattered phenomena that do not obey any rule becomes intelligible when we discover the law that ‘governs’ it. The fact that the law has a ‘regulative’ character makes us justified in believing that other phenomena of the same type will have the same nature.17 But this does not make any sense unless there exists the same type of phenomena involved. When it is the existence of the same universal what determines the presence of the same type of phenomenon, it is rational to suppose that the same universal will determine the same pattern of development for the process in question. When there are objects numerically different from each other collected by their primitive relations of mutual similarity, there do not seem to be sufficient guarantee for an analogous ‘unification’. Assigning the ‘bearer’ of nomological necessitation from objects to similarities does not change the situation since similarities are—ex hypothesi—individual entities. It is a primitive fact for the nominalist that the particular similarities that connect objects to form classes of similarity are, in effect, similarities, and not other relations. There could not be a universal of similarity to which assign nomological necessitation, because a nominalist who postulates a universal is no longer a nominalist. Particular similarities are particular. What a particular resemblance necessitates or does not should have no bearing on what a different particular similarity necessitates or does not. Even accepting that it is—in some way—grounded in the nature of a similarity of those that grounds that something is an F, for example, that it is

17

And it is not necessary that laws are deterministic. Stochastic laws—insofar as determinate objective probabilities can be assigned for the process in question—also allow for such ‘unification’.

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necessary that something then have similarities of those that ground that something is a G, there is no reason to assume that the nature of a particular resemblance has to constrain another particular resemblance. Moreover, according to resemblance nominalism what makes, for example, that an object x is F and is not G is not the intrinsic character of the similarities of x with other objects, but it is the mere fact that x is similar to certain objects rather than others. If the maximal perfect community that fulfills the functions of the property of being F has as elements x, y1, y2,. . . and the maximal perfect community that fulfills the functions of the property of being G has as elements y3, y4, . . ., then what grounds that x is F and is not G is the fact that x is similar to y1 and is similar to y2, etcetera, while it is not similar to y3 or y4, and so on. It is, in other words, the identity of the objects with which one has or not resemblance what makes x possess the nature that it possesses and not the ‘nature’ of the relationship of similarity. It is foreign to resemblance nominalism to maintain that the similarities that ground the nature of being F should have a different qualitative character from the similarities that ground the nature of being G so that, in effect, F 6¼ G. Resemblances are qualitatively uniform.18 The distinction between natures is grounded on the relata of such qualitatively uniform similarities. Then, how is it that some resemblances should ‘nomologically’ necessitate some other resemblances? It would be necessary to introduce qualitative differences of a fairly substantive character between different similarities. Not only the nominalist would require to postulate primitive facts of external similarity, but he should also maintain that, also as a primitive fact, similarities should possess certain qualitative characters by which different similarities necessitate the occurrence of similarities of a certain type rather than another. This multiplication of primitive facts, again, is not a symptom of good philosophical health.19 It is also difficult to know in what sense an ontology of this kind would be ‘nominalist’. It would be postulating objects, but also a plethora of similarities— particular relationships—each endowed with a primitive qualitative character. These primitive similarities endowed with a specific qualitative character seem to be simply tropes. The supposed ontological simplicity of nominalism—which claims an ontology with only one category of entities: objects—would be achieved at the price of

As far as can be supposed. Recall that ‘similarities’ are primitive external relationships, whose intelligibility is already a significant problem, as explained in § 18. 19 Anyone who has some familiarity with the discussion of Armstrong, Tooley, and Dretske’s theory of natural laws will note that an objection of this kind has been directed precisely against it: why would there be a necessary connection between different universals? Calling this relationship “necessitation” does not really make the universals to be connected in the way intended (see Lewis 1983, 40). This lack of connection has been one of the reasons why the most recent defenders of non-Humean conceptions of natural laws have preferred to maintain that laws are simply primitive causal powers. No second-order relationship is required to ‘connect’ universals with each other and to bring about a natural law. It is part of the essence of a universal to confer on its instantiations specific powers to enter certain causal relationships (see Mumford 2004, 143–159). The universal by itself is already the natural law. Defenders of non-Humean conceptions are well aware of the difficulty and have proposed feasible solutions. 18

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the abandonment of nominalism, which is a confirmation of the failure of the nominalist program. It happens, then, that the nominalist cannot replicate the explanatory advantages of universals concerning natural laws, except at the price of distorting nominalism to a point where it is hardly recognizable and in which the theoretical virtues of a nominalist position—advantages of economy for which it might seem attractive at first sight—have vanished. It is certainly more reasonable for the nominalist to place all his hopes on some form of regulating theory of natural laws. For these same reasons, there is no reasonable response to the problem of induction in resemblance nominalism, or in any other form of nominalism. It is highly doubtful how we could get to know, for example, the nature of being F, as it has been explained above (see § 19). Much more dubious is that we get to know with sufficient confidence that, for example, all F is G. The ‘nature’ of being F and the ‘nature’ of being G are grounded on infinite primitive similarities between merely possible objects, necessarily transcendent to our cognitive capacities—or the cognitive capabilities of an angel, for the same reasons. Much less we could have epistemic access to a regularity according to which everything F is G. It is difficult, then, to justify why we are rationally justified in our inductive practices from a nominalist perspective.

3.6

Causal Powers

§ 22. As stated above (see § 13), it has been common to maintain that properties confer ‘causal powers’ on the objects that instantiate them, because the causal relations in which an object enters is a function—among other things—of what properties possess such object. However, the nature of this connection between properties and causality is variable, according to how the causal relationship is understood. The incidence that resemblance classes have for causality will be, for the same reasons, variable. What is expected from a reasonable theory in metaphysics of properties is for it to be in harmony with our intuitions about the dependence of causal relationships on the intrinsic properties of the relata, or at least the dependence on the intrinsic properties of the objects that make up such relata.20 It should not be assumed, of course, that the intrinsic properties constituting an event make necessary the existence of a causal connection between that event and its effect. In principle, a reasonable theory in metaphysics of properties should be neutral about how causality should be understood and, in particular, about whether there are causal 20 Recall that an ‘event’ has usually been understood as the instantiation of a property in an object at a time, or as the instantiation of a relationship in several objects—according to the adicity of the relationship—at a time. This specification of what is an event should be taken cum grano salis. In principle, ‘instantiation’ should be taken here in a way that does not prejudge against nominalist or the defender of tropes. In the same way, ‘object’ should not be taken here in a way that prejudges against the defender of trope bundles.

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relationships that do not make necessary the occurrence of the effect, for example, because they only operate stochastically. What is required, in principle, is that the causal connections are ontologically dependent on the properties involved in the events of cause and effect. There does not seem to be any problem, in the first place, for resemblance nominalism, if causality is understood as regularity with the other usual associated conditions—the spatiotemporal contiguity of cause and effect and the temporal anteriority of the cause. From the perspective of regularist theories of causality, causal connections are grounded on highly disconnected facts, as has been explained. Nor would this conception of causality suffer too much, if one were to replace classes of similarity by mere classes, even if they have very heterogeneous objects as elements. Although each singular causal relationship must occur between events that must be contiguous with each other, the relationship depends on whether each event of the type of the cause is respectively contiguous with events of the type of the effect. These different events can be extremely distant from each other and, of course, occupy spatiotemporal regions disconnected from each other. Then it does not matter much that the nature by which an object is, for example, F is grounded on all other objects with which the respective similarity class is formed. These different objects, of different possible worlds, occupy spatiotemporal regions disconnected from each other so that the fact that x is F is—to put it in some way—‘dispersed’ in different disconnected regions in different possible worlds. From this perspective, causality will be dependent on the classes of similarity involved in the event cause and the event effect, since causality depends on regularities. Regularity between, for example, being F and being G depends on there being the respective natures of being F and being G. Nor does there seem to be a great difficulty for causality to be dependent on resemblance classes if causality is understood as counterfactual dependence. The existence of a causal relation in the possible world w between two events c and e is reduced to the fact that the following counterfactual conditionals are true: if c did not occur, then e could not occur; and if c occurs, then e could occur. The truth of these counterfactual conditionals is grounded on the fact that in all the closest possible worlds to w in which c exists, e exists also, and where c does not exist, neither does e. Causal facts are, therefore, grounded on similarities between possible worlds. The similarity between possible worlds will be a function of the similarities between their respective parts. The usual way of evaluating counterfactual dependencies—which go from the present to the future, and not from the present to the past—is to consider worlds “duplicated” until the moment in which the cause event occurs. If the occurrence of the effect in that class of worlds—which are ‘close’ because they share, at least, an initial segment in common and possess the same natural laws— varies according to whether the event cause occurs, then there is a causal connection between them. It is evident that for two possible worlds to count as duplicates among themselves, they must possess the ‘same’ properties. A resemblance nominalist can comfortably accommodate this requirement. Two worlds will count as duplicates in a given segment if, correlatively, each of the respective parts of those worlds in that segment belongs exactly to the same classes of similarity. Causality here is not a

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‘local’ relationship between events but is grounded on trans-world facts about the regions of possible worlds closest to the world of evaluation. The fact that the nature of an object in resemblance nominalism is not something of a ‘local’ character for the object to which such nature is ascribed is therefore of little importance. Difficulties arise if a non-reductive conception of causality is accepted. The causal relationship in this perspective must be a ‘local’ connection between the cause event and the effect event. An event c which is the cause of the event e must concern what happens in the spacetime region that is occupying c and e, and not with what happens in other disconnected regions.21 The problem is that no nature is something ‘local’ in the perspective of resemblance nominalism. Suppose that there are two objects, x1 and x2 at a distance d from each other. The object x1 has an electromagnetic charge of q1, while the object x2 has an electromagnetic charge of – q2. These facts cause that there is a force of attraction between x1 and x2 directly proportional to the product of the charges and inversely proportional to the square of the distance between x1 and x2 according to Coulomb’s law. From a non-reductivist perspective, this causal connection is not grounded on what happens in the rest of the universe—as it happens in regularist theories—, nor on what happens in nearby possible worlds—as it happens in counterfactual theories. Causality seems to be dependent on the intrinsic natures of x1 and x2 and the distance d between them, but not on what happens or does not happen in other regions disconnected from them. The charge of q1 possessed by x1, nevertheless, is from the perspective of the resemblance nominalist a fact ‘scattered’ in the total region that results from the fusion of all the spatiotemporal regions22 occupied by each of the objects that make up the class of similarity with which that nature should be identified. This disconnected ‘total’ region will have to include regions of different possible worlds. At the very least, the nominalist should suppose that the charge of x1 is something grounded extrinsically in different objects that occupy disconnected regions of spacetime in different possible worlds. Why should it be relevant to the causal interactions between x1 and x2 what may or may not happen in distant regions of our world, not to mention what happens in other possible worlds? Resemblance nominalism imposes a violation of the requirement of locality. Of course, the nominalist could embrace some reductive alternative to causality. If there were independent reasons to prefer any of these options, this should not imply any problem. If there are not, however, nominalism is shown as a theoretically uneconomical position since it closes alternatives that are open to those who postulate universals. On the other

This requirement of ‘locality’ is much weaker than the requirement usually presented under that name. It is not required that the event cause and the event effect be mutually contiguous (see Lange 2002, 1–25). Nothing prevents, for example, phenomena of quantum entanglement (see Lange 2002, 280–299). What is required, however, is that a causal connection between the events c and e involves the spatiotemporal regions occupied by c and e, along—eventually—with the distance between such regions and no other. 22 Or of the union of such regions, if a set-theoretic perspective is adopted and regions are conceived as sets of points with topological properties. A region r is said to be “disconnected” if and only if it is the union of at least two disjoint and closed regions. 21

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hand, if there are independent reasons to prefer a non-reductive conception of causality, there would be reasons to reject nominalism directly. It turns out, therefore, that only with great caution can the nominalist maintain that the ‘nature’ of objects determines causal powers in their possessors.

3.7

A Vicious Regress

§ 23. The form of resemblance nominalism proposed by Rodriguez-Pereyra has a specific problem. As explained above (see § 17), a perfect community is a class in which all the objects that integrate it and all the hereditary pairs of objects that make it up are similar to each other. In Rodriguez-Pereyra’s approach, this requirement allows evading the problem of the imperfect community, since, in every imperfect community, there must be a hereditary pair of objects that are not similar to each other. A class of similarity, then, of those that should satisfy the functions assigned to universals must be grounded on infinite facts. Not only because such classes will be integrated by infinite possible objects, but also because there are infinite facts of resemblance between infinite pairs of hereditary objects of the class. Even if a class of similarity was composed of a finite number of objects, there should be infinite facts of similarity to relate each of the infinite pairs of hereditary objects of that class. There is no doubt, on the other hand, that similarity classes are grounded on primitive similarities between the objects that make them up. It is because there is a similarity between all those objects that there is an authentic class of resemblance of those that could replace universals. The risk of a vicious regress arises here because there is a grounding structure between the different similarities. As Rodriguez-Pereyra formulates the position, similarities of ‘first order’ are not enough to have an authentic perfect community. Resemblances of the nth-order are needed for every hereditary pair of level n. What connection exists, however, between similarities of first order and similarities of a higher order? There are two alternatives: (i) that the similarities of higher order are grounded on similarities of the first order; or (ii) that the similarities of order n are grounded on similarities of order n + 1. Either there is a ‘bottom-up’ structure grounded on the first-order similarities, or there is a ‘top-down’ structure in which each similarity must be grounded on similarities of higher-order. If the situation here were the one indicated in (i), we would find ourselves facing a benign infinite regress. But if the situation were the one indicated in (ii) we would find ourselves facing a vicious infinite regress. It would be, in effect, a situation in which the similarities of the first order should be grounded because they are not enough in themselves to solve the problem of the imperfect community. No grounded entity exists without its ground. First-order similarities demand their grounds, therefore. However, each of the grounds of those first-order similarities must also be a grounded entity that requires its grounds, because they are not sufficient to solve the problem of the imperfect community, etcetera. This sequence is infinite. Then, no entity is fundamental—an ungrounded ground—because all the facts of similarity of

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this series should be grounded on other facts of similarity to generate a perfect community. There being nothing fundamental, none of the grounded entities of the series can be given. And if there are no similarities, there is no class of similarity either. On the one hand, it may seem that this infinite sequence of similarities is benign. As Rodriguez-Pereyra specifies the similarities between hereditary pairs of objects, these seem to be grounded on ‘first order’ properties of the objects that make up the class. Recall that Rodriguez-Pereyra defines a function f that assigns its properties to each entity in the hereditary pair hierarchy, according to these recursive clauses:  f ðxÞ ¼ X 0 1 , X 0 2 , . . . , X 0 m if and only if x is a particular object and the  elements of X 0 1 , X 0 2 , . . . , X 0 m are all the sparse properties of x:  f ðxÞ ¼ X nþ1 1 , X nþ1 2 , . . . , X nþ1 m if and only if x ¼< y, z > and ð f ðyÞ \ f ðzÞÞ ¼ fX n 1 , X n 2 , . . . , X n m g: f ðxÞ ¼ ∅ in any other case: It seems, then, that there could only be a similarity between ordered pairs of the nth order if entities of the n-1th order integrating these pairs previously share properties. And hereditary pairs of the n-1th order share properties only if entities of n-2th order already share properties, etcetera. Ultimately, the fact that an ordered pair of objects has assigned properties X11, X12, . . ., X1n seems to be based on the fact that each of the objects that integrate it has the properties X01, X02, . . ., X0n. Thus, all similarities of hereditary pairs of objects seem grounded on the properties of the objects of the class. It appears to be, therefore, a ‘bottom-up’ structure according to (i). But this ‘bottom-up’ grounding structure is an illusion. No object can be assigned a ‘nature’ if it is not by the similarities that that object possesses with others with which it can form an appropriate class of similarity, that is, a perfect maximal community. And, according to the form of nominalism that is considered here, similarities between objects are not sufficient to generate a perfect community. The entire infinite sequence of similarities between hereditary pairs of objects is also required. The situation, then, is really the following: a nature, be it F0, can be attributed to all the objects that make up a class of similarity α only if: (i) all the objects of α are similar to each other, and (ii) all pairs of objects of α are similar to each other. The nature shared by all these objects is grounded on such similarities. The similarities between objects would be insufficient because they cannot discriminate cases where there is a perfect community from cases in which there is an imperfect community. Similarities between ordered pairs of objects are required. However, now it happens that neither similarities between ordered pairs of objects of α are sufficient to differentiate a perfect community from an imperfect community. For the existence of a similarity between such ordered pairs that generates a perfect

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community, it is required: (i) that all ordered pairs of objects of α are similar to each other, and (ii) that all ordered pairs of ordered pairs of objects of α are similar to each other. But here again, we find that the similarity between ordered pairs of ordered pairs of objects of α is not enough to discriminate between a perfect community and an imperfect community. The similarity between ordered pairs of ordered pairs of ordered pairs of objects of α is required. It turns out, therefore, that the similarities between objects are not enough to generate a perfect community. The perfect community must be grounded on these similarities, but also on the similarities of pairs of objects, the similarities of pairs of pairs of objects, and so on, to infinity. No level in the hierarchy of similarity of ordered pairs is sufficient to ground a perfect community without the help of the superior level of similarity. It is, therefore, a vicious infinite regress. How should the function f be understood? It cannot be seen as describing the grounding structure between facts of resemblance and the respective nature. It is merely a way of making precise what the degree of similarity between arbitrary hereditary pairs is. Recall that the ‘degree of similarity’ between two entities is the number of ‘shared properties’ between those entities. The issue here is that a hereditary pair will have a degree m of similarity with another hereditary pair if and only if those pairs ground albeit partially m ‘properties’ or ‘natures’ in the objects that make up such hereditary pairs. The ‘properties’ or ‘natures’ grounded on such similarities of hereditary pairs help to define their degree of similarity, but should not be seen as grounding, in turn, such similarities.23 There persists, then, the problem of an infinite sequence of similarities, none of which is in itself sufficient to ground a perfect community, as intended. This infinite sequence is a reason to think, therefore, that there is not such a perfect community, because its ground is not possible. A structure of this type in which the similarities of the hereditary pairs ground the ‘natures’ of the objects and, in turn, the ‘nature’ of the objects ground the similarities of the hereditary pairs would be a structure in which both the similarities of the hereditary pairs and the ‘natures’ of the objects would be grounded on themselves. This reflexive grounding is unacceptable if it is a relation of strict grounding that is irreflexive and transitive—and, therefore, asymmetric. It would be acceptable if it were weak grounding, but it would require to assume that ‘natures’ and similarities of hereditary pairs are identical. As indicated above (see § 19) the nominalist must identify the natures with the similarity classes, but not with the similarities which ground such classes. But why not also identify the class of similarity with the similarities that ground it? The problem is that no singular entity—a class—can be identified with a plurality. A statement like “α ¼ the Fs” does not make sense, because “the Fs” designates collectively, in effect, a plurality of entities. A statement like “the Fs ¼ the Gs” in which the sign of identity is flanked by designations of pluralities makes sense, but the meaning of such a statement is derivative of the identity between individuals, because: 23

ðFs ¼ GsÞ¼df ððx1 ¼ y1 Þ ^ ðx2 ¼ y2 Þ ^ . . . ^ ðxn ¼ yn ÞÞ Here, “the Fs” is a plural designation of x1, x2, . . ., xn; and “the Gs” is a plural designation of y1, y2, . . ., yn. Thus, it would make sense to state that, for example, “α ¼ x1” where x1 is one of the Fs, but cannot be identified with the plurality of the Fs. The similarity class α could not be identified either with the plurality of similarities between hereditary pairs.

3.8 Modal Consequences

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Modal Consequences

§ 24. A very well-known characteristic of the forms of resemblance nominalism defended by David Lewis or by Gonzalo Rodriguez-Pereyra is that they depend on an extreme realistic conception—or possibilist conception—of the possible worlds and their inhabitants (see Lewis 1983, 9–10, 1986, 50–53; Rodriguez-Pereyra 2002, 99–104). In the case of Lewis, one could suppose that it is simply a matter of taking advantage of the postulation of possible worlds, as he understands them, that is, as huge objects disconnected from each other spatiotemporally, that have the same nature as the great object that we inhabit, the actual world. The postulation of these possible worlds solves the problem of co-extensive properties, for example. A different question is whether resemblance nominalism requires per se a modal ontology of the kind defended by Lewis. This requirement is what RodriguezPereyra holds. According to Rodriguez-Pereyra, the only way in which resemblance nominalism can solve the problem of co-extensive properties is through extreme modal realism. One could here be inclined to think that this is a hasty conclusion. There are many different conceptions about the nature of modality and possible worlds (see Divers 2002; Alvarado 2006, 2008a). All that is required to solve the problem of co-extensive classes is to leave room for possible objects of a resemblance class whose actual elements are the same as the actual elements of other resemblance class, that belong to one of these classes and not to the other. If, for example, one wants to differentiate the class of similarity associated with ‘animal with heart’ from the class of similarity associated with ‘animal with kidneys’, then all that would be required is a possible non-actual object that belong to the class of animals with hearts and not to the class of animals with kidneys, or vice versa. Some have argued that the modal ontological space of possible worlds is constituted by maximally consistent stories in some language (see Carnap 1947). Others have argued that these are combinations of objects and properties representing possible states of affairs (see Armstrong 1989b). Others have argued that these are maximal structural universals that specify how all things might be by specifying the nature of each of the parts of the world and their mutual relationships (see Forrest 1986; Alvarado 2007, 2008a, 2010). Others have argued that possible worlds are maximal possible states of affairs (see Plantinga 1974). Why could not the resemblance nominalism appeal to any of these conceptions to give an account of possible objects? None of these theories assumes that possible worlds are parallel universes. These are actualist theories in which the actual world has an ontological privilege and can be characterized, without difficulty, as the mereological fusion of everything, without restriction—it is not simply, ‘the mereological fusion of all objects that are at some spatial or temporal distance from myself’. ‘Possible’ worlds are abstract representations of how things might be or ways in which things might be (see § 3). If resemblance nominalism is a reasonable position, it should be, at least, neutral among all these alternatives or most of them.

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It happens, however, that none of these theories would be adequate for resemblance nominalism. Modal theories based on structural universals can be excluded from now on. A nominalist could not admit into his ontology universals while remaining nominalist. Neither would combinatorial theories in the style of what Armstrong proposed (see 1989b, especially, 37–118) be adequate. In these conceptions, the modal space of metaphysical possibilities is determined by the mutual independence between actually existing objects and actually existing universals— which, for Armstrong, as is well known, are immanent universals having at least one actual instantiation. One can conceive variations in this general scheme in which, for example, instead of generating the modal space by the combinatorial of objects and universals, it is produced by tropes independent among themselves.24 The problem is that in no way would such a theory cohere with the fundamental ontology postulated by the nominalist—i. e., an ontology in which the only fundamental entities are particular objects. For a merely possible object to obtain in the modal ontological space, that is, for example, for the existence of a merely possible animal with a heart, even if it is not an animal with kidneys, it would be necessary to postulate an object that does not actually exist. This postulation, however, would not be acceptable to a nominalist.25 Only actual objects would allow—eventually—to generate the modal space in which there is a non-actual object that is animal with heart and without kidneys. Neither would resemblance nominalism postulate possible worlds as ‘maximal possible states of affairs’ (see Plantinga 1974, 44–45).26 Plantinga understands ‘states of affairs’ as Roderick Chisholm has characterized them (see Chisholm 1976, 114–137), that is, as abstract entities of necessary existence, such that some of them obtain and some of them not. A ‘state of affairs’ understood in this way is an entity that can occur recurrently (see Chisholm 1976, 128–130), that is, it is a universal property (see Chisholm 1989, 141–149). A nominalist could not accept an ontology of this type without ceasing to be a nominalist. The situation is a bit more complicated when it comes to the linguistic theories of modality, but neither are they appropriate for resemblance nominalism. In the linguistic theories of modality, possible worlds are identified with maximally consistent sets of sentences of a language. That is, for a language L there is a set of sentences that can be formulated according to the syntactical rules for forming expressions from other expressions of L. In any natural language, given a finite

24

This is the conception of the nature of the tropes defended in authors such as D. C. Williams (see 1953a, b) or Keith Campbell (1981, 1990), but not the one that will be defended in this work. See §§ 87–94. 25 Unless, of course, the nominalist accepts possible worlds and their inhabitants as Lewis conceives them. What is at issue here, however, is to consider whether resemblance nominalism could work without such an ontological commitment. 26 A possible state of affairs S is ‘maximal’, according to the definition of Plantinga, if and only if, for every possible state of affairs S0 , either S includes S0 , or S excludes S0 . In general, a state of affairs S includes S0 if and only if it is not possible for S to obtain and S0 not. A state of affairs S excludes S0 if and only if it is not possible that S and S0 both obtain (see Plantinga 1974, 45).

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number of semantically basic expressions, infinitely many different sentences can be constructed. Each possible world is identified with a set of sentences W such that for each well-formed sentence s of L, either s belongs to W, or “not-s” belongs to W. Every possible world, then, is a story that describes in full detail according to the expressive resources of L how things might be. Since natural languages do not have names to designate all existing objects, nor names to designate all existing properties—however, such properties are understood—a “Lagadonian” language has been proposed in which each object is named by itself and each property is the name of itself (see Lewis 1986, 145–148). A Lagadonian language is a language that, by definition, has expressive resources to designate everything that actually exists. The problem here is that what the resemblance nominalist requires are possible non-actual objects. Non-actual objects should be elements of the nominalist’s similarity classes, together with actual objects. But there are no merely possible objects. Its metaphysical possibility is given by stories in which something is said according to the expressive resources of a language. Several mechanisms have been proposed to represent the possibility of non-actual objects and ‘alien’ properties,27 but it is not necessary to go into the detail of these discussions now. Those mechanisms will be considered below (see § 44; Alvarado 2008b). For what is relevant here, the resemblance nominalist would require the truth of: (1)

It is possible that there exists an x such that: x is F and x is not G and actually for all y: x 6¼ y.28

It should be assumed here that F and G would be the different putative properties actually co-extensive. Since it is possible that something exists that is F and is not G, the similarity classes of being F and being G would not be co-extensive. In a linguistic theory of modality, this metaphysical possibility must be grounded by the existence of a maximally consistent set of sentences, W, which contains the sentence: (2)

There is an x such that: x is F and x is not G.

And it seems very simple to include sentences of this kind in the maximal consistent sets of sentences that are the possible worlds for the defender of a linguistic theory. If the terms “is F” and “is G” exist, why could not there be a sentence like (2)? But the question is much more complicated. The nominalist is interested in a sentence like (2) because “it is F” and “it is G” are predicates that have—or should have— semantic value as authentic properties—of course, understood in the way they are understood by the nominalist, as classes of similarity. The problem is that their semantic values should be fixed by the actual resemblance classes in which such properties consist (see Alvarado 2012). In effect, a class of similarity cannot be fixed

A property is called “alien” if and only if it is not instantiated in the actual world, nor is it constituted by properties instantiated in the actual world (see Lewis 1983, 37). 28 More perspicuously, using quantified modal logic: [⋄∃x (Fx ^ ØGx ^ A8y (x 6¼ y))]. The operator A of actuality restricts the value of what is within its scope to the actual world. 27

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but by the objects that belong to such a class, and such cannot be but existing objects. It happens, then, that from the perspective of a resemblance nominalist a sentence like (2) would be unintelligible. One may suppose that the classes with which the natures of being F and being G are identified might be different from how they are. But this is metaphysically impossible. No class can have different elements from those it has because the conditions of identity of a class are its elements. Admitting a sentence like (2) would imply ignoring the assignment of semantic value that has already been made to the predicates “is F” and “is G”. It turns out, then, that there do not seem to be ways of accommodating resemblance nominalism to an actualist metaphysics. At least, it does not seem to be able to accommodate any of the known forms of actualism. It cannot be excluded a priori that some different form of actualism appears in the future, but as long as that does not happen, the most reasonable thing is to come to this conclusion: resemblance nominalism only seems to work with extreme modal realism. It is not necessary to explain here why an ontology of possible worlds, as postulated by David Lewis, is extremely implausible.29 If resemblance nominalism can only work with these ontological commitments, then it is also an extremely unbelievable philosophical position. Here concludes the examination of resemblance nominalism. As has been shown, it is an unlikely position in which the—eventual—advantage of ontological economy is achieved at a very high cost. Obscure ‘primitive facts’ are required that seem to be merely relationships. It requires to renounce a reasonable conception of natural laws and causal powers. It makes it difficult to explain the necessary systematic connections between determinate and determinable properties. It turns all the knowledge that we claim to have about properties—a great part of what natural science is about—unjustified. In the version defended by Rodriguez-Pereyra it leads to a vicious infinite regress. Finally, it seems to work only accepting implausible ontological commitments with parallel universes.

References Allen, S. R. (2016). A critical introduction to properties. London: Bloomsbury Academic. Alvarado, J. T. (2006). ¿Qué es el espacio ontológico modal? Philosophica, 29, 7–44.

29

A useful presentation of the criticisms against Lewis’s extreme modal realism can be found in Pruss (2011), 63–123. Lewis argues that the modal facts—i. e., the facts about what is necessary, possible or contingent—are grounded on a plurality of ‘possible worlds’, understood as entities of the same nature as the actual world. A possible world is a mereological fusion of all objects that are at a spatial or temporal distance between them. These are concrete entities (pace Lewis 1986, 81–86). How could we know such concrete entities if there is no causal connection with them? Furthermore, how does Lewis claim that such worlds obey a ‘principle of recombination’ by which anything can exist together or separated from any other (see Lewis 1986, 86–92)? Indeed, it cannot be known that the space of possible worlds satisfies that principle by inspection of those worlds.

References

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Alvarado, J. T. (2007). Mundos posibles como universales estructurales máximos. Una conjetura ontológica. Análisis filosófico, 27 N 2, 119–143. Alvarado, J. T. (2008a). ¿Qué son los mundos posibles? Intus legere, 2(1), 1–23. Alvarado, J. T. (2008b). Teorías modales lingüísticas refinadas. Pensamiento, 64(240), 315–343. Alvarado, J. T. (2010). Espacio modal y universales estructurales máximos. Estudios de filosofía, 41, 111–138. Alvarado, J. T. (2012). Referencia directa en los términos de clases naturales. Reflexiones ontológicas. Areté, 24(2), 231–262. Alvarado, J. T. (2014). Propiedades determinadas, propiedades determinables y semejanza. Discusiones filosóficas, 15(24), 129–162. Armstrong, D. M. (1978a). Universals and scientific realism, Volume I, Nominalism and realism. Cambridge: Cambridge University Press. Armstrong, D. M. (1978b). Universals and scientific realism, Volume II, A theory of universals. Cambridge: Cambridge University Press. Armstrong, D. M. (1980). Against “ostrich” nominalism: A reply to Michael Devitt. Pacific Philosophical Quarterly, 61(4), 440–449. Armstrong, D. M. (1983). What is a law of nature? Cambridge: Cambridge University Press. Armstrong, D. M. (1989a). Universals. An opinionated introduction. Boulder: Westview. Armstrong, D. M. (1989b). A combinatorial theory of possibility. Cambridge: Cambridge University Press. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Armstrong, D. M. (2004). Truth and truthmakers. Cambridge: Cambridge University Press. Bergmann, G. (1967). Realism. A critique of Brentano and Meinong. Madison/Milwaukee: The University of Wisconsin Press. Bigelow, J., & Pargetter, R. (1990). Science and necessity. Cambridge: Cambridge University Press. Bird, A. (2007). Nature’s metaphysics. Laws and properties. Oxford: Clarendon Press. Bradley, F. H. (1897). Appearance and reality. Oxford: Clarendon Press. Campbell, K. (1981). The metaphysic of abstract particulars. Midwest Studies in Philosophy, 6, 477–488. Campbell, K. (1990). Abstract particulars. Oxford: Blackwell. Carnap, R. (1928). The logical structure of the world (p. 1969). Chicago: Open Court. Carnap, R. (1947). Meaning and necessity. Chicago: University of Chicago Press. Chisholm, R. M. (1976). Person and object. A metaphysical study. Chicago: Open Court. Chisholm, R. M. (1989). On metaphysics. Minneapolis: University of Minnesota Press. Divers, J. (2002). Possible worlds. London: Routledge. Dretske, F. (1977). Laws of nature. Philosophy of Science, 44, 248–268. Edwards, D. (2014). Properties. Cambridge: Polity. Fales, E. (1990). Causality and universals. London: Routledge. Forrest, P. (1986). Ways worlds could be. Australasian Journal of Philosophy, 64(1), 15–24. Goodman, N. (1966). The structure of appearance. Indianapolis: Bobbs-Merrill. Imaguire, G. (2014). In defense of Quine’s ostrich nominalism. Grazer Philosophische Studien, 89, 185–203. Imaguire, G. (2018). Priority nominalism. Grounding ostrich nominalism as a solution to the problem of universals. Cham: Springer. Lange, M. (2002). An introduction to the philosophy of physics. Locality, fields, energy, and mass. Oxford: Blackwell. Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell.

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Melia, J. (2005). Truthmaking without truthmakers. In H. Beebee & J. Dodd (Eds.), Truthmakers. The contemporary debate (pp. 67–84). Oxford: Clarendon Press. Mumford, S. (2004). Laws in nature. London: Routledge. Paseau, A. C. (2015). Six similarity theories of properties. In G. Guigon & G. Rodriguez-Pereyra (Eds.), Nominalism about Properties. New Essays (pp. 95–120). London: Routledge. Plantinga, A. (1974). The nature of necessity. Oxford: Clarendon Press. Pruss, A. R. (2011). Actuality, possibility, and worlds. London: Continuum. Rodriguez-Pereyra, G. (2002). Resemblance nominalism. A solution to the problem of universals. Oxford: Clarendon Press. Schulte, P. (2019). Grounding nominalism. Pacific Philosophical Quarterly, 100, 482–505. Strawson, P. F. (1959). Individuals. An essay in descriptive metaphysics. London: Methuen. Tooley, M. (1977). The nature of laws. Canadian Journal of Philosophy, 7, 667–698. Tooley, M. (1987). Causation. A realist approach. Oxford: Clarendon Press. Williams, D. C. (1953a). On the elements of being: I. The Review of Metaphysics, 7, 3–18. Williams, D. C. (1953b). On the elements of being: II. The Review of Metaphysics, 7, 71–92.

Chapter 4

The Superiority of Universals Over Classes of Tropes

Abstract The second alternative to universals that is considered in this work is the ontology of classes of tropes. Defenders of these ontologies have sustained that classes of tropes are free from the problems that affect resemblance nominalism while still evading universal entities. It is argued here that these supposed advantages are illusory. Resemblance classes of tropes have the same difficulties of resemblance classes of objects, because the relation of resemblance relevant is also in this case ‘external’ to tropes. Besides, an ontology of tropes without universals is inadequate for an actualist modal metaphysics. If resemblance classes of tropes are substituted with ‘natural’ classes, the situation does not improve. § 25. From their first formulations in the twentieth century, the ontologies of tropes have tried to dispense from universals through classes of tropes that should fulfill their same functions (see Stout 1923; Williams 1953a, b; Campbell 1981, 1990; Denkel 1996; Maurin 2002; Ehring 2011). The ontologies of tropes have had as central motivation the same search for ontological economy that has motivated the nominalists since they are only postulating the existence of particular entities. Unlike the nominalist, however, it is about the postulation of properties and not of objects. An object is multiply characterized. On the other hand, a trope possesses or grounds a unique characteristic—at least, in principle. A particular object has, for example, a shape, a size, a mass, and so on. A trope of mass, however, would only be characterized by its mass and nothing else. Supposedly, this fundamental difference would bring a series of theoretical advantages for the friends of the tropes over the nominalists. The nominalists must explain how it is that the same object has different characteristics or natures different from each other—the problem of the many over one (see § 11). For the friends of the tropes this does not constitute any difficulty, because if an object has a certain mass and a certain electromagnetic charge, these properties will be tropes numerically different from each other. However, the advantages that an ontology of tropes has in the abstract when it comes to the problem of the many over one are compensated by special explanatory demands when it comes to offering a plausible theory of the nature of particular objects. The existence of particular objects does not offer any difficulty for nominalism. For the © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_4

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defenders of tropes, however, objects must be explained as some form of ‘construction’, as a ‘bundle’ or ‘cluster’, or as a mereological fusion of tropes satisfying certain requirements. It is interesting to consider here the advantages and disadvantages of trope theories to substitute universals by fulfilling their theoretical functions. Issues relating to trope bundles or mereological fusions will be considered in detail in Part III (§§ 78–94). There are two great ways in which it has been proposed that classes of tropes replace universals. Most proponents of trope ontologies have proposed resemblance classes of these entities (see Williams 1953a, 9–11; Campbell 1981, 483–485, 1990, 27–51; Denkel 1996, 158–171; Maurin 2002, 78–116). In a recent work, however, Douglas Ehring proposed natural classes of tropes for the same functions (see Ehring 2011, 173–241). There are important disadvantages of natural classes of tropes versus resemblance classes, so attention will be concentrated on the latter, although observations will also be made regarding natural classes when appropriate (see § 31). Proponents of trope ontologies have argued that these entities do not present the classic problems of resemblance nominalism, that is, the problem of the imperfect community and the problem of the companionship (see, in particular, Campbell 1981, 484–485, 1990, 32–34). Recall that the problem of the imperfect community arises because a set of objects can satisfy the requirement of being all similar to each other, but not in the same ‘respect’. A class of similarity of objects that are spherical and with 10 gr of mass, or with 10 gr of mass and 10 cm of height, or of 10 cm of height and spherical shape, is a class in which any two objects are similar to each other. In effect, an object of the class will be similar to any other because both will have a spherical shape, or both will have 10 gr of mass, or both will be 10 cm tall. However, when it comes to tropes, it seems—at least prima facie—that heterogeneous resemblance classes like this cannot arise. A trope of spherical shape, for example, does not have mass and length. It is only characterized by its shape or by being that which grounds the shape in something. It can only be similar to another object concerning its shape, therefore. Similarly, the problem of the companionship would not arise for trope ontologies either. This problem occurs in a case where, for example, all F is G, although not all G is F. One can assume that the natures of being F and of being G are made up of classes of objects that are all similar to each other. It happens in a case of this type that there are objects that do not belong to the class of Fs, but that are similar to all Gs. Since, in effect, every F is G, any object that is G will be similar to any object that is F, since that object is also G. This is in violation of the requirement that no object that does not belong to a class of similarity—of those that would be appropriate to fulfill the functions of a universal—should be similar to all objects in the class. When it comes to trope classes, however, it seems that this difficulty is resolved automatically. Tropes have only one characteristic. It cannot happen that, for example, all tropes of mass have a certain shape and that not all tropes with a certain shape have such a mass. Tropes of mass only have the characteristic of having mass or ground in something certain mass, and can only be similar to other tropes by their mass. At least, this is what it seems prima facie.

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Nor does the problem of co-extensive classes arise for trope theories. Suppose that the class of animals with hearts is exactly the same class as the class of animals with kidneys. For the nominalist, without further refinements, this would lead to the identification of the property of having a heart with the property of having kidneys. However, the trope or tropes that make up the property of having a heart are completely different from the trope or tropes that make up the property of having kidneys. So, there is no danger here of identifying the two properties. Recall that the problem of co-extensive classes was the central motivation for the nominalists of similarity to postulate classes of possibilia as the entities that substitute universals and to accept extreme modal realism. Theories of tropes, then, are not obliged to accept the unfortunate modal consequences that resemblance nominalism seems to have. It appears, therefore, that trope theories have substantive advantages over resemblance nominalism. If one is inclined to reject universals—and one is not inclined to accept concepts in the mind of God, for example, because one is not inclined to accept that God exists—then it would seem that the best alternative is an ontology of tropes. This advantage of tropes over resemblance nominalism is, however, a mere appearance. There is no such advantage or, at least, it is not obvious that there is such an advantage. If there are many reasons to reject resemblance nominalism because of its explanatory ineptitude in almost all areas in which it must compete with universals, there are also such reasons when it comes to resemblance classes of tropes. First, friends of the tropes must postulate a primitive external relation of similarity as resemblance nominalists must do. Secondly, resemblance classes of tropes are exposed to the same difficulties of the imperfect community and the companionship to which resemblance nominalism is exposed. They must, therefore, deploy the same theoretical resources that the nominalist must deploy, with all its problems. Third, although the friend of tropes does not need to resort to extreme modal realism, it is still very difficult to make a trope theory coherent with modal actualism. This section will conclude with some considerations about the natural classes of tropes.

4.1

Primitive Resemblances

§ 26. What should satisfy the functions of a universal property for the friend of the tropes is a resemblance class of tropes specified in the same way as a resemblance class of objects of the nominalist, that is, as a class of tropes, be α, which satisfy the following two requirements: (A) Any two tropes that belong to α are similar to each other. (B) No trope that does not belong to α is similar to all tropes that belong to α. That is, it must be a class of tropes, all of which satisfy the requirement to be all similar to each other and that is the ‘largest’ class of tropes that satisfies such a

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condition because if there were a trope that satisfies such a condition, it must belong to such class. What allows to unify these collections of tropes are the relations of similarity between them. As in the case of nominalism, the question arises here as to how such a relationship of similarity should be taken. There is, in effect, a risk of a vicious regress. It must be assumed that what grounds that a trope of F is a trope of F is the similarity of that trope with all the others with which it forms the respective class of similarity. If the resemblance, however, is a relationship like any other, then it must be a trope. Indeed, a defender of an ontology of tropes could not maintain that, for example, such a relationship was a universal, without abandoning its metaphysical position. If it is a trope like any other, however, what makes such a trope of similarity being what it is, is its resemblance+ with all the other tropes of the respective class of similarity+.1 But this resemblance+ must be taken again here just like any other relationship. So what grounds that the tropes of resemblance+ are such should be their resemblance++ with all the other tropes of the respective class of similarity++ and so on. What alternatives does the defender of tropes have here? Of course, he could turn to ‘primitive external facts’ of resemblance as the nominalist has done. Before considering this option, however, it will be convenient to consider another strategy tried by Keith Campbell that, if it works, would make everything much easier for the defender of tropes. Campbell argued that relations of similarity between tropes could be taken as internal relationships—as common sense assumes (see Campbell 1990, 30–32, 34–37; also, Maurin 2002, 87–116). An internal relationship between x and y, as explained (§ 4) is a relationship that is grounded on the intrinsic natures of x and y. That is, it is a relationship that obtains automatically once the intrinsic nature of the relata is given. For an object, its ‘intrinsic nature’ is the collection of its intrinsic properties, however such ‘properties’ are understood. When it comes to tropes, however, it does not make sense to talk about “intrinsic ‘properties”. There are no intrinsic properties of a trope because a trope is a property. One can speak, however, of certain “intrinsic nature” as that which a trope is by itself, regardless of what other entities exist or do not exist (see § 4). The idea of Campbell is that tropes have by themselves, independently of what other entities exist or do not exist, a qualitative character. There is nothing on which a trope of F should be grounded to be, indeed, a trope of F. The character by which a trope of F is a trope of F is a fundamental fact, not grounded on anything or—with the precautions of the case—’primitive’. In this way, tropes of F are similar to each other because each of them is F. There is no risk of a vicious regress in a conception of this type. The qualitative nature of tropes is not grounded on their mutual similarities—as is the case with the natures of objects in resemblance nominalism. Qualitative natures of tropes are something fundamental. Similarities are not an ‘addition of existence’ to tropes with their qualitative characteristics.

Resemblance between ordered pairs of resemblance is designated as “resemblance+”. It may be the same relation applied to itself or a relation of higher ‘logical type’. In any of these cases, a vicious regress is generated.

1

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Tropes, on the other hand, are ontologically simple entities (see Maurin 2002, 11–15).2 A trope of mass does not result from the combination of a qualitative character of mass and a character of individuality. The trope by itself is a particular quality or, at least, something particular that grounds a quality. Of course, nothing hinders one from being able to distinguish the qualitative character of the trope from its individual character conceptually, but there is no real difference between them. The situation is similar to the difference between Hesperus and Phosphorus. It is the same celestial body, but there are two different ways in which such a celestial body is offered to our knowledge. A trope contrasts with a state of affairs, as this type of entity is understood by, for example, David Armstrong (see Armstrong 1997, 113–127). States of affairs are here structures of universal properties and particular objects at a time (see Alvarado 2013). One can assign a qualitative character and an individual character to a state of affairs, but these characters result from the contribution of numerically different ontological constituents. The qualitative character of a state of affairs is given by the universal that constitutes it. The individual character of a state of affairs is provided by the particular ‘thin’ object, ‘bare’ particular or substratum that also constitutes it. In a trope, on the other hand, there is no ontological distinction between different constituents. This conception of tropes as possessing a fundamental and intrinsic qualitative character has been much criticized. Some of these criticisms have come from defenders of trope ontologies (see Ehring 2011, 175–184). It has been argued that the simplicity of the tropes is incompatible with their intrinsic qualitative character and their individuality. J. P. Moreland has been the one who has noted these difficulties in the most persistent way (see Moreland 2001, 53–73). It will be convenient to consider the difficulties of the first position of Campbell (Campbell 1981) on this point to deal afterward with the difficulties of his later position (Campbell 1990). In his first work, Campbell maintained that the individuality of a trope is grounded on its spatiotemporal location (see Campbell 1981, 486). What grounds that a trope is an individual entity is the fact that it is located in a specific spatiotemporal region. The tropes are in this conception essentially ‘qualities in a location’. Moreover, since a spatiotemporal location is essential to a trope, it is also essential for tropes the shape and the volume of the region they are occupying, with which a shape and a volume are essential to tropes. Any spatiotemporal region must

2

It should be noted, however, that there are forms of internal complexion that a trope can have. For example, a trope can be the mereological fusion of other tropes that are its parts. There are Gestalt or ‘structural’ tropes that result from the parts that make up that trope and from their mutual external relations—which must also be tropes and ordinarily consist of external relations between the objects that instantiate these tropes or of which those tropes are a part. If a macroscopic object has a certain length of n cm, this length is grounded on tropes of length of each of the parts of that object. The sum of the lenghts of each of these parts—that is, the respective tropes of those parts—must be the length of the whole, assuming the connection and continuity of all those parts. There are different ways in which the mereological structure of the particular objects that make these ‘structural’ tropes has been understood (see, for example, Campbell 1990, 135–155; Simons 1994, 569–574), but it is not necessary to enter into these differences here.

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possess, at least, a shaped volume. Any trope that is essentially a ‘quality in a region’ will also have to be, therefore, a ‘quality in a shaped volume’. It has been shown that in a conception of this type all the tropes that make up an object—because they are ‘co-present’ in the same region of spacetime—must be identified as the same trope (see Moreland 2001, 58–59). Suppose there are three tropes ‘co-present’ forming an object. Each of these tropes is a simple entity in which there is only a conceptual distinction between its qualitative character and its individuality. But its individual character is grounded on its spatiotemporal location. It happens, however, that being these three tropes co-present, they must possess exactly the same location.3 Then, those three tropes must be the same trope. Against appearances, the difference between the qualitative characteristics of these three tropes would be an illusion. The difficulties are not limited to this identification of all the tropes forming a bundle. Assume that two tropes t1 and t2 of mass and electromagnetic charge, respectively, are co-present. Each of these tropes integrates certain classes of similarity. Given the previous result, it cannot be that t1 and t2 belong to different classes of similarity. It cannot be that the same trope is and is not similar to other tropes. What happens, then, is that not only must co-present tropes be identified by forming an object, but also the classes of similarity to which those tropes belong must be identified. Thus, in a case such as the one mentioned above, the property of mass and the property of electromagnetic charge should be identified in a general way. These results would be disastrous for an ontology of tropes. It is assumed that in an ontology of this type, particular objects are bundles of a plurality of properties, but it would turn out that they could only be made up of a single trope. On the other hand, there would be no way to solve the problem of the many over one, because all the properties of an object could only be a single property.4 These difficulties led to several changes in Campbell’s position in the formulation of 1990. Now the individual character of a trope does not come from its spatiotemporal location, but it is a primitive character, not grounded on anything (Campbell 1990, 68–69). The idea that tropes have an essential shaped volume is maintained. It is also maintained that tropes have a fundamental qualitative character and that, therefore, relations of resemblance that tropes may have with each other are internal. The modification related to the spatiotemporal location brings with it several

3

In the ontologies of tropes defended by D. C. Williams and by Keith Campbell, it is precisely the ‘co-presence’ or ‘co-location’ in the same spacetime region that makes different tropes to form an object (see Williams 1953a, 9–10; Campbell 1981, 485–486). 4 There are several additional difficulties that have to do with modal problems that are not relevant to discuss now. In the classical theories of tropes, such as those of Williams or Campbell, tropes are entities fundamental and independent among themselves. If one supposes, however, that the individuality of a trope is grounded on its location—with the other conditions indicated above, i. e., tropes are simple entities with a fundamental intrinsic qualitative nature—a trope would be essential for all tropes with which it is co-present. Thus, if an object is a perfect sphere of 10 gr of mass, it could not have more or less mass than it has, it could not have a different shape from the one it has, and it could not have a spacetime location different from the one it actually has, although the difference was infinitesimal.

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theoretical advantages. It is no longer essential to a trope the location it actually has.5 It is no longer necessary to identify all the tropes co-present in the same location since their location is not essential to a trope, and it is not that in which their individual character consists. But important problems remain. These problems have to do with the supposed simplicity of the tropes and their intrinsic individual and qualitative character (see Ehring 2011, 177–182). The central problem has to do with the following principle: [Simplicity]

If x and y have two or more arbitrarily different internal relations, then x and y are not ontologically simple.

Suppose there are two objects a1 and a2. Object a1 has a mass of 10 gr and is red. Object a2 has a mass of 10 gr and is green. These two objects have two different internal relationships. On the one hand, they have the relation of having the same mass. On the other, they have the relationship of having different colors. It is evident that in such a case, neither a1 nor a2 are ontologically ‘simple’ entities because, in each of these objects, there is a difference between their mass and their color. Any reasonable position in metaphysics of properties should explain, when dealing with the problem of the many over one, how such a plurality of natures is given in the same object. This principle requires that internal relationships be ‘arbitrarily different’ (see Ehring 2011, 177–178). Two objects of 10 gr and 20 gr of mass respectively, for example, could have the internal relations of having both a mass and having more mass than. It is evident, however, that these two internal relationships—even though both are based on the intrinsic nature of the relata—are connected, since an object that has a determinate mass must also have the determinable property of having mass (see § 12). These two internal relationships are grounded on the same intrinsic natures. Internal relationships count as ‘arbitrarily different’ from each other when it is not the case that one of them necessitates the occurrence of the other. It seems that there are arbitrarily different internal relations between tropes, if they are conceived as indicated, that is, with a simple character at the same time qualitative and individual. Let two tropes of mass t1 and t2. Trope t1 is a trope of 10 gr of mass. Trope t2 is a trope of 20 gr of mass. Tropes t1 and t2 have the relation of having less mass than and also have the relation of being both individuals (see Ehring 2011, 180–182). Supposedly, these are internal relationships grounded on the intrinsic nature of t1 and of t2. These are relations arbitrarily different from each other since none of them necessitates the occurrence of the other. Ehring also presents the case of internal relations arbitrarily different from each other of having mass and of being numerically different (see Ehring 2011, 177–180). If tropes t1 and t2 are in 5 In effect, our commonsense conception is that the same object could be located in regions different from those in which it is actually located. In these ontologies of tropes, particular objects are mereological fusions of tropes co-present with each other (see Williams 1953a, 9–10). If each of these tropes could not have a different location from the one it has, then the object that has those tropes—which is, essentially, the mereological fusion of those tropes—could not have another location than it has.

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arbitrarily different internal relations, then, according to the principle of Simplicity, they are not simple entities.6 There could be reasons for doubt about this argument as to whether the relationships of being both individuals and being numerically different are internal relationships. In a general way, an internal relationship is a relationship grounded on the intrinsic properties of the relata. These intrinsic properties must be authentic properties. It is not enough that something can be said truly of an entity for a correlative property to be assigned to that entity, as has already been repeatedly emphasized. What happens here is that one could doubt that there is something like a property of being an individual. If there is no such intrinsic property, there would be no correlative internal relation either. Analogously, if there is no authentic property of being a non-cat, a supposed internal relation of being both non-cats cannot be attributed to a dog and a canary. I believe, however, that there is no problem in accepting as an intrinsic ‘character’ of an individual being an individual, even though it cannot be admitted as a ‘property’. The intrinsic character by which an individual is an individual grounds the internal relations of being both individuals and of being an individual different from other individuals. The crucial consideration concerning the character of being an individual is that, from the outset, what grounds internal relationships are not the intrinsic properties of tropes, because tropes are already properties, but what such tropes are ‘in themselves’. When one is inclined to maintain that being an individual is not an authentic property it is because one is not inclined to suppose that being an individual of an individual is something that must form a complex ontological structure with other entities to form the individual. A different issue is that it is not intrinsic to an individual to be an individual. Clearly, it is intrinsic to an individual to be an individual. Under any of the ways in which the difference between intrinsic and extrinsic properties has been analysed (see § 4), being an individual is intrinsic. Recall that it has been defined that a property F is combinatorially intrinsic if and only if the fact that an object x instantiates F is indifferent to whether x is alone or accompanied (see Lewis and Langton 1998). In a general way, x is ‘alone’ in a possible world w if and only if there is no object y different from x in w. And x is ‘accompanied’ in w if and only if x is not alone in w. Could the fact that a trope t be an individual vary because there are other entities other than x? Clearly not. The fact that a trope exists in possible worlds more or less populated by other entities does not generate any difference concerning its individual character. Another way to analyse the distinction between intrinsic and extrinsic properties is based on grounding relationships. Thus, it has been argued that the property F is grounding intrinsic if and only if: (i) if the fact that x is F is grounded on y, then y is (an improper) part of x; and (ii) if the fact that x is not F is grounded on y, then y is (an improper) part of x. Remind yourself that everything is—trivially— Moreland presents a different argument. He argues that it would be ‘unintelligible’ for two simple entities to be simultaneously under internal and external relationships (see Moreland 2001, 64). But there does not seem to be any difficulty with two substrata—something about which there are few doubts about their simplicity—that have the internal relation of being both substrata and that have also the external relationship of being 10 m away. 6

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improper part of itself. The fact that a trope is an individual is not grounded on anything mereologically disjointed from that same trope. It is, therefore, an intrinsic character of such a trope.7 Then, the individual characters of tropes t1 and t2 ground that they have the internal relationship of being both individuals and also that they have the internal relationship of being numerically different. Therefore, tropes conceived as simple entities endowed with an individual and at the same time with a qualitative fundamental intrinsic character have several arbitrarily different internal relations among themselves.8 Then, they cannot be simple entities, according to the Simplicity principle. This problem implies that the defender of tropes cannot assume that the relations of resemblance between tropes are internal. The defender of tropes must suppose that similarities are, as for the resemblance nominalist, external ‘primitive facts’, not grounded on anything. As it happens for the resemblance nominalist, the friend of tropes must maintain that the tropes possess the qualitative character that they possess because they are similar to certain tropes and unlike others. This brings several important consequences. It has already been seen how the external and primitive character of the resemblance to which the nominalist should appeal has great costs in a variety of areas. It becomes very difficult to explain the systematic relationships between determinate and determinable properties. It becomes very difficult to explain natural laws. It becomes very difficult to explain the reliability of our inductive practices. It also becomes very difficult to explain the knowledge we claim to possess of properties, characteristic of the natural sciences. There are explanatory requirements also not solved regarding the ‘external’ character of the resemblance relation. Not all the considerations relating to these difficulties will be repeated here. What is interesting to note is that trope theories do not have an advantageous position in any of these areas. The absence of universal properties makes the modal character of natural laws mysterious. Of course, the friend of tropes can try to reject this modal character as an illusion and resort to some theory of regularity. The point is that it does not have the same range of options that friends of universals have. The ‘disconnection’ of different tropes also makes it very difficult to explain why we should trust our inductive practices, why should we assume that the range of causal interactions associated with one trope should also be valid for another, in the case of different existences. Not only that. For reasons similar to

7 As can be seen, in 1981, Campbell’s first conception, the situation is different. The location of something in the spacetime region r is dependent on that region. Regardless of whether one adopts a substantialist or relationalist conception of space and time, this involves other entities than the localized object. It would be, then, that the individuality of a trope would be something grounding extrinsic for the trope. Even in this case, however, the individual character of a trope would be combinatorially intrinsic. 8 Could not this same argument apply to any pair of individuals, such as two substrata, for example? In effect, two substrata would be with each other in the internal relation of being both substrata and in the internal relation of being numerically different. But these internal relations are not ‘arbitrarily different from each other’, as the numerical difference between two entities of a certain category C must be grounded on facts about each of these entities being of category C.

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those that operate for resemblance nominalism, it is difficult to understand how we could get to know some property—in this case, some class of similarity of tropes— because for this, we should know its elements and their mutual similarities. Although in this type of theories the modal consequences do not seem as serious as for resemblance nominalism, and it does not seem necessary to suppose that the classes of similarity will be constituted by merely possible tropes, the requirement of knowing a class of actual tropes seems sufficiently exorbitant. Among all this plethora of difficulties, attention will now be concentrated on one. A primitive external relation of similarity opens the risk of a vicious infinite regress again. The postulation of internal similarities between tropes seemed to block this risk. The failure of this strategy brings the risk back. A ‘primitive fact of similarity’ is a similarity. A similarity is a relationship. One cannot here claim that ‘it is not an addition of being’ since it is an external relation and fulfills a crucial role because, without such relations, no trope would have any qualitative character. As defenders of tropes admit relationships of similarity, it does not have the costs it has for a nominalist. After all, tropes are properties, and similarity is one more property, only relational. The problem arises here because the qualitative character of a trope is grounded on its resemblance to other tropes—just as the nature of an object for a nominalist is grounded on its resemblance to other objects. This is true for monadic tropes, but it must also be true for relational tropes. Then it should also be true for the relational tropes of similarity. That is, a similarity between tropes is such because it is similar+ to other relational tropes. But here the similarity+ is also a relationship. Then the character of similarity+ is such, is because it is similar++ to other relational tropes, and so on. The qualitative character of each trope is grounded on similarities, which are only similarities because they are grounded on other similarities, and then to infinity. It is, of course, a vicious infinite regress. Perhaps here the defender of the tropes could argue that, although the qualitative character of a ‘normal’ trope is grounded on its similarities with other tropes, the qualitative character by which a similarity is a similarity would be fundamental or ‘primitive’. A maneuver of this kind brings with it an important cost in qualitative economy if two types of tropes so different from each other are postulated. The only motivation for such a maneuver would be the intention of avoiding the regress, which makes the distinction ad hoc. The most serious problem, however, is that it would be necessary to postulate tropes—the fact that they are relational is here indifferent—that have an intrinsic character at the same time qualitative and individual. Although ‘normal’ tropes do not possess an intrinsic qualitative character, similarities must possess it. But it has already been shown with latitude that to suppose such an intrinsic character clashes with the—supposedly—simplicity of tropes. The vicious regress, then, remains.

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4.2

91

Imperfect Communities and Companionships

§ 27. As stated, proponents of trope ontologies have assumed that trope resemblance classes are free from the problems of the imperfect community and the companionship. But there is no such thing. Ideally, tropes should be mono-characterized entities which would prevent the indicated problems from being generated. This assumption rested on a still very general understanding of the existing theoretical alternatives concerning the nature of tropes. It is interesting to note, in a preliminary way, that resemblance classes of tropes with an intrinsic qualitative and individual nature— i. e., classes of tropes understood in the way that has been criticized in the previous section—have already generated the problems of the imperfect community and the companionship. The difficulties that will be presented here are not the result of admitting primitive external similarities between tropes. They are independent problems. It has been postulated that tropes possess essentially a volume and a form (see Campbell 1981, 486). Initially, this has been derived from the fact that tropes possess an essential spacetime location. If one assumes that tropes have an intrinsic qualitative character, however, this requirement must be maintained, even if one postulates the independence of tropes from their location. Suppose, in effect, that the form and volume were contingent to a trope. Then, it would be possible for a color trope, for example, in a perfect square with sides of 1 km each to be a trope of color in a perfect circle with a radius of 10 cm. Although one supposes that the same type of color is kept constant in different possible worlds—which requires that the tone, the brightness and the saturation of the color be kept constant—the identification of a color trope in a square of 1 km2 in volume with a color trope in a circle of 2π
100 cm2 (ffi 0.006 km2) in volume seems implausible. If its volume and shape were contingent for a trope, then such identification should be made. A trope t1 that in the world w1 characterizes a square of 1 km2 would be identical to a trope t2 in the world w2 characterizing a circle of 0.006 km2. At this point, it could perhaps be replicated that, although it is not possible for a trope that characterizes a volume of 0.006 km2 to characterize a volume of 1 km2, it would seem to be possible for that same trope to characterize a volume of 0.0061 km2. Although large variations in a trope are not modally admissible, small variations should be admissible. But the sum of small variations makes a great variation. If it is possible with respect to w1 a world w2 in which t characterizes a volume of 0,0061 km2, and then it is possible with respect to w2 a world w3 in which t characterizes a volume of 0,0062 km2, then it is possible with respect to w1 the world w3 in which t characterizes a volume of 0.0062 km2. The reiteration of this process of slight variations as often as necessary brings it finally possible concerning w1 a world wn in which the same trope t characterizes a volume of 1 km2. Assuming the admissibility only of

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small variations, then, does not improve the situation.9 Volume and form seem essential to a trope. The introduction of a volume and a form for each trope opens the door for cases of imperfect community and cases of companionship. Suppose a class of similarity of tropes that either ground the character of having something 10 gr of mass in a perfect cube, or ground a character in a perfect cube of 10 cm of side—therefore, with a volume of 1000 cm3—or they ground the character of having something 10 gr of mass in something with 1000 cm3 of volume. Any two tropes of the class will be similar to each other, either by grounding the character of having 10 gr of mass, or by grounding some character in a perfect cube, or by grounding some character in a volume of 1000 cm3. All tropes fulfill the condition of being similar to each other, but it is—intuitively—a very heterogeneous class of tropes. Similarly, companionship cases arise. Suppose that all the tropes that ground the character of having k gr of mass do it in a volume of m cm3, although not everything that ground a character in m cm3 grounds the character of having k gm of mass.10 The class of similarity of the tropes that ground some character in something of m cm3 of volume is a companion of the class of similarity of tropes that ground the character of having k gr of mass. One can assume at this point that these difficulties are a consequence of postulating for tropes an intrinsic qualitative nature, but the situation will not improve, as will be explained, if a qualitative nature grounded on external similarities is postulated for the tropes. Before considering these cases with detention, however, it will be useful to introduce a fundamental distinction as to how a trope can characterize an object. This distinction has only recently been highlighted by Robert Garcia (see Garcia 2014, 2015a, b). A trope, being a particular property, makes the object that possesses it to have a certain nature. A green trope, for example, makes the object that owns it green. Or, in other words, the object in question is green because it has a

9

Faced with this type of problems, some have argued that the transitivity of the accessibility relations between possible worlds should be rejected (see Salmon 1989, 2005, 238–240). That is, although w2 is accessible to w1, and w3 is accessible to w2, it is not necessary that w3 be accessible to w1. This assumes that modal logics of type S4 or stronger would not be valid, and the characteristic axiom [□p ! □□p] must be rejected. For many, a scenario in which something is metaphysically necessary, but it is metaphysically contingent that it is necessary, that is, a scenario in which something necessary might not have been necessary (see Plantinga 1974, 51–54; § 42) is very counterintuitive. Another alternative would be to postulate that modal facts relating to tropes are grounded on counterparts (see Lewis 1968). A counterpart of x is an entity y numerically different from x, but which is similar to x in important respects. Thus, it would be possible for the trope t1 in w1 to characterize a volume of n cm3 if and only if there is a trope t2 in w2 that characterizes a volume of n cm3 and that counts as ‘counterpart’ of t1. But many reasons have also been presented for rejecting the counterparts as determining modal facts in which one cannot enter here (see Plantinga 1974, 88–120; Fara and Williamson 2005). 10 The examples that come to mind, in this case, are a bit more sophisticated. Every elementary physical particle in the Standard Model of particle physics has a volume of 0. They are treated as point particles. But they differ among themselves by their respective masses, their respective charges, and their respective spins. Thus, any trope of mass of 0.511 MeV/c2 (the mass of an electron) characterizes something with volume 0, although not every trope that characterizes a volume 0 grounds the character of having 0.511 MeV/c2 of mass.

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trope of green color. There are two radically different ways in which a trope can contribute to such a nature. The trope can ground that the object is green without being the trope green itself, or the trope can ground that the object is green because the trope is green. The tropes of the first type have been designated “modifier tropes”. The tropes of the second type have been designated “modular tropes”. A modifier trope does not have the character that it grounds. The modular trope possesses the character which, then, is attributed to the object. Modifier tropes are more appropriate—at least, in principle—for ontologies in which there is a substratum that is the bearer of the tropes (see, for example, Heil 2003, 169–178; 2012, 12–32). Indeed, since modifier tropes have no character, in an ontology in which particular objects were understood as trope bundles, without a substratum, nothing would possess the character grounded by the tropes. No modifier trope has the grounded character. In the bundle, there are only tropes that ground a character in something else, but none of them is characterized. Modular tropes, on the other hand, are more appropriate—in principle—for ontologies in which particular objects are bundles of tropes. If there were a substratum, an over-determination problem would be generated, because if a substratum instantiates a mass trope of n gr, then there would be two entities with n gr: the substratum and the trope. The trope is something with n gr of mass because it is a modular trope that has the character that it grounds. The substratum would also be something with n gr of mass because it instantiates the property of having n gr of mass. One should explain, then, how the structured entity possesses only n gr of mass and not 2n gr of mass. If there were a causal interaction in which that mass was relevant, there would be two entities that could be that which interacts, since both the mass trope and the substratum have such a mass. It is much more reasonable, on the other hand, to think about modular tropes as forming trope bundles. The complete bundle can, for example, have n gr of mass by owning a part that has n gr of mass. There are also very marked differences between resemblance classes of modifier tropes and resemblance classes of modular tropes. In both cases, although in different ways, the problems of the imperfect community and the companionship appear, as will be shown.

4.2.1

Resemblance Classes of Modifier Tropes

§ 28. A first alternative would be to suppose that the classes of similarity that fulfill the functions of a universal are formed by modifier tropes connected by a primitive external relation of similarity. One should suppose that this alternative should be preferred by philosophers who defend ontologies with tropes and substrata that instantiate them (see, for example, Martin 1980; Heil 2003, 151–168; 2012, 84–116). Modifier tropes, as highlighted, do not have the characters that they ground in the object that instantiates them. It is the object that possesses the character. Thus, a trope of color does not have any color. The object that instantiates that trope of color does possess it, precisely because it instantiates such a trope. In a conception of this kind, it is assumed that the classes of tropes that should satisfy the functions of a

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universal are classes selected to have as elements tropes all of which are similar to each other and because it is the ‘largest’ class of tropes that satisfies such condition. It is the similarity between the tropes that establishes that the tropes ground, in turn, a certain character in their owners. But the tropes have no character. How could they, then, be grouped by their mutual similarity? If it is the class of similarity of the tropes of electromagnetic charge of q, it should be constituted by the similarities between such tropes. But none of those tropes has a charge of q. The comparison of any trope with any other—concerning its intrinsic nature—will show the same. None of these tropes has any character of mass, electromagnetic charge, shape, length or volume. They cannot be similar to each other by mass, or by electromagnetic charge, or by shape, or by length, or by volume. Nor can they be unlike each other in these respects. Its only intrinsic character is to be that which grounds a character in an object. Then, what kinds of similarities could be established between them? One could at this point be inclined to suppose that classes of similarity of tropes could be fixed by the similarity between the objects whose character has been grounded by the tropes they possess. Thus, the class of similarity of the mass tropes of n gr is determined because it is the class of all the tropes that ground on objects the character of having n gr of mass. All these objects are similar to each other by having n gr of mass. The tropes that ground such character in these objects are those that make up the class of similarity. These are tropes similar to each other, but vicariously. They are similar to each other in grounding a character in objects that are similar to each other. The problem with this alternative is that the objects have multiple characteristics, and the classes of similarity of objects cannot discriminate the cases of imperfect communities. Also, there are cases of companionship for classes of similarity. Then, we have back the problems of the imperfect community and the companionship. In effect, it is assumed that the tropes must be grouped into similarity classes generated by the similarities between the objects that possess the characters that such tropes ground. For everything that primitive external relations between objects determine as a nature, there must be a trope that grounds such a nature. All these tropes should be grouped as a class of similarity that will satisfy the theoretical functions of universals. But if similarity classes are formed between objects by satisfying the two already long-cited requirements—that all the objects of the class are similar to each other, and that no objects that do not belong to the class are similar to all the objects of the class—then we will select classes of objects that, according to our intuitions, share a property, but we will also select classes of heterogeneous objects that, according to our intuitions, do not share a property. Due to the companionship problem, also, there will be classes of objects that, according to our intuitions, share the same property that will not be selected. But there is another, more serious problem with this alternative. If one defends an ontology of tropes, it is assumed that the natures of the objects are grounded on the tropes that these objects possess—it does not interest if the object is conceived as a substratum or as a bundle of tropes. This grounding relationship would be reversed here. In effect, a trope can ground a characteristic in an object because, in some way, it ‘has’ it. It has already been seen in the previous section that it cannot be assumed that tropes possess an intrinsic qualitative nature. Such a nature is grounded on the

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primitive external similarities of those tropes with other tropes. If things worked here reasonably well, what we should find is that the characteristics of the objects are grounded on the tropes that these objects possess and that the qualitative characters of those tropes are grounded, in turn, on the primitive external similarities between tropes. Thus, by transitivity of the grounding relationship, it should turn out that the characteristics of the objects are grounded on the external similarities between tropes. It happens, however, that the external similarities between tropes are ‘vicarious’ to the external similarities between objects. This is the same external similarity that would be sufficient—with all the limitations presented previously—to generate the nature of the objects. It turns out, then, that tropes are only suitable for grounding characteristics in objects if they have inherited their external similarities from those objects. Recall that tropes do not have an intrinsic qualitative character. As we are dealing here with modifier tropes, such ‘qualitative character’ is the character by which such tropes ground a qualitative nature on an object. This aptitude to ground a qualitative character is grounded on external similarities with other tropes. But such similarities are the similarities between objects. The natures of the objects—generated independently—ground the character of the modifier tropes by which these are what establish a qualitative nature in the objects that possess them. Then it would turn out that the qualitative nature of objects is what grounds the qualitative nature of objects. Therefore, in a conception of this style, there is no need to introduce tropes. They are perfectly dispensable because for these tropes to have the aptitude to ground qualitative characters in objects, there must previously be a qualitative character in these objects.11

4.2.2

Resemblance Classes of Module Tropes

§ 29. The second alternative is to suppose that the classes of similarity of tropes that fulfill the functions of universals are classes of module tropes. One can assume that this alternative should be preferred by all defenders of ontologies of trope bundles as what fulfills the functions of a particular object. This has been the most frequent position among defenders of trope ontologies. Modular tropes have the character that they ground in the object that has them. The objects are in these theories usually mereological fusions of tropes, so that an object ‘possesses’ a property in these theories by having a part that is such property. The similarity relation is, again, a fundamental external relation. In this alternative, the problems that affect resemblance classes of modifier tropes do not occur, because the external similarity is a

11

The situation would be different if modifier tropes had a fundamental intrinsic nature. This fundamental intrinsic nature would be what grounds, at the same time, the qualitative character of the objects that instantiate such tropes, as well as the internal similarities between tropes. Resemblances between objects would only be epistemologically prior to resemblances between tropes. But, as we have seen, tropes do not possess a fundamental intrinsic qualitative nature.

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relation that obtains directly on the modular tropes that possess the qualitative character that they ground. It also happens here, however, that the resemblance classes of modular tropes have the problems of the imperfect community and the companionship. Suppose a class of tropes including: (i) all tropes co-localized with something of n gr of mass and co-localized with something with electromagnetic charge of q, (ii) all tropes co-localized with something with electromagnetic charge of q and co-localized with something perfectly spherical in shape, and (iii) all tropes co-localized with something perfectly spherical in shape and co-located with something of n gr of mass. Recall that these are modular tropes, so they have the characteristics that they ground. Trivially any trope is co-localized with itself. Therefore, a trope of mass of n gr is co-located with something with mass of n gr, as well as a trope of electromagnetic charge of q is trivially co-located with something with an electromagnetic charge of q. It results that any two tropes of this class are similar to each other either because both are co-located with something of n gr of mass, or because both are co-located with something of electromagnetic charge of q or because both are co-located with something perfectly spherical in shape. It is a class that satisfies requirement (A) indicated above, since all tropes are similar, but intuitively it is a class of very heterogeneous tropes. The objection that will probably generate such a scenario is that being co-located with an F is an extrinsic property for that which possesses it or, at least, an extrinsic characteristic, if one does not want to commit oneself to properties. It is assumed that tropes are grouped into classes according to their mutual similarities concerning their ‘intrinsic’ qualitative natures so that everything that could be attributed extrinsically to them would fall out of consideration. Therefore, the similarities in terms of ‘being co-located with an F’ would not be relevant. The problem is that tropes do not have intrinsic qualitative natures, as explained in detail above. The qualitative nature of a trope is grounded on the external similarities of that trope with others. That is, one can only attribute an ‘intrinsic’ nature to a trope given the relationship of that trope with others. Thus, only after the similarities and dissimilarities between the tropes have been established a certain qualitative nature of such tropes will result. If tropes are similar to each other by being both ‘co-localized with an F’, then being co-located with an F will be part of the qualitative nature of such tropes. What is proposed here is a problem of a general nature. As explained above, one of the ways in which the distinction between intrinsic and extrinsic properties has been made is based on grounding relations: the property F is grounding intrinsic if and only if, for all x, if the fact that x is F is grounded on y, then y is (an improper) part of x, and if the fact that x is not F is grounded on y, then y is (an improper) part of x. The qualitative nature of a trope—whatever it may be—is grounded on other tropes with which it has relations of external similarity. Following this way of making the distinction between the intrinsic and the extrinsic, no qualitative nature of a trope will be intrinsic. Another way to make the distinction between intrinsic and extrinsic properties concerns ‘enlargements’ or ‘restrictions’ of worlds. A property F is combinatorially intrinsic, according to this conception, if and only if, the possession of F by x is indifferent to being x alone or accompanied in a possible world. So, if

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F is an intrinsic property, the fact that an object x has F will not be modified by the fact of existing or not more objects together with x in a possible world. Considered from this point of view, it also seems that the qualitative nature of a trope is extrinsic, because in worlds where there are no other tropes that make up the class of similarity the trope will not have a qualitative character that actually has. It would be curious, however, a conception in which tropes have qualitative characters contingently. In such a conception, a trope of green color could be a red color trope. This has to do with the modal consequences of a metaphysics that substitutes universals for classes of tropes, which will be dealt with in the next section (see § 30). Anyway, for what interests here, if the qualitative character of a trope is essential to it—what seems most likely—then this qualitative character will count as intrinsic to each trope, although it is grounded on its fundamental external similarities with other tropes. If in the previous hypothesis, all attributions are equally extrinsic, here all would be equally intrinsic. Thus, the attributions of ‘being co-located with an F’ cannot be disqualified. In the same way that scenarios of imperfect community arise for the classes of similarity of modular tropes, companionship scenarios also arise. Suppose a class of similarity of tropes co-located with something that is F and a class of similarity of tropes co-located with something that is G. If all F is G, although not all G is F, it will result that there are tropes similar to all tropes co-localized with an F. In effect, any trope co-located with an F is also co-located with a G and will, therefore, be similar in this respect with the tropes co-located with a G.12 Again, the character by which a trope is co-located with a G cannot be disqualified as ‘merely extrinsic’. This concludes the argument. No matter which conception is defended, resemblance classes of tropes generate the problems of the imperfect community and the companionship—against what one might have initially assumed. This is not a fatal flaw, of course. The defender of tropes can appeal to the same resources that the nominalist has. It must be emphasized, however, that the supposed advantage of resemblance classes of tropes versus resemblance classes of objects disappears. This was an important advantage of ontologies of tropes over nominalism. Without it, the initial motivation to accept tropes instead of postulating a form of nominalism also disappears. After all, nominalism has advantages of economy in its favor that cannot be compensated by other comparative disadvantages of tropes. It should also be noted that, just as it happens for resemblance nominalism, the introduction of the resources used by the nominalists opens serious flanks for trope ontologies. In the first place, a primitive external relation multigrade and contrastive of resemblance*—à la Lewis—is difficult to understand and makes unknowable the

12

Here too, there are no simple examples that come to mind. In the Standard Model, all elementary particles have volume 0. They are point particles. An electron has a mass of 0.511 MeV/c2, while— for example—a quark of the type ‘above’ has a mass of 2.4 MeV/c2. Any mass trope of 0.511 MeV/ c2 is co-localized with something of volume 0, but not all that is co-localized with something of volume 0 is also co-localized with something of mass of 0.511 MeV/c2. Thus, some tropes do not belong to the class of mass tropes of 0.511 MeV/c2 that are similar to all tropes of this class, because they are co-located with something of volume 0.

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facts of similarity that ground all qualitative character. Second, appealing to perfect maximal communities of tropes—à la Rodriguez-Pereyra—brings with it the problem of vicious regress noted in § 23. Then, the problems of the imperfect community and the companionship for resemblance classes of tropes are not only the appearance of a small dose of additional complication to the theory. It brings with it serious costs that—as it happens with nominalism—make the entire project unlikely.

4.3

Modal Consequences

§ 30. As for resemblance nominalism, there are also undesirable modal consequences for theories that seek to substitute universals by classes of tropes. These are less serious consequences but serious enough to constitute an important theoretical cost. These costs appear when trying to make coherent an ontology of tropes, without universals, with an actualist metaphysics of modality (see Alvarado 2011). The difficulties disappear if one accepts an extreme possibilist conception for the modal facts, but these conceptions are so incredible, that this is not consolation for the friend of the tropes. A first aspect that must be considered is that resemblance classes of tropes that are going to fulfill the functions of universals should have as elements actual and merely possible tropes, at least for the great majority of cases. As explained above (see § 10), almost all determinate properties have only a single actual instantiation. Although superficially, for example, it seems that there are many objects with the same shape as my computer if one examines with enough detail such shape it would appear that none of the computers in question—manufactured by the same brand of the same model, etc.—have the same shape. There will be differences, perhaps imperceptible for our perceptive capabilities, and for what a very powerful microscope can show, but no less real. An atom of more or less makes a difference of shape. An electron of more or less makes a difference in shape. These differences are not relevant for practical purposes, of course, but here it does not matter how useful a difference is, but simply whether it exists or not. What happens for the shape is valid for any other property. There are probably no other instances of the same color that my computer’s surface has. Most likely, there are no other objects with exactly the same mass—at least not if they are objects occupying a connected and continuous region. Most likely, there are no other objects with exactly the same electromagnetic charge. But it is assumed that by postulating resemblance classes of tropes we are looking for an answer to the problem of the one over many. There is no sense in a singleton resemblance class with a single element. Nothing like this would be apt to satisfy the functions attributed to universals and to be something one over many. A reasonable conception of possible tropes that can be elements of resemblance classes becomes indispensable. If there are no possible tropes, then there are no trope resemblance classes that can fulfill the functions of a universal. The entire program of an ontology of tropes will have failed.

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As already indicated, if one postulated an extreme modal realism like that of Lewis, in which the possible worlds are entities of the same nature as the actual world, only that disconnected spatiotemporally, then there would be no problem with this demand. If there are tropes that are inhabitants of our world, there will also be tropes inhabitants of other worlds. These tropes will have relations of similarity between them—or not—and the required similarity classes will be established. But, as it happens for resemblance nominalism, to resort to these parallel universes is a very high ontological cost that should be avoided. Trope theories should be compatible with different forms of modal actualism or, at least, with some of them. It is this, however, that is doubtful. Recall that four different forms of modal actualism have been postulated: (i) linguistic theories, in which possible worlds are complete stories in some language; (ii) combinatorial theories, in which possible worlds are sets of ‘possible states of affairs’ that result from abstract combinations of objects and properties; (iii) theories based on universals, in which possible worlds are maximal structural universals; and (iv) theories in which possible worlds are maximal possible states of affairs or maximal propositions (see Divers 2002, 169–180; §§ 24, 42–44). There are, of course, many variations in each of these general types, but for this examination it will suffice to consider them in their main features. It’s obvious, for once, that the theories of tropes are incompatible with modal theories based on universals. The goal of the friends of the tropes is to offer substitutes for universals. If one generally maintains that modal facts are facts about maximal structural universals, then the only way in which resemblance classes of tropes could be suitable for this task would be to presuppose universals—the universals required to form the facts about merely possible tropes—which is the failure of the theoretical program of the friend of the tropes. Something similar happens with the theories that ground modal facts on ‘possible maximal states of affairs’ (see Plantinga 1974, 44–46) or in ‘maximal propositions’ (see Fine 1977; Adams 1974, 335–230). A ‘possible state of affairs’ is a universal, as explained above (see § 24) as well as a ‘proposition’—it has usually been understood as a 0-adic property, that is, without free variables.13 When it comes to the combinatorial modal theory, it seems that the situation is more auspicious for the friend of the tropes. In their traditional form (see Armstrong 1989), possible worlds are sets of possible states of affairs, which—in turn—are ntuples of universal properties and objects that represent states of affairs that do not actually exist but could exist. Possible worlds are here, therefore, complete sets of ntuples that represent how everything could be and that are grounded on the mutual independence between different objects and different universals. Of course, it would be unacceptable for a defender of an ontology of tropes to accept universals as 13

Propositions have also been understood as sets of possible worlds (see Lewis 1986, 53–55), but such a conception would be completely useless if the same possible worlds are reduced to maximally consistent sets of propositions or maximal propositions. A ‘maximal proposition’ or ‘world proposition’ is a proposition q such that: [8p (☐(q ! p) _ ☐(q ! Øp))], in a way analogous to how a ‘maximal state of affairs’ is a state of affairs S such that, for all state of affairs S’, either S includes S’, or S excludes S’.

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generators of modal facts, but one might think that nothing should prevent that independent tropes among themselves were the generators of modal facts. Something of this style is what the defenders of the classic theories of tropes have envisaged (see Williams 1953a, 7–9; Campbell 1981, 479, 1990, 71–72). Since tropes are entities independent of each other, they can vary freely in different possible worlds, existing together in some or separately. Not in all conceptions of tropes does this independence exist. In this work, tropes are going to be posited, but with relations of ontological dependence among them crucial for the formation of individual objects (see §§ 87–94). Nor does this independence exist in the conceptions that posit tropes and substrata that instantiate these tropes (see Heil 2003, 140–142; 2012, 106–109), since it is part of the conditions of identity of a trope the substratum on which is instantiated We should consider the situation with respect to more liberal theories of tropes that seem better disposed to a combinatorial conception. A combinatorial modal theory that posits tropes instead of universals and particular objects should hold that the modal space of all metaphysically possible worlds is generated by the combinations of tropes independent of each other. If there is actually a perfect green sphere and a perfect red cube, then—by combinatorics of the respective tropes—there could be a perfect red sphere and a perfect green cube, just as there could be only a green sphere, only a red sphere, only a red cube, only a green cube, only a sphere and a cube, only a green color, and only a red color, etc. The problem presented here, however, is that it is assumed that the modal space must be generated by actually existing entities. This is what happens in traditional combinatorial theories in which possibilities result from a given totality of entities—actual objects and universals. A combinatorial theory of tropes should depend on actual tropes. But it happens that this totality does not include merely possible tropes that can form the classes of similarity that the defender of tropes requires to replace the universals. We are back to the initial problem, because for the great majority of determinate properties, there would be no resemblance classes, since there would only be a single actual trope for each of those—supposed—classes. One way in which this problem could be solved is to expand the base of entities that should generate the modal space of possible worlds. Instead of restricting it to actually existing tropes, it could be assumed that it is also constituted by merely possible tropes. Among the merely possible tropes should be the tropes to form the required classes of similarity. This would be, however, an ontologically extravagant theory. What are such ‘merely possible tropes’? What independent justification would exist for them? Would they be abstract entities? Concrete entities? If they were concrete entities such as actual tropes, then we do not see how such a theory would differ from extreme modal realism. And if they were abstract entities, it would be necessary to explain what relation the ‘abstract tropes’ have with actual concrete tropes. Is it that every concrete trope is an ‘instantiation’ or ‘realization’ of an abstract trope? This seems too close to the relationship that a universal has with its particular instances. Is it that tropes are contingently concrete, that is, concrete in some possible worlds and abstract in others? Propose such thing would add more extravagance where it already abounds. It could be objected at this point that the postulation of a totality of merely possible tropes should not cause surprise. If other

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modal theories postulate, for example, transcendent universals, why not tropes?.14 But there are big differences at this point between tropes and universals. Universals are repeatable entities. A universal actually instantiated is the same numerically to one whose possible instantiation is contemplated. A merely possible trope is a numerically different entity from an actual trope. There is no reason to suppose that the knowledge of an actual trope must be a reliable source of knowledge of the merely possible. It would be useless here to argue that it is enough that the merely possible tropes are of the same ‘type’ as actual tropes, for merely possible tropes are needed in the first place precisely to make sense of a ‘type’ of tropes, that is, a resemblance class of tropes. In effect, tropes are of the ‘same type’—in a theory in which tropes replace the universals—if they belong to the same class of similarity. But here we do not have resemblance classes without having previously merely possible tropes. Merely possible tropes, then, are extravagant entities. Universals are not.15 Another alternative to which the defender of a trope ontology could appeal is to argue that the only classes of similarity are those formed by tropes of those that make up the fundamental physical particles. As indicated above, at the quantum level, the same determinate property in each of the particles of the same type is observed. All electrons have exactly the same mass. All quarks of the type ‘above’ have exactly the same charge, etc. What the defender of a trope ontology could argue is that only these tropes are those that form the basis of the complete modal space. Actual tropes that make up the fundamental physical particles form resemblance classes that can fulfill the functions of a universal. The modal space is formed with the combinatorics of these same tropes and, eventually, tropes that would belong—if they existed—to the same classes of similarity. Everything that could be in a possible world, then, are tropes of precisely these types—belonging to these classes of similarity—and, of course, structural or Gestalt tropes resulting from mereological fusions of tropes of these types. The problem with this maneuver is that it implies adopting an extreme form of micro-physical reductionism as a metaphysically necessary fact. As it has been indicated, in effect, it is sustaining not only that all tropes are of those that make up the fundamental physical particles. It is also contended that there could not be tropes of another type. There would be no space, for example, for mental properties not reducible to physical properties. It would be metaphysically impossible for non-physical entities of any kind to exist. And even if one adopts the crudest physicalism, it is excessive that metaphysical possibilities are not admitted, for

14

I owe this objection to Axel Barceló. To these difficulties must be added those that result from less ‘liberal’ conceptions about the mutual dependence of tropes among themselves. If, for example, the substratum in which it is instantiated is essential to a trope, then there are not many combinations in which a trope can enter. This trope could not be instantiated in another substratum. The only combinations in which it could enter is to be accompanied by other tropes instantiated on the same substratum. If one maintains, on the other hand, that the remaining tropes with which a trope form a bundle are essential to it, these combinatorial possibilities are reduced to zero. 15

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example, for fundamental tropes of mass—mereologically atomic—larger than the mass trope of a quark. It remains to consider, however, the viability of some of the modal linguistic theories. In these theories, modal facts are grounded on the possibilities of expression offered by the sentences of a language. A state of affairs S is metaphysically possible in a conception of this type if and only if there is a sentence s of the language L—the language in which the maximally consistent stories are formulated—such that: (i) s expresses that S, and (ii) s belongs to a maximally consistent set of sentences—which is what a possible world is identified with in this conception. Then, the metaphysical possibility of a merely possible trope may seem unproblematic, because it is sufficient that there are sentences that express the existence of such trope to generate its possibility.16 But it is not. The problem that occurs here is analogous to what has appeared for resemblance nominalism. Suppose there is a mass trope of the entire universe. There is no other actual mass trope resembling it—and there could not be, by definition. The class of similarity of mass tropes of exactly that type must be formed with merely possible tropes. In a linguistic theory of modality, the possibility of such a trope would have to be given by a sentence of some possible world, that is, belonging to any of the maximally consistent sets of sentences of L. Let “tm” be the proper name of the mass trope of the universe. It is a mass trope of n gr—for an n very large, but finite. Let the sentence be: (1) There is an x such that: x 6¼ tm and x is a trope of n gr of mass.17 The merely possible trope cannot simply be designated by a new name, because a proper name has as its semantic value its referent, and there is nothing to refer to. It is impossible for there to be a ‘chain of uses’ that links speakers with an ‘initial baptism’ since there is no baptism. There is nothing to baptize. The merely possible trope—different, therefore, from all tropes actually existing—must be given in a ‘descriptive’ way. The metaphysical possibility of a merely possible trope must be given by an expression of the type “there is an F” in which “F” is the appropriate description. It is also essential that the description indicates that it is a trope of n gr of mass. An arbitrary trope of any kind would not serve to form the class of similarity of the tropes of n gr. At this point, a problem arises analogous to that which has been seen to affect resemblance nominalism. The defender of an ontology of tropes is interested in the predicate “is a trope of n gr of mass” because it is a predicate that all and only the tropes that make up the relevant class of similarity satisfy. The semantic value of the predicate “is a trope of n gr of mass” should be precisely such class of similarity. The problem here is that there is no such class. There is only one mass trope of n gr, and this is insufficient to give semantic value to the predicate “is a mass

16

And, precisely, how easy it seems to generate possibilities in this conception is evidence of how unlikely these conceptions are as alternatives in modal metaphysics. 17 More precisely, assuming that L is a well-regimented first-order language: [∃x ((x 6¼ tm) ^ (x is a trope of n gr of mass))].

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trope of n gr” Without such a predicate, there is no metaphysical possibility of a merely possible trope of mass of n gr. And without such metaphysical possibility, there is no class of similarity of mass tropes of n gr. Conversely, if there were a class of similarity of mass of n gr, it should be grounded on a sentence such as (1), which depends on the predicate “is a mass trope of n gr”. But this predicate depends ontologically on the class of similarity of tropes of n gr of mass that should be its semantic value. But there is no such class. Assuming, therefore, that things are as the defender of the tropes assumes they are, a sentence like (1) would be unintelligible, without truth conditions and, therefore, would not be a ‘sentence’ in any reasonable sense of the term. Therefore, there seems to be no way of accommodating the idea that resemblance classes of tropes can satisfy the functions of universals with the most reasonable conception of modal facts. Holding an ontology of tropes without universals implies huge costs in modal ontology. And this is, of course, another reason to reject such an ontology.

4.4

Natural Classes of Tropes

§ 31. Douglas Ehring has proposed ‘natural classes’ of tropes to satisfy the functions of universals (see Ehring 2011, 175–202; Alvarado 2014). For any object x and any nature F, x is F because x belongs to the natural class of Fs. It must be assumed that the expression “the class of the Fs” is simply a way of designating that class, whatever it is, that fulfills the functions that the universal of F should fulfill, without presupposing such property—as it seemed by the term “class of the Fs”. Of course, any class of tropes would not serve to fulfill the functions of a universal property, for there is an exuberant abundance of classes. For every plurality of tropes—no matter how heterogeneous they may be—there is a class of precisely those tropes. The classes must be, therefore, ‘natural’. This ‘natural’ character of a class is a fundamental fact, not grounded on anything. Probably, the most important inspiration for Ehring has been what David Lewis argued about resemblance classes of objects: that they are not more acceptable than the nominalism of natural classes of objects, in which the character of ‘natural’ is something primitive and fundamental (see Lewis 1983, 14–15). This comparison, however, could be valid concerning the primitive of ‘resemblance*’ explained above (§ I.2.1), a primitive multigrade and contrastive relation of similarity that has as range the totality of possible objects. A predicate so complicated and distant from our cognitive practices does not seem much better than simply declaring some classes as ‘natural’, without further explanation. This is, indeed, evidence of how unlikely a resemblance nominalism that appeals to similarity* would be, rather than evidence of the likelihood of a nominalism of natural classes. Ehring believes that there are certain advantages in a theory of natural classes of tropes in relation to theories of resemblance classes of tropes. Theories of resemblance classes of tropes, as explained, should assume that resemblance facts are

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‘primitive’, fundamental, and that the formal characteristics of the resemblance relationship—its reflexivity, symmetry, and intransitivity—are brute facts. If one postulates natural classes of tropes, instead, facts of resemblance, as well as the formal features of the relationship, would be grounded on the facts about natural classes of tropes (see Ehring 2011, 187–193).18 But this conception of natural classes of tropes have great difficulties of its own. With all the problems that the theories of resemblance classes of tropes have, they try to explain ontologically the one over many by ‘similarities’—what has some connection with our communicative practices and our cognitive transactions with the world, despite all the difficulties with such notion (see § 18). Where there is an attempt at explanation by the defender of resemblance classes of tropes, there is nothing offered by the defender of natural classes. One offers a bad explanation, while another offers none. Everything must be thrown under the rug of a primitive ‘natural’ character. Besides, a large part of the difficulties already presented remain. How are the relationships between determinate and determinable properties explained? What kind of causal power confers a trope to its possessor? As it happens in the theories that ground the nature of things on relations of external similarity, here the nature of things would be based on belonging to natural classes. Why would something extrinsic to an object have relevance to the causal interactions in which that object can enter? From an epistemological perspective, how could we get to know a property if it is a natural class of tropes? To know a class is to know its elements. Knowing the ‘natural’ character of a class requires knowing such a class. How can we achieve reliable inductions in this way? How can we be justified in believing that there are natural laws? As it happens for the theories of resemblance classes of tropes, merely possible tropes are required here, and there do not seem to be resources from a perspective with natural trope classes to generate the appropriate modal facts if one wants to remain a modal actualist. Ehring considers the problem that there could not be more or fewer tropes to preserve the identity of a natural class (see Ehring 2011, 203–226). In effect, it is essential to a class what its elements are. If there is a natural class of tropes, this class would not be the class it is, if one of the tropes that are its elements were missing, or—for the same reasons—if there were an additional trope as an element of the class. Ehring proposes ‘counterparts’ of such natural classes in response. The natural class that actually fulfills the functions of the putative universal of being F could not have more or fewer elements, but the sentence: “the actual natural class of being

18

It is intriguing, on the other hand, that Ehring says that natural classes of tropes do not have the problems that the resemblance classes of tropes have, if the relation of resemblance is supposed to be not an internal relation, grounded on the intrinsic qualitative nature of the tropes (see § 26; Ehring 2011, 184–186). He cannot claim that tropes have an intrinsic qualitative nature for the same reasons. If tropes had an intrinsic qualitative nature—which grounded the ‘natural’ character of the classes that will fulfill the functions of a universal—then there would be more than one arbitrarily different internal relationship between two tropes, which is at odds with its simplicity. Ehring must maintain that a trope possesses the qualitative character that it possesses because it belongs to one natural class rather than another.

References

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F could have an additional trope” is true because there is a counterpart of the actual natural class of being F that has such an additional element. The problem here, as it has been explained in some detail in § 30, is that there is no way to generate the merely possible tropes that should be elements of the natural class to function as counterparts from an actualist perspective. Furthermore, Ehring seems completely unaware that without merely possible tropes there are not even actual natural classes of tropes for the same reasons that without merely possible tropes there are actual resemblance classes of tropes. These considerations seem sufficient to set aside the theoretical alternative of natural classes of tropes. This also concludes the examination of ontologies of tropes to replace the universals. It is not a more attractive conception than pure and simple nominalism.

References Adams, R. M. (1974). Theories of actuality. Noûs, 8, 211–231. Alvarado, J. T. (2011). Clases de tropos como universales ersatz. Trans/Form/Açao, 34(1), 87–114. Alvarado, J. T. (2013). Estados de cosas en el tiempo. Revista de humanidades de Valparaíso, 1(2), 83–104. Alvarado, J. T. (2014). Clases Naturales de Tropos. Filosofia unisinos, 15(2), 148–160. Armstrong, D. M. (1989). A combinatorial theory of possibility. Cambridge: Cambridge University Press. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Campbell, K. (1981). The metaphysic of abstract particulars. Midwest Studies in Philosophy, 6, 477–488. Campbell, K. (1990). Abstract particulars. Oxford: Blackwell. Denkel, A. (1996). Object and property. Cambridge: Cambridge University Press. Divers, J. (2002). Possible worlds. London: Routledge. Ehring, D. (2011). Tropes: Properties, objects, and mental causation. Oxford: Oxford University Press. Fara, M., & Williamson, T. (2005). Counterparts and actuality. Mind, 114, 1–30. Fine, K. (1977). Prior on the construction of possible worlds and instants. Postcript to Arthur N. Prior & Kit Fine, Worlds, Times, and Selves. London: Duckworth, 116–168. Reprinted in Kit Fine, Modality and Tense. Oxford: Clarendon Press, 2005, 133–175. Garcia, R. K. (2014). Bundle’s theory black box: Gap challenges for the bundle theory of substance. Philosophia, 42, 115–126. Garcia, R. K. (2015a). Two ways to particularize a property. Journal of the American Philosophical Association, 1(4), 635–652. Garcia, R. K. (2015b). Is trope theory a divided house? In G. Galluzzo & M. J. Loux (Eds.), The problem of universals in contemporary philosophy (pp. 133–155). Cambridge: Cambridge University Press. Heil, J. (2003). From an ontological point of view. Oxford: Clarendon Press. Heil, J. (2012). The universe as we find it. Oxford: Clarendon Press. Lewis, D. K. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, 65, 113–126. Reprinted with postscripts in David Lewis, Philosophical Papers, Volume I. New York: Oxford University Press, 1983, 26–46.

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Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell. Lewis, D. K., & Langton, R. (1998). Defining ‘intrinsic’. Philosophy and Phenomenological Research, 58, 333–345. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 116–132. Martin, C. B. (1980). Substance substantiated. Australasian Journal of Philosophy, 58, 3–10. Maurin, A.-S. (2002). If tropes. Dordrecht: Kluwer. Moreland, J. P. (2001). Universals. Montreal: McGill-Queen’s University Press. Plantinga, A. (1974). The nature of necessity. Oxford: Clarendon Press. Salmon, N. (1989). The logic of what might have been. The Philosophical Review, 98, 3–34. Reimpreso en Metaphysics, Mathematics, and Meaning. Philosophical Papers (Vol. 1, pp. 129–149). Oxford: Clarendon Press, 2005. Se cita por esta última versión. Salmon, N. (2005). Reference and essence (2nd ed.). Amherst: Prometheus. Simons, P. (1994). Particulars in particular clothing: Three trope theories of substance. Philosophy and Phenomenological Research, 54, 553–575. Stout, G. F. (1923). The nature of universals and propositions. Proceedings of the British Academy, 10, 157–172. Williams, D. C. (1953a). On the elements of being: I. The Review of Metaphysics, 7, 3–18. Williams, D. C. (1953b). On the elements of being: II. The Review of Metaphysics, 7, 71–92.

Chapter 5

The Superiority of Universals over Theological Nominalism

Abstract The third alternative to universals discussed in this work is theological nominalism. This form of nominalism postulates concepts in the mind of God to replace universals. There are many advantages of this position over regular concept nominalism. Nevertheless, it is argued here that the conception seems unfeasible if one supposes divine simplicity. It is also affected by a problem of regress specific to this form of nominalism. The chapter finally considers if Platonic universals are incompatible with divine aseity. They are not. § 32. The third alternative that will be considered in this Part I is theological nominalism. It is a form of nominalism of concepts, but concepts of a very special entity, not of any finite thinker. Although there have been variations in the way in which the nature of God is conceived, it has traditionally been held that God is an entity omnipotent, omniscient, and perfectly good. Of course, it is not possible here to make an even superficial review of the philosophical positions that have been adopted regarding the nature of God, or of the motives adduced to justify or to reject his existence.1 For what matters here, the point is that the postulation of the existence of God seems to bring theoretical advantages of importance to those who are inclined 1 In recent years, all the traditional forms of argumentation for the existence of God have been explored, as well as the problem of evil and the ‘hiddenness of God’. A form of argument that I find especially convincing depends on modal and causal principles. ♠x ¼ df [◊∃y (y ¼ x) ^ ◊Ø∃y (y ¼ x)], i. e., [♠x] is an abbreviation of “x is contingent”. The denial of [♠x] is —strictly— [□8y (y 6¼ x) _ □∃y (y ¼ x)], that is, it is either impossible or necessary. Since it does not make any sense to say of something (that has to be in the range of quantification) that is impossible, [Ø♠x] will be taken as [□∃y (y ¼ x)], abbreviated [♣x]. ‘x causes y’ will be abbreviated as [x ) y]. ‘|’ designates the mereological connective ‘is disjoint from’. For simplicity, it will be assumed that quantifiers have states of affairs as their range.

1. ∃x(♠x ^ (x ¼ c)) 2. 8x ∃ y( ♠ x ! ((y| x) ^ (y ) x)) This principle (2) has been called sometimes as the ‘principle of sufficient reason’. By universal instantiation of (2) and modus ponens with (1), it follows: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_5

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to a nominalist ontology. Theological nominalism is probably the most formidable theoretical adversary of the theories of universals. Nominalism of concepts without qualification, on the other hand, has been considered as a rather weak nominalist alternative. Concepts are constitutive of the thoughts that a thinker has or can have. There is a huge controversy about the nature of concepts, but it is an assumption accepted by all that they are ‘mental’ entities (see Vallejos 2008). That is, they are entities whose existence depends ontologically on the existence of a thinker. Whether one understands them as ‘mental representations’, or as ‘abilities to judge’, they are entities of such a nature that, if there is no thinker of whose thoughts they are a constituent, then they do not exist. What concepts a thinker possesses is a contingent fact (see Armstrong 1978, 21–22, 25–27; Edwards 2014, 88–94). A different historical-cultural development would have led us to have different concepts that we have. Concepts also are acquired in an instant of time and can be lost in an instant of time. If the scientific revolution had not occurred in the seventeenth century, for example, then we would not have come to form the concept of ‘electron’. Before the end of the nineteenth century, we did not have such a concept. If in the year 2050 there is a great nuclear conflagration that would make human beings disappear, perhaps such a concept would also cease to exist. Concepts are variable, then, between different possible worlds, and between different times in the same possible world, at least if they are concepts of finite rational thinkers like us. For a nominalist of concepts, the fact that, for example, x is F is grounded on the fact that there is a concept CF such that x falls under CF. From now on, the concept of ‘being an F’ will be designated as “CF”. Then, something is an electron because the concept of ‘electron’ exists. But, then, before the nineteenth century, there were no electrons. In possible worlds that are physically like ours, but without rational thinkers, there would be no electrons because the concept of

3. ∃y((y|c) ^ (y ) c)) 4. 8x((x ) c) ! ♠ x) _ ∃ x((x ) c) ^ ♣ x)) This statement (4) is simply an instance of the principle of the excluded middle. But: 5. Ø 8 x((x ) c) ! ♠ x) From (5) and (4), it follows, by modus tollendo ponens: 6. ∃x((x ) c) ^ ♣ x)) And, then, by simplification of (6): 7. ∃x ♣ x Why would the premise (5) be accepted? One reason is that [c ¼ (xι) 8y ((y ◯ x) $ ∃z (♠z ^ (y ◯ z)))], i. e., c is the mereological fusion of all contingent states of affairs. Any disjoint cause of c must be necessary, by definition. Probably the most discussed premise in this argument is (2). For a detailed discussion of the principle of sufficient reason, see Pruss 2006. Note also that what is concluded in (7) is that there is a necessary state of affairs. A necessary state of affairs must have as a component a necessary object if this state of affairs is a causal relatum, so it follows that there is at least one necessary concrete object.

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‘electron’ would not exist. For the same reasons, there would be no objects with mass, or objects with shapes, or trees, or galaxies, because there would be no concepts of ‘mass’, ‘form’, ‘tree’ or ‘galaxy’. And if there were a great nuclear conflagration that would make human beings disappear—assuming there are no other rational thinkers—then there would be no more electrons in the world. All these consequences are, of course, absurd. There are also problems of regress for the nominalism of concepts (see Armstrong 1978, 18–21, 27). In the first place, why should the concept of ‘electron’ in the mind of the thinker S1 be the same concept as the concept of ‘electron’ in the mind of the thinker S2? These are different mental entities since S1 and S2 have different minds. The intuition is that they are concepts of the same ‘type’, but this could not be understood as instantiations of the same universal without abandoning nominalism. What a consistent nominalist should hold is that two concepts in different minds are the ‘same’ if they fall under the same concept. That is, both the concept-of-electronin-the-mind-of-S1 and the concept-of-electron-in-the-mind-of-S2 fall under the concept ‘concept-of-electron’. But now the problem arises again: why the concept of ‘concept-of-electron’ in the mind of two different thinkers, counts as the same concept? Appealing now to the ‘concept-of-the-concept-of-electron’ is not going to solve the problem. Traditional nominalism of concepts must also face the classical problem of regress that has already been explained in detail above (see § 18, also § 68). What grounds that x is F is that x falls under the concept CF. But then there is a relationship of ‘falling under the concept –’ that has a crucial ontological relevance. The facts of the world are dependent on such relationships. But if one is a coherent nominalist, one could not sustain that such relations were universal. As it holds for any other case, x falls under the concept CF because falls under the concept ‘x falls under the concept X’. But if one here considers the fact of falling under the concept ‘x falls under the concept X’, this must be grounded on the fact that falls under the concept ‘x falls under the concept X’, etc. Another traditional difficulty is the problem about how objects can acquire causal powers because they have certain nature (see Armstrong 1978, 22–27). The fact that an object falls under a concept seems somewhat extrinsic to the object. Why concepts that possess or not rational thinkers could have an impact on the causal relationships in which an object can enter? This problem must be taken with the precautions indicated above (see § 13). There are no difficulties at this point if one defends some regularist or counterfactual conception of causality. The problem arises if you want to explain how an object acquires causal powers by the nature it possesses, assuming that causality is an ontologically fundamental relationship. Nominalism of theological concepts seems to be immune to almost all these difficulties. The peculiarities of the divine nature are crucial for this. As indicated, it has been normally assumed that God is the only omnipotent, omniscient, immutable, and morally perfect entity. There has been recent controversy regarding some characteristics of God such as his simplicity or eternity (see Swinburne 1993, 216–229; Mann 2005, 47–57), but these issues have no relevance in what follows. For what interests here, God is an entity that satisfies the following principles:

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[Omniscience]

It is necessary that for every proposition p: if p is true, then God knows that p.

Knowledge also generally satisfies a principle of factivity. That is if someone knows that p, then p is true. Both principles imply: [Omniscience*]

It is necessary that for every proposition p: p is true if and only if God knows that p.

[Omnipotence]

It is necessary that for every state of affairs Q: Q is metaphysically possible if and only if God can cause Q to obtain.

No decision should be made here as to the order of dependence between the righthand side and the left-hand side of the principle of Omnipotence. It could be argued that the fact that a state of affairs is metaphysically possible is ontologically independent of God’s being able to cause such a state of affairs. It just happens that the causal power of God covers the entire modal space. One could also maintain, however, that the fact that a state of affairs is metaphysically possible is grounded on the fact that it could be caused by God. For what is going to be examined in this section, either alternative is indifferent. God is also a necessary and eternal entity.2 That is, God exists in every possible world, and in every moment of time. [Necessity] [Eternity]

It is necessary that: if God exists, then it is necessary that God exists.3 It is necessary that for all time t: God exists in t.

To these principles is added the principle of immutability that requires careful formulation: [Immutability]

It is necessary that for all intrinsic attribution F: if God is F at a time t, then God is F at all times.

Here an ‘intrinsic attribution’ is whatever is conceived as the nature of something considered in itself (see § 4). It does not prejudge against the nominalist concerning what such a nature is. Extrinsic attributions are excluded. God would mutate if such respects were admitted for Immutability. When Napoleon is 10 years old, God is such that Napoleon is 10 years old. When Napoleon come to be 11 years old, God ceases to be such that Napoleon is 10 years old. There is a Cambridge change in God between these times because it has ceased to have an extrinsic attribution. Extrinsic variations, however, do not count here. Concepts in the mind of God do not seem to have the difficulties indicated for the usual nominalism of concepts. God must possess mental states in the first place.

2 ‘Eternity’ is the absence of temporal distension. As indicated above, some have questioned the coherence of the idea of ‘eternity’ and, therefore, prefer to speak of “omni-temporality” (see Swinburne 1993, 223–229). For what matters here, it is sufficient with the supposition that God exists at all times, which is consistent both with his eternity and with his omni-temporality. 3 More precisely: [□(∃x (God ¼ x) ! □(∃x (God ¼ x)))].

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Otherwise, he could not know. If God knows that p, then he must possess the constitutive concepts of the thought that p. Nobody can know that p without thinking that p. God is a necessary entity, so his existence is invariant in all possible worlds. When it comes to the concepts of finite rational thinkers like us, there are possible worlds in which we do not exist, but it seems absurd to think that in those worlds there are no electrons or galaxies. When it comes to God, however, in all worlds, there are divine concepts correlated with the different natures of objects. It does not matter for this that not in all possible worlds, it is true that there are electrons. In the worlds where there are electrons, God knows that there are electrons. In the possible worlds in which there are no electrons, God knows that there are no electrons. In any case, God must think the propositional content there are electrons, only that in some possible worlds he thinks that this propositional content is true, while in other possible worlds he thinks that this propositional content is false. In all possible worlds, therefore, there is the concept of ‘electron’ in the divine mind. On the other hand, God exists at all times. When it comes to our concepts, there are times when they did not exist, because we did not exist, but it is absurd to think that in those times, there were no electrons or galaxies. God, on the other hand, existed in all those times and will exist in all future times. Since God is immutable, the beliefs of God and the thoughts of God do not vary between different times. What happens in this case with the states of affairs that come into existence or cease to exist? It could not be sustained—in a way analogous to what happens for the modal case—that when there are tyrannosaurs, God knows that there are tyrannosaurs, and when there are no tyrannosaurs, God knows that there are no tyrannosaurs, because that would be a change in an intrinsic attribution in God—what would be in conflict with the principle of Immutability. What happens is that God knows in all instants of time that in t1 there are tyrannosaurs, and God also knows in all instants of time that in t2 there are no tyrannosaurs.4 The concepts of finite rational thinkers like us are multiple. As the concepts are mental entities, they differ numerically between different thinkers. The question arises, therefore, why the concepts of ‘electron’ in the minds of different thinkers are the ‘same’ concept. This problem generates, as it has been explained, a vicious regress. But in God, there would only be a unique concept of an electron. There is, therefore, no problem about how different mental entities in different minds are the ‘same’ concept because there is only one of those entities. Theological nominalism

4

Several issues arise concerning this response to the problem. For Immutability to work in the mental states of God, it is required that all moments of time —past, present, or future— are equally available to his knowledge. But this does not seem reasonable if there are no future times. Nor does it seem reasonable if the character by which some time is present is perpetually changing. There could not be a ‘static’ knowledge of the entire time series since the entire series would always be in change. These questions cannot be discussed here with all the detail that it would require. It seems that the principles of Immutability and Omniscience work with a tetra-dimensional or eternalist ontology of time in which all times have the same ontological status. We also have to take care of the coherence of Omniscience and the freedom of the will of created persons, if this freedom is incompatible with determinism.

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does not seem to be affected by the problem of causal powers either. It seems reasonable to maintain that the causal powers of objects, according to their respective natures, depends causally, in turn, on God’s action. And the concepts in the divine mind have causal relevance to the action of God. There is, therefore, no disconnection between concepts and causal powers as it appears for the usual nominalism of concepts. There are a series of difficulties for the forms of nominalism presented above that have to do with epistemological consequences. Neither in resemblance nominalism nor the theories of tropes can we explain how we have knowledge of authentic properties or how we can carry out reliable inductions. These difficulties seem much less formidable when it comes to theological nominalism. The classes of similarity of objects and the classes of tropes do not allow to ‘unify’ reality in the way that universals do. To know one of those classes would require knowing infinite objects or tropes that, of course, escape our finite cognitive abilities. For the same reasons, there is no way to ensure that the type of causal connections that involve an object or a trope must be replicated in other objects or tropes. There are no reasons why we should trust our inductive practices. There is also no reason to think that there are natural laws other than as regularities between types of events. It does not seem reasonable to attribute a regulatory role to natural laws in these conceptions. When it comes to theological nominalism, however, there is something that grounds that the same nature is present in different objects. Be that x1 is F and x2 is F. The fact that both x1 and x2 have the same nature is grounded on the fact that they fall under the same concept CF in the divine mind. And this concept is the ‘same’ in the strongest sense because numerically it is only one. Concepts in the divine mind also have the virtue of ‘unifying reality’, as do universals. In this way, knowledge of a concept in the divine mind would be a way of knowing everything that falls under such a concept. It should not be surprising either that everything that falls under the same concept obeys the same laws. Natural laws would be inscribed in the concepts of the divine mind. Everything that falls under such concepts would be ‘regulated’ by them. It might seem strange here that one has epistemological access to concepts in the mind of God. Would not this be as extravagant epistemologically as knowing an infinite class of objects or tropes that includes actual and merely possible entities? But it happens that we would know such concepts by their cases. And to know what falls under a concept does not seem epistemologically more extravagant than to know a universal by its instantiations.5

5 Also, it should be noted that there is an entire Augustinian epistemological tradition that understands knowledge as divine illumination. In this tradition, we can develop our superior cognitive abilities precisely because we can assimilate the concepts in the mind of God and interpret perceptual information from the ‘point of view of God’. Again, this may seem epistemologically extravagant, but I do not see that it is more extravagant than knowing infinite merely possible objects or tropes that make up a class of similarity. Certainly, it is no more extravagant than the traditional forms of nominalism, if one admits at the outset that God exists. Illuminating the mind of a finite rational thinker is within his causal powers.

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These motives have led several philosophers to postulate that concepts in the divine mind are what could satisfy the theoretical functions usually attributed to universals (see Leftow 2006, 2012, 299–316; Pruss 2011, 205–208, 216–217). It is, moreover, a conception defended since late Antiquity, at least.6 Leftow also has argued that theological nominalism can be seen as an important theoretical reason to justify the existence of God (see Leftow 2006, 355–356). Theological nominalism seems the most reasonable form of nominalism. To postulate the existence of God would allow us to achieve a great ontological economy since only objects would be required as a fundamental category. If one wants to avoid universals, this is a promising alternative. Theism would be the most appropriate way to solve the deepest theoretical concerns of the nominalists, lovers of “desert landscapes”. For this, it is indispensable that theological nominalism is capable of fulfilling, in effect, all the functions that are required of a property, and that also it does not bring other excessive theoretical costs. Unfortunately, as will be shown, theological nominalism fails on both points. In the remainder of this chapter, these different difficulties will be presented. In the first place, theological nominalism has rested—at least until this moment—in a quite vague idea as to what should be understood by a ‘concept in the divine mind’. When one examines in more detail what such concepts may be, the explanatory virtues of theological nominalism seem to vanish (see Alvarado 2011). Second, theological nominalism has not solved a regress problem analogous to Bradley’s regress. Traditional solutions to this regress can no longer be considered valid, and theological nominalism has nothing in its place. This chapter will be concluded by explaining why a frequent objection to theories that do not assign the roles of universals to divine concepts, motivated by the so-called “divine aseity”, should be no worry.

6

It is a conception already defended by the Neo-Platonists. Plato (1982) had maintained in the Timaeus that the demiurge eternally contemplates forms (see Timaeus, 28a-29b). In Middle Platonism ideas are located in the divine mind (see Philo, About the creation of the world according to Moses, 4; Alcinous (1993), Handbook of Platonism IX, 1). Plotinus interprets the Timaeus holding that the act of contemplation and the contemplated form are identified (see Enneads, V, 5). An important background of this thesis is Aristotle (1998) Λ 7, 1072b 19–23 and Aristotle (2010) III 4, 430a 3–6 of Aristotle. Plotinus differentiates between the One and the Intellect-Intelligible. If one were to identify God with the One, it happens, then, that the forms are not identified with concepts of the first hypostasis, but with concepts of the second hypostasis. St. Augustine, on the other hand, identifies the forms with concepts in the mind of God, in such a way that our cognition of forms is produced by divine illumination (see Octoginta trium quaestionum, Q. 46, ML 40, 30). St. Thomas Aquinas later rejects divine enlightenment as a cognitive mechanism but argues that there are ideas that are divine forms in whose likeness the world has been made (see Summa Theologiae, I-II, q. 15). I thank Marcelo Boeri and José Antonio Giménez for illustrating me about this historical background.

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Concepts and Mind of God

§ 33. There are two fundamental problems concerning concepts in the mind of God. In the first place, it is not at all peaceful how the nature of a concept should be understood—in God or any rational thinker. Second, the mind of God is not like the mind of a thinker like us. We do not know what it is for God to be aware of his thoughts, just as we do not know what a thought of God is. Depending on the conception one has of what a concept is, and the conception one has of the nature of divine thought, the conditions of identity of his concepts will also vary. This is crucial here, because if what we are inclined to think of as properties are concepts in the mind of God, then the conditions of the identity of such concepts should be projected onto the conditions of identity of what—intuitively—appear to be properties. There is a requirement here that theological nominalism must satisfy—just like the other alternatives in metaphysics of properties—: our intuitions about what authentic properties exist cannot be completely inadequate, or strongly inadequate. To be more precise, if it were to follow from theological nominalism that our intuitions about what authentic properties exist were very inadequate, then a reasonable explanation should be offered as to why we suffer from this inadequacy. It would be a gross inadequacy, in effect, that would affect much of the knowledge we claim to have about the world by our best theories. It would constitute, in fact, a rather radical form of skepticism. Brian Leftow has been concerned about the intentional nature of concepts in the mind of God (see Leftow 2006, 333–335; 2012, 309–310) and the question about ‘broad content’ (see Leftow 2006, 331–333), but the problems that will be explained here are orthogonal to those raised by Leftow. The question of whether or not there is ‘broad mental content’, whether there is ‘narrow mental content’ or both, cannot be resolved here.7 The question whether the only way in which a mental state has an intentional character is if it has as ‘content’ something different from the mind itself and its states cannot be resolved here, either. Of course, if there is only broad mental content, or if intentionality requires an ‘external’ content, then nominalism of divine concepts would be in a serious problem. Indeed, under any of these alternatives, there could only be concepts in the divine mind if there were independent natures

It has been designated as “narrow content” the type of content of mental states that a rational thinker can possess regardless of whether there is something ‘external’ to the mind that constitutes what that content refers to or on what such content depends. Narrow content corresponds to how mental states have been conceived in the Cartesian tradition. One might think that there are elephants without there being elephants, nor an independent property of being an elephant, nor a chain of causal connections and/or chain of communicative transmission that ultimately is linked with elephants or with the property of being an elephant. For those who propose, instead, “broad content”, nobody could think that there are elephants without elephants or without the property of being an elephant. The content of this concept involves the environment in which that thinker is since it consists of the natural kind of elephants or the property of being an elephant. Without elephants or without the property of being an elephant, there simply would not be such content (see Kim 2006, 254–272).

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that constituted the content of those concepts. One could not, then, suppose that things are as they are because they fall under certain concept because the concept in question would be the concept that it is because, in turn, it has a certain nature as a content. It would turn out that things are as they are because they have an independent nature, which hardly gives any explanatory value to the concepts in the mind of God. But these problems are not going to be solved here. There are much more basic issues that must be considered.

5.1.1

Concepts

§ 34. Concepts have been understood in various ways: as sets of necessary and sufficient conditions, as dispositions to judge, as mental representations, as ‘prototypes’ or ‘stereotypes’, as ‘ways for something to be given’, and so on. We may not even be designating a single ‘natural kind’ of entities when we talk about them. The view called “Fregean” can be left aside here from now on (see Vallejos 2008, 97–107). A ‘concept’ in this family of views is an abstract entity that does not depend on the existence of a mind, and that can characterize a plurality of different objects. It is difficult to differentiate it from a universal. So, if concepts in the divine mind were understood in this way, theological nominalism could be a roundabout way to propose a theory of universals. Throughout the panoply of non-Fregean theories, concepts are mental entities whose identity conditions are epistemic. Two concepts are the same concept if they satisfy the same functions in the ‘cognitive transactions’ of a thinker. Among these cognitive transactions are categorization practices, the descriptive individualization of objects, inferential practices, testing and confirmation of hypotheses—crucial for learning—and the generation of content (see Vallejos 2008, 59–60). Concepts must obey a requirement of generality (see Evans 1982, 100–105), i.e., if a rational thinker is capable of thinking about the content of the structure Fa —that is, the content formed by the concept ‘F’ and the conceptual individualization of a—and possesses the resources to conceptually identify b, then that rational thinker must be able to think the content Fb—that is, the content formed by the concept of ‘F’ and the conceptual individualization of b. Concepts also present stability in their content between different rational thinkers. Many thinkers can think about the ‘same’ content. Let [. . . C . . .] be a content whose constituent in one or more places is the concept C. Let m1, m2, . . ., mn be ‘contexts’ in which a thinker has an attitude towards a content. For simplicity, we will assume that the attitude towards the content is a ‘belief’, but the same could be said about other attitudes. A context includes the set of all the mental states that the thinker possesses at a time and until that time, that is, the perceptive evidence that he possesses, the other beliefs he has about the world, his preferences and intentions for all instants of time previous and simultaneous to the moment in which he possesses the attitude concerning the content in question. It must also include all its ‘environment’. In order not to establish here restrictions that may seem arbitrary, this ‘environment’ includes the complete possible world in

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which that thinker dwells. A context is, therefore, a centered possible world on a thinker and a time. So: [Concepts]

It is necessary that, for every thinker S, for every context m, for every concept C1, C2: C1 ¼ C2 if and only if: S believes that [. . . C1 . . .] in the context m if and only if S believes that [. . . C2 . . .] in the context m.8

All the difference between [. . . C1 . . .] and [. . . C2 . . .] is that in all the places where the concept C1 occurs in [. . . C1 . . .], in the content [. . . C2 . . .] occurs the concept C2. It does not matter what justification the rational thinker has to believe this content. It may be the possession of certain perceptual evidence. It can be inferential links with other beliefs. These conditions of identity are neutral between ‘atomic’ conceptions of concepts and more ‘holistic’ conceptions.9 What is relevant is that the identity of a concept is determined by the role that the concept has in the mental states that are justification for the contents in which such concept occurs, and the role it has in the contents that are justified by those in which it occurs. It includes all the inferential links of the concept with others. Therefore, what concepts a rational thinker has is correlated with what the thoughts and cognitive processes of that same thinker are. The fact that certain concepts are possessed—given, also, a space of combinations of such concepts to form complex contents—grounds the space of thoughts that such a thinker can think. No thinker could think a content without understanding the concepts involved in it. When it comes to nominalism of concepts, one may be inclined to think that concepts have to be correlated with those traits that we consider constitutive of an authentic property. But, given the grounding connection between the natures of things and concepts in this form of nominalism, it is crucial that concepts ground natures. For any object x, the fact that x is F is grounded on the fact that x falls under the concept CF. Therefore, concepts ground what nature things possess. In this case, it is divine concepts that ground the nature of objects. And what concepts God has is determined by the cognitive processes that God has—if it is possible to talk about ‘processes’ here without violating the principle of Immutability. One cannot directly project to God the types of cognitive processes that we have and our concepts. It is

8

More precisely, keeping the same variables: □8S □8m □8C1 □8C2 □((C1 ¼ C2) $ ((S believes that [. . . C1 . . .] in m) $ (S believes that [. . . C2 . . .] in m))) 9 In some theories, the identity of a concept is determined by the inferential connections of that concept with others. No concept can here, therefore, exist in isolation without the ‘cluster’ of which it is a part. Similar situations have been adduced to reject the adequacy of other identity conditions. These questions are not going to be discussed here. The identity of concepts must be seen here as fixed in a block for all the concepts that are connected inferentially. In other theories, however, there are ‘atomic’ concepts whose identity is independent of the others. One can then postulate complex concepts constituted from these simple concepts and dependent on them. In any of these cases, the conditions of the identity of a concept can be characterized by a principle like Concepts.

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crucial, therefore, to try to specify better what kind of cognition or cognitions God has.

5.1.2

The Mind of God

§ 35. There is a long and venerable philosophical tradition according to which the object of the thought of God is God himself (see Aristotle, Metaphysics Λ 71072b 19–23). Moreover, it has also traditionally been held that God is a perfectly simple entity, i.e., (i) God has no spatial or temporal parts, (ii) God does not instantiate intrinsic accidental properties, and (iii) there is no distinction in God between his essence and his being. Then, the act of thought by which God thinks of himself and the concept that God has of himself, are identical. As indicated above, the thesis of divine simplicity has been the subject of some controversy (see, for example, Plantinga 1980, 26–61). It has been argued that a property is an abstract entity and that, therefore, divine simplicity would imply that God would also be an abstract entity, which seems false. It is not possible to enter here to ponder these controversies.10 What is interesting is that under the assumption of divine simplicity in God, there could not be more than a single concept. This unique concept would be identical to God himself. For there to be two different concepts in God, be C1 and C2, it would be required that there are at least thought acts of a content [. . . C1 . . .] that were not thought acts of content [. . . C2 . . .]—assuming that all the difference between [. . . C1 . . .] and [. . . C2 . . .] is that all occurrences of C1 in [. . . C1 . . .] are replaced by occurrences of C2 in [. . . C2 . . .]. According to the principle Concepts, this would guarantee that there were at least two different concepts in God. But in God, there is only one act of thought. It must be an act of eternal and necessary thought. No entity less perfect than God could be the object of such thought. The content of that act of thought must be a necessary entity and of such a character that the mere contemplation of that entity guarantees Omniscience. Since God is the causal principle of everything, it is reasonable to suppose—in a conception of this kind—that the contemplation of the divine essence is all that is required to understand and know everything. That same divine essence is the only divine concept. It is clear that under this assumption, theological nominalism would have disastrous consequences. If there is only one single concept in the divine mind, then there would be only one nature. Recall that for any object x, x is F in a theory of this type because x falls under the divine concept CF. Since the natures of objects are grounded on the concepts under which these objects fall, what concepts exist will 10

God has to be an abstract entity if the only way a property could exist is as a universal. If one admits tropes, however, the problem does not arise. The thesis of divine simplicity must be seen as the thesis that God is a unique trope and that the expressions “the omnipotence of God”, “the wisdom of God” or “the goodness of God” are all designations of it. Olof Page has also pointed out to me that it is difficult for a nominalist of concepts to adduce this argument against divine simplicity since from the perspective of a nominalist there are no abstract properties of any kind.

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ground what the authentic natures of such objects are. If there is really only one concept in the divine mind, there should be only one nature. Thus, although we have the intuition that the mass is something different from the electromagnetic charge, this should be an illusion. The mass and the charge would really be the same. The same goes for any other ‘authentic property’ in whose existence we believe by natural science. It would be, therefore, that everything we think we know about the world by the theories that we think are better justified, would not be justified. We could not trust any of the best scientific theories, because they erroneously differentiate what appear to be different properties that are not. It is not difficult to see that this path leads to radical skepticism. The least that is expected of a reasonable metaphysics of properties is that it is in harmony with our best evidence of what the natures of things seem to be, as those natures are presented in natural science and our ordinary access to the world. Theological nominalism would not respect this minimum requirement, at least if there is divine simplicity. It is true that in the same tradition that postulates divine simplicity, it has been held that there are different ‘ideas’ in the divine mind that work as ‘exemplars’ of the action of God on which everything depends causally (see St. Thomas Aquinas, Summa Theologiae, I, q. 15). But such ‘ideas’ are simply God’s comprehension of everything created by contemplating his essence. It is not a plurality of concepts, but a single concept that is the very being of God perfectly self-transparent to himself and sufficient for the understanding of everything. One might suppose at this point that the defender of theological nominalism should seek refuge in some conception of God’s nature in which it is not a simple entity. I am afraid, however, that these theories do not offer much solace to the nominalist either. A form of credible theological nominalism should postulate concepts in the mind of God that correspond to what appear to us to be authentic properties, as highlighted. There should be a plurality of concepts in the mind of God and, with it, a plurality of complex intelligible contents that can be formed compositionally by such concepts. How is it that God would come to have such concepts? Of course, it would be incompatible with the Immutability thesis to sustain that God comes to temporarily acquire such concepts when entering into cognitive transactions with created entities. Note also that if one were to suppose that God ‘learns’ from ‘experience’ and comes to acquire the mastery of concepts from such ‘experience’, however this ‘experience’ is understood, it is difficult to argue that objects must have the nature that they have because they fall under certain concepts in the divine mind. The concepts in question would be dependent on the natures of the objects that they have allowed to understand because they are concepts that God acquires because he understands how those objects are independently. Or, in other words, if the objects of the world did not have the nature that they have, God would not come to form the concepts under which such objects fall. It cannot be, then, that the nature of objects is ontologically grounded on divine concepts. Instead, divine concepts are ontologically derivative to the nature of such objects. The function of divine concepts in this scheme would be perfectly idle. There would be some alternatives that could be explored by the theological nominalist compatible with Immutability and Omniscience. In any of these

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alternatives, God is from all eternity and for all eternity endowed with a plurality of adequate concepts to think everything. If these concepts have epistemic identity conditions, as stated in the principle of Concepts, the difference of these concepts must be reflected in a plurality of thoughts different from God. This plurality of thoughts is immutable. It is not, therefore, that God comes to learn something that at some previous time, did not know. In every moment of time, God possesses all these thoughts at once. Two concepts, C1 and C2, differ from each other because for some content [. . . C1 . . .] God believes that [. . . C1 . . .] and does not believe that [. . . C2 . . .]—where all the difference between [. . . C1 . . .] and [. . . C2 . . .] is that in all the places where C1 occurs in [. . . C1 . . .] C1 is replaced by C2 in [. . . C2 . . .]. A theory of this kind should be accompanied by a plausible explanation of how God is endowed with such a plurality of correlative concepts and thoughts if he has not acquired them by ‘experience’. A first alternative is to suppose that it is a domain of sui generis entities of necessary and eternal existence, and which by their nature can be attributed to a plurality of objects. But this sounds too much like the postulation of a domain of transcendent universals to be acceptable to the theological nominalist. A contingent domain created of sui generis entities, on the other hand, seems an important fault of ontological economy. God would have to create the concept of ‘electron’ and then create electrons. These concepts would be numerically different both from God and from what falls under them. These entities would also be of such a nature that they could be attributed to a plurality of objects, which makes them suspiciously similar to immanent universals. The most reasonable alternative, therefore, would be to postulate that the concepts are part of the intrinsic nature of God. God omni-temporarily possesses a totality of thoughts and concepts, making up the content of such thoughts. Since this theory must be given in a general nominalist scheme, it should not imply the postulation of properties of God, numerically different from himself. It must be ‘part’ of his intrinsic nature in the same way that form and mass are ‘part’ of the nature of an object that is spherical and has 10 gr of mass at the same time. God’s acts of thought are not numerically different entities from God either. It is a ‘part’ of his intrinsic nature, just like any other ‘property’ from the perspective of a nominalist. That is, it should be an ontologically primitive fact, not grounded on anything, that God is such that he possesses such a plurality of concepts. It is simply a fundamental fact that God, for example, possesses the concepts of ‘electron’ and ‘centaur’, and it is a fundamental fact that he thinks that there are electrons and that there are no centaurs, thoughts in which such concepts are involved. This seems to be the most promising form of conception of the mind of God suited to the purposes of theological nominalism. Is it, however, a credible theory about divine thought? It would not be justification for the plurality of divine concepts the need to offer a nominalist alternative to theories of universals. That would be petitio principii. Indeed, theological nominalism only becomes a feasible alternative to theories of universals if God has a plurality of concepts that correspond to what intuitively appear to us to be authentic properties, as they appear in our best-justified theories. If that is the only reason to make theological nominalism a credible alternative in metaphysics of properties, then theological nominalism would be an

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acceptable alternative to theories of universals because it should be an acceptable alternative to theories of universals. The postulation of a plurality of concepts in the mind of God and, with it, the postulation of a plurality of different thoughts among themselves in the mind of God, must have independent justification in their favor. The only reason I can imagine is the rejection of divine simplicity. If God is not a perfectly simple entity, then what appear to be authentic properties in God must be either universal, or tropes, or whatever is offered to fulfill their functions from a nominalist perspective. If this is so, then if God has a thought, this thought should be understood analogously to how it is understood for a rational thinker like us. If for a rational thinker like us to think that p consists in a certain state of affairs that involves authentic properties, then the fact that God thinks that p must also be a state of affairs that involves authentic properties, or what fulfills its functions. But the analogy cannot be pressed much more. It is one thing to maintain that God is not simple and that, for example, his omniscience is not identical with his omnipotence, or that his goodness is not identical with his eternity, but it is quite another that he must possess a plurality of concepts correlated with what seem to us to be authentic properties. Compared with how divine thought is understood traditionally, this alternative seems an anthropomorphic projection. Be that as it may, there are other more important difficulties for theological nominalism that should be considered.

5.2

A Vicious Regress, Again

§ 36. It has been shown above that resemblance nominalism, and theories of tropes are affected by problems of infinite vicious regresses (see §§ 18 and 26). The traditional strategy of postulating ‘primitive’ facts of external resemblance is not a solution to these difficulties since a ‘primitive relational fact’ is simply a relationship. The nominalist is faced with the dilemma of accepting relationships in his ontology—if he admits primitive external similarities—so he would cease to be a nominalist, or treat resemblances in the same way as any other relationship, which generates a vicious infinite regress. In the case of theological nominalism, an analogous difficulty appears. As has been shown, under the most plausible form of theological nominalism, God possesses a plurality of concepts and thoughts. In a nominalist conception, it could not be maintained that such concepts or thoughts are the instantiations of universals or tropes. It could not be argued either that God is a trope bundle—in which thoughts and concepts are tropes of the divine bundle. Any of these alternatives would imply abandoning nominalism. The fact that God possesses a plurality of concepts must simply be a fundamental part of the divine nature. For a resemblance nominalist, for example, the fact that an object x has the nature of being an F must be grounded on the similarity of x with all the other objects that make up the class of similarity of the Fs. For a theological nominalist, an object x is F because x falls under the divine concept CF. If this applies to the nature of any object, it must also apply to the divine nature. How theological nominalism would work is

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assuming that the divine nature is such that God has concepts C1, C2, . . ., Cn that corresponds to what appear to be authentic properties. This is part of the divine nature. Just as for any other case, the nominalist should hold here that God has the concept CF because God falls under a divine concept. One could here be inclined to think that the fact that God possesses the concept CF could be grounded simply on the fact that God falls under the same concept CF, but this cannot be. God should have, for example, the concept of ‘tadpole’, but God does not fall under such a concept. If God fell under the concept of ‘tadpole’, God would be a tadpole. From a coherent nominalist perspective, the fact that God possesses the concept CF, whatever it may be, must be grounded on the fact that God falls under the divine concept of ‘possessing the concept CF’. And the fact that God possesses the concept ‘to possess the concept CF’ must be grounded on a divine concept. The same concept does not serve here to ‘own the concept CF’, because the fact that God possesses the concept ‘possesses the concept CF’ cannot be grounded on the fact that God falls under the concept ‘possesses the concept CF’. What this fact grounds is that God possesses the concept CF, but not that he possesses the concept of ‘possesses the concept CF’. The fact that God possesses the concept ‘has the concept CF’ must be grounded on the fact that God falls under the divine concept ‘has the concept of having the concept CF’. It is part of the divine nature that God has the concept of ‘has the concept of having the concept CF’, and this fact must be grounded on the fact that God falls under the concept ‘has the concept of having the concept CF’, and so on. This is a vicious regress. One may be less inclined to consider it as such because it is a regress that involves God, who is already an “infinite” entity, but it is equally problematic. In effect, the fact that God has a certain concept C must be grounded on God having another concept C0 , and this fact must be grounded on God having another concept C00 and so on. There is no fundamental fact in which each of the nodes of the infinite sequence is grounded. It is evident that the theological nominalist will be inclined at this point to try the strategy of postulating divine concepts as ‘primitive facts’. It would be the same maneuver that has been attempted by resemblance nominalists, defenders of universals and defenders of tropes. Just as it has been argued that external similarity or that instantiation are ‘primitive’ to avoid the emergence of vicious regresses, it would be argued in this case that the fact that God possesses a plurality of concepts— corresponding to our intuitions about what authentic properties exist—would also be a ‘primitive fact’ not grounded on anything. As in the previous cases, however, this strategy must be rejected. If God possesses a nature not grounded on the concepts in which he falls, at least as regards what concepts he has, why should it be different in all other cases? An unjustified exception is being made only to avoid the emergence of the regress. To postulate such a thing would be ad hoc. It should also be noted that this regress does not seem to arise if divine simplicity is accepted and, therefore, that there is only one divine concept that would be identical with God himself. In effect, a coherent theological nominalist should hold that God possesses the nature he possesses because God falls under a divine concept. If one admits the thesis of simplicity, the divine nature is presented to us conceptually in various ways such as ‘omnipotence’, ‘omniscience’, ‘moral

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perfection’, and so on. Let the unique nature of God be D. The fact that God is D must be grounded on the fact that God falls under the divine concept CD. God must be such that he must possess such a concept. The fact that God possesses the concept CD must be grounded on the fact that God falls under the concept of ‘owning the concept CD’. But are CD and ‘owning the concept CD’ different concepts? Recall that the concept CD is a concept that should be identified with the divine essence. Understanding such a concept is something like ‘contemplating the face of God’. It cannot be identified with our concept of ‘God’. I could not even be identified with the concept that an angel had of ‘God’. A subject with finite cognitive abilities could understand that something falls under the concept of ‘God’ without understanding that something falls under the concept ‘has the concept of God’.11 We can, therefore, distinguish between our concept of ‘God’ and our concept of ‘owning the concept of God’. When it comes, however, to CD it does not seem that such a distinction can be made. According to the principle of Concepts, two concepts are different if there is a possible context in which someone could assent to the content integrated by one of those concepts without assenting to the content integrated by the other—assuming that those contents differ, at most, by the occurrence of such concepts. In the case of CD, there does not seem to be any possible context in which someone believes that something falls under CD and does not believe that the same thing falls under the concept ‘has the concept CD’. Only God could possess CD. Only God falls under the concept CD. And in all the scenarios in which God thinks that God falls under CD, God also thinks that God falls under the concept of ‘owning the concept CD’. According to the principle of Concepts, therefore, there is no difference between CD and the concept ‘has the concept CD’. It turns out, then, that if the theological nominalist admits divine simplicity, he manages to evade the problem of regress, but he does so at the price of having to postulate a single nature. If, on the other hand, he rejects divine simplicity and postulates a plurality of divine concepts, the disastrous result of identifying what seem to be authentic properties different from each other is avoided, but at the price of generating a vicious regress for his position.

5.3

Divine Aseity

§ 37. “Divine aseity” is the fact that God is an entity absolutely a se, not dependent on anything, nor grounded on anything, and on which everything depends causally. In the Jewish, Christian, and Islamic religious traditions, everything has been created by God. Theological nominalism seems a philosophical position following this long tradition. If God has created all things “visible and invisible”, “in the heavens and on 11

A scenario of God without God understanding the nature of God may seem very strange given our traditional way of understanding him, but we must think of a scenario in which we have a non-personal concept of God or a ‘processual’ conception of God in which God ‘falls in time’ and does, for example, abandon its attributes. Something of this style is what has been defended by the ‘kenotic’ theories of the Incarnation.

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the earth”, then God should have also created universals—if there were such things. A transcendent universal, however, does not begin to exist in a moment of time, nor can it cease to exist in a moment of time. There are no contingent vicissitudes that make it to exist or to cease to exist. It is not a category of entity of which it makes sense to say that “has been created”. The postulation of transcendent or platonic universals, therefore, seems to be in opposition to the traditional doctrine of divine aseity. The reasonable thing for a theist should be, therefore, to adopt some form of nominalism (see Bergmann and Brower 2006; Craig 2012). And the postulation of concepts in the mind of God seems a form of nominalism that avoids most of the difficulties that affect other forms of nominalism, as explained above.12 There are several reasons to doubt this supposed incompatibility. Divine aseity does not oblige us to seek refuge in nominalism. The question should be considered with more detention. The thesis of divine aseity can be formulated in the following way (see Bergmann and Brower 2006, 361): [Aseity]

(i) God does not depend ontologically on anything, and (ii) everything different from God depends ontologically on God.

It has sometimes been added to clause (ii) of Aseity that everything different from God has been created by God. If God has created something, it follows that it is ontologically dependent on God. What is interesting here is that the converse does not hold. Ontological dependence does not require a causal connection on which it is grounded. There is ontological dependence, for example, of a state of affairs on its constituents—objects, property or relation, and time—but it does not make sense to say that the state of affairs has been “caused” by its constituents. The thesis of Aseity must be seen as a thesis about ontological dependencies and not as a thesis about modal covariations. The problems that have been posed concerning divine aseity seem to have been motivated precisely by the confusion of ontological dependence with modal covariation (see § 4). One could be inclined to analyse the ontological dependence of x on y as the strict conditional: it is necessary that, if x exists, then y exists. If one identifies dependence with this conditional, the thesis of Aseity should be understood as: [Aseity*]

12

(i) It is possible that God exists and that nothing different from God exists; and (ii) it is not possible that something different from God exists and does not exist.13

But it is not the only possible way in which this adjustment could be made. In abstract, it would not be incompatible with divine aseity to postulate tropes and classes of tropes that fulfill the functions of universals. Nor would it be incompatible with divine aseity to postulate immanent universals. God would create these universals by creating their instantiations. Nor would it be incompatible with other forms of nominalism. What has happened, however, since Antiquity, is that these philosophical positions have been considered—correctly—far inferior to the postulation of Platonic universals. A form of nominalism of divine concepts has seemed the most reasonable way to accommodate the advantages of Platonic universals without affecting divine aseity. 13 The thesis (i) of Aseity, that is, that God does not depend on anything, is analysed as: [8x ((x 6¼ God) ! Ø□((God exists) ! (x exists)))]. From this thesis it follows that [8x ((x 6¼ God) ! ◊((God

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Creatures satisfy, naturally, Aseity*, because creation is contingent. Created entities exist only by the action of God, so they only exist in possible worlds where God exists—that are all possible worlds—but God does not create in all possible worlds,14 so God could exist without there being anything different from God. On the other hand, transcendent universals exist equally in all possible worlds, so the clause (i) of Aseity* would be false. What is interesting to consider here, however, is that it is not obligatory to interpret the thesis of Aseity as Aseity*. Ontological dependence should not be analysed as a strict implication. Thus, the fact that God and transcendent universals exist in the same possible worlds—that is, in all metaphysically possible worlds— does not make God ontologically dependent on these universals. The fact that universals exist only in possible worlds in which God exists is no guarantee, in and of itself, that they are ontologically dependent on God. The fact that God and the transcendent universals exist equally in all possible worlds does not prejudge the further question about what the direction of ontological dependence between them is. Thus, the thesis of Aseity is compatible with the postulation of Platonic universals. It happens, also, that the preoccupation concerning the thesis of Aseity —understood as Aseity*— brings with it very contra-intuitive consequences. One should suppose that the concepts in the mind of God should be what mathematical truths are about, just as for those who defend the existence of transcendent universals, mathematical truths should be about universals.15 All mathematical truths should, therefore, have God as its truthmaker. If the theological nominalist also intends to replicate the conception of natural laws as relations of a higher order among universals by concepts in the mind of God, it will also turn out that God will be the truthmaker of all statements about natural laws. According to how the mind of God is conceived, as explained above (see § 35), divine concepts will be ‘parts’ or ‘aspects’ of the divine nature, or it will be the simple essence divine. None of these two alternatives to conceive the concepts in the mind of God is very promising in itself, as has been indicated, but even putting the difficulties aside, it would seem that talking about such concepts cannot be about something that is numerically different from God. The truthmaker of the proposition 2 + 2 ¼ 4 is God, therefore, in the same way as God is the truthmaker of Gödel’s incompleteness theorem. To some, it may seem extravagant that all mathematics is, after all, a department of theology. What is

exists) ^ Ø(x exists)))] The thesis (ii) of Aseity, that is, that everything different from God depends on God, is analysed as: [8x ((x 6¼ God) ! □((x exists) ! (God exists)))], which is equivalent to [8x ((x 6¼ God) ! Ø◊((x exists) ^ Ø(God exists)))]. 14 At least, this has been the supposition made by the Judeo-Christian tradition in which the creation is a contingent and free action of the Will of God. In other traditions, the ‘emanation’ of the creatures from God is a necessary process. Others, additionally, have argued that it is necessary that God creates the best of all possible universes. What is argued here about the Aseity of God applies in any of these conceptions about God and its creatures. 15 A structuralist point of view in the philosophy of mathematics according to which what mathematics deals with are ‘abstract structures’ that can have multiple instantiations has gained a lot of acceptance. These ‘structures’ are transcendent universals (see Shapiro 1997, 84–106).

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more problematic than the preceding, however, is not this eventual extravagance, but the fact that all mathematical truths and natural laws have the same truthmaker. Anyone willing to accept this consequence should also offer a detailed explanation of why, although arithmetic seems a very different theory from topology, ultimately the two are about ‘the same’. It is not claimed here that this would be a fatal problem for theological nominalism. However, it is a difficulty that adds to all the previous ones to make the theory unlikely and very theoretically inferior to the postulation of universals.

5.4

Conclusive Summary of Part I

§ 38. With these considerations about theological nominalism, Part I closes. It has been shown that the postulation of universals is theoretically much better than any of its alternatives. We have examined, in particular, three theories that have seemed the most plausible, if one wants to dispense with universals: resemblance nominalism, the postulation of tropes and theological nominalism of concepts in the divine mind. These three theories have been shown deficient for a variety of reasons. In the case of resemblance nominalism and the theories of trope classes, the introduction of ‘primitive relational facts’ is required, which are, in one case, simply relations — which betrays the whole nominalist program— or which generate a vicious regress. Neither of these two conceptions can deliver a reasonable ontology of natural laws in which they have an authentic regulative function. What’s worse, none of these theories can explain how we carry out successful inductions so often, or how we simply have cognitive access to authentic properties —or to whatever works as such. Any of these positions would force us to adopt a rather radical skepticism about the complete natural science as a rational enterprise. Both positions are affected by the problems of the imperfect community and the companionship, and the alternatives they have available to avoid these problems generate additional difficulties. Neither resemblance nominalism nor the theories of classes of tropes can work with a reasonable modal metaphysics. Theological nominalism of concepts in the mind of God evades most of these difficulties, but neither is a reasonable alternative to the postulation of universals. The existing defenses of theological nominalism have not made a careful elucidation of what is a ‘concept in the mind of God’. When one examines what might be such a thing in more detail, however, it appears that theological nominalism would lead either to the identification of all natures in one or to a vicious regress. It turns out, therefore, that neither resemblance nominalism, nor the theories of classes of tropes, nor theological nominalism can replicate the explanatory virtues of the universals in our ontology. The most reasonable thing to do, therefore, is to postulate the existence of universals.

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References Alcinous. (1993). The handbook of platonism. Translated with an introduction and commentary by John Dillon. Oxford: Oxford University Press. Alvarado, J. T. (2011). Dios y el problema de los universales. Aporía. Revista internacional de investigaciones filosóficas, 1, 42–61. Aristotle, De anima (2010). Acerca del alma. Traducción, notas, prólogo e introducción de Marcelo D. Boeri. Buenos Aires: Colihue. Aristotle, Metaphysica (1998). Metafísica. Introducción, traducción y notas de Tomas Calvo Martínez. Madrid: Gredos. Armstrong, D. M. (1978). Universals and scientific realism, Volume I, Nominalism and realism. Cambridge: Cambridge University Press. Bergmann, M., & Brower, J. (2006). A theistic argument against platonism (and in Support of Truthmakers and Divine Simplicity). In D. W. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 2, pp. 357–386). Oxford: Clarendon Press. Craig, W. L. (2012). Nominalism and divine aseity. In J. L. Kvanvig (Ed.), Oxford studies in philosophy of religion (Vol. 4, pp. 43–64). Oxford: Oxford University Press. Edwards, D. (2014). Properties. Cambridge: Polity. Evans, G. (1982). The varieties of reference. Oxford: Clarendon Press. Edited by J. McDowell. Kim, J. (2006). Philosophy of mind. Cambridge, MA: Westview. Leftow, B. (2006). God and the problem of universals. In D. W. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 2, pp. 325–356). Oxford: Clarendon Press. Leftow, B. (2012). God and necessity. Oxford: Oxford University Press. Mann, W. E. (2005). Divine sovereignty and aseity. In W. J. Wainwright (Ed.), The Oxford handbook of philosophy of religion (pp. 35–58). Oxford: Oxford University Press. Plantinga, A. (1980). Does god have a nature? Milwaukee: Marquette University Press. Plato, Timaeus (1982). Timeo. In Diálogos VI. Introducción, traducción y notas de Francisco Lisi. Madrid: Gredos. Plotinus, Enneads (2007). Enéadas. Textos esenciales. Traducción, notas y estudio preliminar de María Isabel Santa Cruz y María Inés Crespo. Buenos Aires: Colihue. Pruss, A. R. (2006). The principle of sufficient reason. A reassessment. Cambridge: Cambridge University Press. Pruss, A. R. (2011). Actuality, possibility, and worlds. London: Continuum. Saint Augustine. De libero arbitrio (2009). El libre albedrío en Obras filosóficas. 2ª Parte. Traducción, introducción y comentarios de Victorino Capánaga. Madrid: Biblioteca de Autores Cristianos. Saint Thomas Aquinas. Summa theologiae (1952). Cura et studio Petri Caramello cum texto ex recensione leonina. Roma: Marietti. Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. Oxford: Oxford University Press. Swinburne, R. (1993). The coherence of theism (2nd ed.). Oxford: Clarendon Press. Vallejos, G. (2008). Conceptos y ciencia cognitiva. Santiago: Bravo y Allende.

Part II

Transcendent Universals

Chapter 6

Transcendent Universals and Modal Metaphysics

Abstract This chapter is the first of the Part Two of the book in which Platonic universals are defended. Here it is argued that only transcendent universals are adequate for an actualist modal metaphysics. This argumentation requires a detailed examination of how transcendent universals work in comparison to immanent universals in each of the main actualist conceptions of possible worlds: possible worlds as maximal structural universals, possible worlds as set-theoretic ‘combinations’ of objects and universals, and possible worlds as ‘complete’ maximally consistent novels. In all cases it results that the supposition that only instantiated universals exist is unable to explain the metaphysical possibilities concerning ‘alien’ properties. § 39. It has been defended that there are universals. These are the entities that best satisfy the theoretical requirements that have been considered for the metaphysics of properties. Universals allow solving the problem of the one over many and the many over one. They allow grounding objective similarities and the systematic relations between determinate and determinable properties. They deliver an ontology for natural laws. They allow us to explain why our inductive practices are epistemologically reliable and allow us to explain why the properties determine the causal interactions in which an object can enter. Many philosophers have been inclined to accept the existence of universals for these reasons and others, but it has seemed to them that the most reasonable alternative is to accept the existence of immanent or ‘Aristotelian’ universals. Probably, the one who has defended this position in the most notorious way has been David M. Armstrong (see 1978a, b, 1989a, 1997, 2010, among others), who throughout more than thirty years of work has had an enormous influence not only in the development of the metaphysics of properties, but, more generally, in the vindication of metaphysics as a philosophical discipline. In this Part II, I will defend the existence of transcendent or ‘Platonic’ universals, pace Armstrong. After the defense of universals that have been made in the previous chapters, the postulation of transcendent universals may seem surprising for several reasons. From the beginning, it has been suggested that what we are dealing with in this © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_6

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work are ‘authentic properties’, whose existence is justified—as a general rule—a posteriori, by empirical research. These are properties that determine causal powers and ground objective similarities. So far it has been usual that those who postulate universals that are ‘authentic’ properties in this sense have postulated immanent universals. In effect, a basic motivation to introduce authentic properties has to do with the pretension of developing an adequate ontology for what natural science seems to show. It is more in line with a general ‘naturalistic’ point of view not to postulate entities that are above or outside the spacetime system. Immanent universals can be seen as nothing ‘above’ or ‘outside’ their instantiations and, therefore, more appropriate to be ‘authentic properties’. Traditionally, too, those who have defended the existence of transcendent universals have done so because of the function that such universals should fulfill in order to constitute the content of our thought and the semantic value of the expressions of our language (see, for example, Bealer 1982, 1993; Chierchia and Turner 1988; Jubien 1989; van Inwagen 2004; Carmichael 2010). “Property” has designated a ‘predicable’, i. e., the semantic value of a predicate, whatever it may be, which must be able to integrate complex contents. Usually, ‘propositions’ have been understood as properties in this sense—‘closed’, 0-adic properties, whose variables have been bounded by quantifiers or completed by the occurrence of, for example, an object if it is a ‘singular proposition’. These properties, with the qualifications of the case, can be the object of propositional attitudes of belief, desire, and so on. Someone could wish for there to be unicorns, which requires the existence of the proposition: there are unicorns, for which the existence of the universal to be a unicorn is required that has no instance. Therefore, it seems reasonable in this perspective the postulation of transcendent universals to fulfill these theoretical functions. Those who have defended the existence of transcendent universals have usually postulated ‘abundant properties’, just as those who have defended the existence of immanent universals have normally postulated ‘sparse properties’. The reasons that have been adduced in Part I are justification for accepting ‘sparse’, not ‘abundant’ properties. These reasons would make one suppose that the properties that are being postulated are immanent universals. Contrary to the expectations that someone could have, however, it is going to be shown here that universals exist independently of having or not instances. Universals are transcendent and not immanent. Despite being transcendent universals, they are universals whose conditions of identity are determined by causal powers conferred to their instantiations. What differentiates the defenders of immanent universals from the defenders of transcendent universals is the attitude towards the principle of Instantiation, that is: [Instantiation]

It is necessary for all n-adic universal U that there are n objects instantiating U.

Those who defend the existence of immanent universals endorse this principle. Those who defend the existence of transcendent universals reject it. It will suffice, therefore, to justify the thesis defended here to show that it is metaphysically

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possible that there is at least one universal possibly not instantiated.1 It is formally consistent with the existence of non-instantiated universals that those universals are contingent, or that some of those universals are contingent and others are not. It is not very plausible, however, to assume any of these hypotheses. There should be some reason why transcendent universals exist in some possible worlds and not in others, or that some of these universals are necessary entities and other contingent entities. But there seems to be no such reason. It is a requirement of economy not to introduce substantive differences in the nature of universals without a positive justification, as already explained (see § 2). The most plausible view, for this reason, is that transcendent universals are all necessary entities that exist invariably in all metaphysically possible worlds. This Part II, in what follows, will have the following structure. In the first place, the reasons for postulating the existence of transcendent universals will be explained (chapters VI to VIII, §§ 41–59). These arguments must be understood, bearing in mind that it has already been justified that universals must be admitted. The question that will be addressed here is what kind of universals should be admitted. There are three central arguments that will be developed for this conclusion: (i) transcendent universals fulfill functions in a reasonable modal metaphysics that cannot be replicated by their alternatives (chapter VI, §§ 41–44); (ii) transcendent universals fulfill functions for a reasonable metaphysics of natural laws that cannot be replicated by their alternatives (chapter VII, §§ 45–52); and (iii) the postulation of immanent universals reverses relations of grounding and ontological dependence (chapter VIII, §§ 53–59). These positive arguments for the acceptance of transcendent universals would be very unpersuasive if the most frequent objections against transcendent universals were not also answered. For centuries these criticisms have been decisive for many philosophers that have quickly dismissed transcendent universals even as a reasonable alternative that deserves to be examined with detention. This work aims, at least, to dispel this superficial impression. The arguments most frequently adduced against transcendent universals come from confusions or assumptions accepted hastily. Three of these traditional objections will be considered, which are not all the objections that have been presented historically, but they are the ones that seem most important and that have had more relevance in the discussion (Chapter IX, §§ 60–74): (i) it has been argued that only those entities that produce a variation in the causal powers of something should be admitted as existing. Transcendent universals would be violating this requirement; (ii) it has been argued that the postulation of transcendent universals is less economical than the postulation of immanent universals since in the first case it is necessary to introduce tropes additionally in the ontology. A transcendent universal, in effect, will only have an impact on the causal interactions by its instances or tropes. A trope has a spatiotemporal location and is a

1 In fact, in higher-order quantificational modal logic, the principle of Instantiation can be formulated as: [□8X□∃x1 . . . ∃xn (x1 . . . xn instantiate X)]. Here ‘X’ is, of course, a higher-order variable that has universals as range. The negation of the principle of Instantiation is [◊∃X◊8x1 . . . 8xn Ø(x1 . . . xn instantiate X)].

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causal relatum. A transcendent universal not. An immanent universal, on the other hand, seems to be located in the states of affairs that it integrates and, therefore, the additional introduction of tropes would not be necessary; (iii) it has been argued that a transcendent universal is a type of entity with which we could not have reliable cognitive contact, unless one is willing to incorporate ‘special’ epistemological capabilities—and suspicious ones from a naturalistic perspective. The introduction of transcendent universals seems to go hand in hand with a dualistic conception of the mind in which there are things that are visible to the ‘eye of the soul’ and not to the ‘bodily’ senses. For many, these associations make the postulation of transcendent universals unreasonable. Each of these objections will be answered. It will be shown that transcendent universals are perfectly coherent with a naturalistic vision of the world—even though I do not share such a global perspective. It is to be argued that universals are the ordinary object of our knowledge or our justified beliefs, both in our ordinary relationship with the world and in the natural sciences (see §§ 71–74). Transcendent universals also confer causal powers on the objects that instantiate them (§§ 61–66). And its introduction does not entail a violation of ontological economy since the introduction of tropes is required whether one admits transcendent universals or that one admits immanent universals (§§ 67–70). In chapter X (§§ 75–77), the question about what are the conditions of identity of a universal, as they have been postulated here, will be considered. It has been usual for the defenders of transcendent universals to maintain that they have epistemic identity conditions.2 Against this usual position, it is going to be argued that the conditions of identity of universals are given by the causal powers that they confer to the objects that possess them. This assumption has been the object of several criticisms. It has been said that a position of this kind leads to a form of ‘causal structuralism’ in which the identity of a property is inherited by its position as the ‘node’ of a structure. It is going to be argued that this should not be seen as a problem.

6.1

Arguments to Accept Transcendent Universals

§ 40. As indicated, three central arguments will be developed to justify the existence of transcendent universals (see Alvarado, 2010a, b, c, e, 2012b). The first has to do with the theoretical functions that universals must satisfy in modal metaphysics; the second has to do with the theoretical functions that universals must satisfy in the ontology of natural laws; and the third has to do with the type of ontological priority

2 The central intuition that has guided many developments has been that what differentiates, for example, the universal of being a three-sided polygon and the universal of being a polygon with three internal angles is the fact that someone could believe that something is a polygon with three sides and not believe that it is a polygon with three internal angles or vice versa.

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relationships that universals must have regarding their instantiations. There are wellknown arguments, however, that are not going to be considered here. The arguments that rest on premises related to questions of philosophy of language or, if you like, related to questions about the ‘content of thought’ will not be discussed. A way of defending the existence of transcendent universals that has been frequent has rested on the hypothesis that the objects of beliefs and other attitudes that a thinker can adopt, as well as the content of linguistic acts of assertion, are ‘propositions’ that are constituted by universal properties together with—eventually—other entities, as has already been indicated above. So that, whether it is true or false, the proposition there is a unicorn requires the existence of the universal to be a unicorn as a constituent of the proposition. A proposition is typically understood as a set-theoretical entity that depends on its elements such as any other set.3 In all possible worlds in which the proposition exists, there must exist the entities on which it is ontologically dependent. In possible worlds where there are no unicorns, the proposition there is a unicorn is false. But how can a proposition be false if it does not exist? And if the proposition exists, its elements must exist, or the elements of its elements must exist. It seems, then, that in the possible worlds in which there are no unicorns, there must be the universal of being a unicorn (see, for example, Carmichael 2010).4 This line of argument crucially depends on the premise that connects the existence of propositions in a possible world with the fact that they are true or false when stating what 3 Thus, the proposition that is enunciated by the sentence “a is a unicorn” is an ordered pair whose elements are the object a and the universal of being a unicorn. The proposition in question is the set {{a}, {a, being a unicorn}}, if Kuratowski is followed for the definition of ordered pairs. The set {{a}, {a, being a unicorn}} depends ontologically on its elements {a} and {a, being a unicorn}. In turn, the set {a, being a unicorn} depends ontologically on its elements a and the universal of being a unicorn. By transitivity, it follows, then, that {{a}, {a, being a unicorn}} depends ontologically on the universal of being a unicorn. 4 And this is not the only argument that has been adduced to justify the existence of ‘abundant’ transcendent universals. It has also been argued that the only adequate regimentation to capture the logical validity of certain derivations requires the assumption that the expressions “that ---”, where “that” is complemented by a complete sentence, are referential, and refer to propositions (see Bealer 1993). Consider the reasoning:

(i) (ii) (iii)

It is true that Cicero denounced Catilina. That Cicero denounced Catilina ¼ that Marco Tulio denounced Catilina. It is true that Marco Tulio denounced Catilina.

This seems to be an instance of a reasoning scheme of the type: (i) Fa, (ii) a ¼ b, then (iii) Fb. Reading the reasoning in this way requires supposing that the expression “that Cicero denounced Catilina” refers to the same thing as the expression “that Marco Tulio denounced Catilina”. George Bealer maintains, moreover, that this is a sophistication of Alonzo Church’s considerations (see Church 1950). More recently, Timothy Williamson has argued that the Barcan Formula and the Converse Barcan Formula in higher-order quantificational modal logic are valid. The validity of these formulas allows him to display a justification of ‘necessitism’, that is, the thesis that everything is necessary. If the higher order variables are interpreted as having universals as range, this will imply the existence of transcendent universals (see Williamson 2013, 221–235, 254–261; Alvarado 2013b, 2017). None of these other arguments will be discussed here, for the same reasons that the argumentation explained above will not be considered.

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happens (or not) in such a world. That is, this line of argument crucially depends on the fact that only a proposition existing in the world w can be true or false of w. Many philosophers, however, have shown that there is a difference between being true a proposition at a possible world w and be a true proposition in a possible world w (see Fine 1985, 191–213). Although the prepositions “at” and “in” are not in themselves very explanatory, there is a substantive distinction that is intended to be expressed with the contrast. It is the distinction between the fact that a proposition p is true in describing what happens in the world w existing as one of its inhabitants and the fact that p simply describes what happens in the world w correctly regardless of whether it exists or not as part of w. In the first case, it is said that p “is true in w”, while in the second, it is said that p “is true at w”. In the second way, it is not necessary that the proposition in question exists as part of the world described to be true or false in describing it. The proposition Fido is a unicorn, for example, is a true or false proposition in the possible worlds in which Fido exists and there is the property of being a unicorn—for in those worlds its constituents exist—but it will be false at the worlds where there is no Fido, or there is no universal being a unicorn. Consider the analogous case of the sentence “no existe el idioma español”. The sentence is true when it describes possible worlds in which Spanish does not exist, but in those worlds, the sentence “no existe el idioma español” does not exist, because it is a sentence in Spanish.5 Some have rejected the intelligibility of the notion of ‘being true at a possible world’ (see Carmichael 2010, 375–383), for reasons that seem to me not very convincing (see Alvarado 2013c, 2017), but this is something that will not be discussed here. This line of argument to justify the existence of transcendent universals depends on the idea that every predicate must have a property as semantic value, just as any belief—or another propositional attitude—must have as its object a property. This work intends to be neutral regarding this assumption. The properties considered here are ‘sparse’ properties whose justification can be done by their causal roles in natural laws. Of course, although this line of argument is not going to be accepted to justify the existence of transcendent universals, it is obvious that there is a philosophical problem here that an ontology of universals must face. In some way, the domain of transcendent universals must be able to offer ‘contents’ for our thought and our language. The semantic content of the expressions of our language, as well as the content of our thoughts, must have to do with existing universals. There is, however, no immediate, one-to-one relationship between universals and ‘propositional contents’. Determining what such a relationship is or what such relationships are is an important theoretical challenge, but it is a challenge that must only be faced once the

5 The same argument that has been used to justify the existence of non-instantiated universals has been used to justify the necessary existence of objects. Socrates should exist in all possible worlds, because in the possible worlds in which he does not exist, the proposition Socrates does not exist is true, and Socrates does not exist ontologically depends on Socrates (see Williamson 2013, 288–300). If there are reasons to resist this argument, there must also be reasons to resist the argument for transcendent universals (see Alvarado 2017).

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existence and nature of universal properties have been justified independently. It is this previous task that is intended to be addressed here.

6.2

The Function of Universals in Modal Metaphysics

§ 41. It has been shown above that resemblance nominalism can only work with a possibilist modal conception (see § 24), in which possible worlds are understood as ‘concrete things’ of the same nature as the universe we inhabit. It has also been shown that the theories of classes of tropes are also not in a good position to form classes of possible tropes if an actualist modal conception is adopted (see § 30). The modal theories that incorporate universals are in much more favorable conditions, but there are important differences if the universals in question are immanent or transcendent. What are going to be considered here are precisely such differences. It is going to be argued that the postulation of immanent universals brings about intuitively unacceptable modal consequences (see Alvarado 2010e). It should be noted, first of all, that the postulation of both immanent and transcendent universals is consistent with modal possibilism. Although the proponents of possibilism have been typically nominalist—as is the case with David Lewis (see Lewis 1986a, 50–63)—one could perfectly well maintain that modal facts are facts about a totality of ‘things’ of the same nature that the actual world that are not related to each other neither spatially nor temporarily, with the proviso that some of the parts of those “big things” are universals. Universals would inhabit all the possible worlds in which they are instantiated or all possible worlds without exception. Possibilists have typically argued that objects cannot exist in more than one possible world, otherwise different possible worlds would be mereologically overlapped and would cease to be spatiotemporarily disconnected from each other. Indeed, if x is at some spatiotemporal distance from y and y is at some spatiotemporal distance from z, then x is at some spatiotemporal distance from z. If there is an object that inhabits two different possible worlds, then it will be possible to connect spatially or temporally all the objects of the two worlds and, according to the Lewisian definition of ‘world’, those worlds would cease to be two different worlds. Universals could inhabit more than one possible world, but they are not objects, so they would not put possibilism at risk. A universal can be located entirely in spatial or temporal regions disconnected from each other—at least an immanent universal— so that objects at a certain distance from the instantiation of a universal are therefore not necessarily at a distance from other objects that are, in turn, at a distance from other instances of the same universal. What is interesting here, however, is how different actualist theories work with universals, since actualist theories are the more reasonable modal theories. As indicated above (§ 24), actualist theories have usually been divided into four types: the combinatorial theory, the theory based on universals, the theory of possible worlds as maximal possible states of affairs or maximal propositions, and the linguistic theories (see Divers 2002, 169–180; Melia 2003, 123–172; Alvarado

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2006b, 2008a, b). The theories of possible worlds as maximal possible states of affairs or as maximal propositions are a form of the modal theory based on universals, so special consideration will not be made of them. The examination will be restricted, therefore, to the combinatorial theory, the theory based on universals and the linguistic theories. Although all of these alternatives are actualist, there are important differences between them. The linguistic theories seem much less reasonable than the others as modal theories because they make the modal ontological space to depend on our languages. It is an extraordinarily anthropocentric theory, therefore, in which fundamental dimensions of reality are determined by the vicissitudes of our cultural history. Given, however, that it is still a position of a certain popularity, it will be treated with attention. Modal metaphysics must offer explanations of the facts about what is possible and what is necessary. According to our pre-theoretical intuitions, although things are in a certain way, they could be different. It is not simply that we can imagine or conceive entities and configurations of entities different from the entities and configurations of actual entities. According to our ordinary intuitions, things— regardless of what we can or cannot believe, conceive or imagine—could be different. Formally, these possibilities can be conceived as a modal ‘space’ in which not only what can happen is given, but also what cannot but happen, because it is invariant in all the ‘positions’ in this space (see Alvarado 2006b, 2007a, c, 2008a, b, 2010b). From an actualist perspective, the actual world is the only concrete world.6 The ontological modal space must be grounded on actual abstract entities and ‘constructions’ of these actual entities. It seems reasonable to suppose that it is possible that there were fewer objects than the ones that actually exist, or that there were more objects than the ones that actually exist. Any theoretically acceptable modal metaphysics should be able to explain on what these facts are grounded.7 However, for what matters here more relevant are possibilities analogous to those 6 With more precision. Let the world be the mereological fusion of everything —understanding “everything” unrestrictedly. That is, the world is: [(xι) 8y (y < x)], the only entity of which everything is an improper part. A mereological fusion, one of whose parts is concrete, is also concrete. The actual world has as its parts concrete and abstract entities —such as universals— so it is a concrete entity. A ‘possible world’ is an abstract entity, which must have as constituents entities that are part of the actual world, which represents how all things could be —understanding “all” unrestrictedly. Therefore, there is exactly one ‘possible world’ that represents how things are actually. It can be called the “actual world” if you like, but it generates an ambiguity because the expression “actual world” is also used to designate the only concrete world. In what follows, “actual world” will designate the concrete world and “possible world” will designate an abstract construction. 7 Not only should it be possible that there are objects that do not actually exist. Modal variations in such merely possible objects should also be possible in a reasonable modal metaphysics. If it is possible that, for example, a cat is on the roof, it should also be possible that this cat is not on the roof. Formally, in quantified modal logic with the operator ‘actually’: [◊∃x (A8y (x 6¼ y) ^ Fx ^ ◊ØFx)]. It is necessary, then, to ‘trace’ the same merely possible object in different worlds, since it is the same object that is possibly F and is possibly not F. Accommodating an actualist modal metaphysics to these demands brings with it very serious problems that will be dealt with only tangentially here (see Alvarado 2006a, 2007b, 2008c, 2009, 2010b, d, 2016).

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indicated, but related to ‘alien properties’. An alien property (see Lewis 1983, 37) is a property not instantiated in the actual world, and that cannot be ‘constructed’ from properties instantiated in the actual world. Then, any reasonable modal metaphysics should explain that, or at least should not be incoherent with: [Alien properties]

It is possible that some property that is not actually instantiated is instantiated.8

The ‘properties’ in question are, of course, universal. This principle of Alien properties seems independently credible. It will be useful to consider a couple of examples to see the likelihood of the principle of Alien properties, even though it is debatable that the cases that are going to be considered are really of ‘alien’ properties, that is, of properties not instantiated and that cannot be ‘constructed from’ properties actually instantiated. There is some mass that the mereological fusion of all existing objects must have. It must be a mass equal to the sum of all the masses of all objects in the actual world. Let this mass be of n gr. It seems prima facie metaphysically possible that there is something with a mass of n + 1 gr, but such mass is not instantiated. There also seems to be an indenumerably infinite continuum of colors in the chromatic space. It is very doubtful, however, that every one of those infinite different shades in the chromatic space is instantiated in some actual surface. It is even more doubtful that all determinate colors are instantiated in some or other surface if it is considered that brightness and saturation are also relevant for the chromatic space. There also seems to be a property of being a perfect cube. It is very doubtful, however, that there is actually a perfect cube. Everything that we have seen as a cube has been close to a cube, but not a cube. Our intuition is that there could be objects with a mass of n + 1 gr, and surfaces with each of the indenumerable infinite colors of the chromatic space, and perfect cubes. How are any of these hypotheses, however, metaphysically possible? Of course, our ordinary intuitions are not untouchable. It could be rational to leave them aside, if they had unacceptable consequences not initially foreseen, or if there were large theoretical advantages that would be obtained only at the cost of rejecting them. The point is that whoever holds that none of these hypotheses is metaphysically possible must give a detailed explanation about why they are not possible, against our ordinary intuitions. The argument that is going to be developed here coincides with what was raised by Matthew Tugby (2015; also, Ingram 2015), who states that there is an alien paradox, that is, a paradox that has to do with the requirement to find truthmakers for the metaphysical possibility of non-instantiated properties. Unlike Tugby, however, a detailed examination will be made here of how this problem could be addressed in each of the three most reasonable actualist theories—or families of actualist theories. The conclusion will be negative: if one postulates only immanent universals, alien properties are not metaphysically possible. We will now consider, therefore, how it

More precisely, in higher-order quantified modal logic: [◊∃X◊∃x ((x instantiates X) ^ A8y Ø(y instantiates X))]. As in other cases, the variable ‘X’ has universal as range. The operator ‘actually’ (A) makes that what is within its scope should be evaluated in the actual world. 8

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is that each of the three actualist theories works concerning the principle of Alien properties.

6.3

Modal Theories Based on Universals

§ 42. Modal theories based on universals hold that possible worlds are maximal structural universals (see Forrest 1986; Bigelow and Pargetter 1990, 165–213; Alvarado 2006b, 2007a, 2010b, 2011). A structural universal is a universal that specifies how an object is by the specification of: (i) what its parts are, (ii) what intrinsic properties such parts possess, and (iii) what relationships exist between those parts (see Armstrong 1978b, 67–71, 1997, 31–38; Bigelow and Pargetter 1990, 82–92; Alvarado 2011). For example, the property of being a water molecule is the property of having something three parts, one of which is an oxygen atom, two of which are hydrogen atoms, and there is a chemical bonding relationship between each of the hydrogen atoms with the oxygen atom. The central idea of this theory is that possible worlds are understood in the same way, as great structural universals that specify: (i) what would be the intrinsic nature of each of the parts of the world, and (ii) what relations exist between these parts. Of course, the intrinsic nature of a part can be given by another structural universal. A possible world is a maximal structural universal because it makes the specification of how everything would be— understood “everything” unrestrictedly. The idea of a structural universal has been called into question since it would not be a form of a mereological composition nor of a set-theoretic structure (see, in particular, Lewis 1986b; but, Alvarado 2011). It is obvious, however, that not every form of composition must be either mereological or set-theoretic. It is a form of composition by operations on properties that yield other properties. The treatment of these operations on properties has been done frequently in the developments of property logics (see, for example, Swoyer 1998). For example, the structural universal UE is the property that instantiates everything that satisfies the following predicate: λx∃y1 ∃y2 . . . ∃yn ððy1 < xÞ ^ ðy2 < xÞ ^ . . . ^ ðyn < xÞ ^ P1 y1 ^ P2 y2 ^ . . . ^ Pn yn ^ Ry1 y2 . . . yn Þ

That is, the predicate of “being an x such that y1, y2, . . ., yn are improper parts of x, y1 is P1, y2 is P2, . . ., yn is Pn and y1, y2, . . ., yn are with each other in the relation R”. It must be assumed here that P1, P2, . . ., Pn and R are predicates that have authentic properties as semantic value. According to the usual ways of specifying a complex property, the operations of ‘conjunction’, ‘reflection’ and ‘existentialization’ would be required on the properties that are the semantic value of P1, P2, . . ., Pn and R (see Swoyer 1998, 302–303). Suppose that the semantic values of P1, P2, . . ., Pn and R are respectively the universals U1, U2, . . ., Un and UR. It is assumed that U1, U2, . . ., Un are monadic and that UR is an n-adic relation. The operation of ‘conjunction’—hereinafter “Conj”—on properties is the operation of

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generating an n + m-adic property from an n-adic property and an m-adic property. That is, if Ui is n-adic and Uj is m-adic, then Conj (Ui, Uj) is n + m-adic. For UE it is necessary to generate a 4n-adic property resulting from applying 2n + 1 times the Conj operation on the relation of being improper part of, U1, U2, . . ., Un and UR first. Then, the operation of ‘reflection’—hereinafter “Ref”—is the operation of ‘merging’ variables. If there is a binary relation Sx1x2, what results from Refx1, x2 (Sx1x2) is the reflexive relation Sx1x1. The Ref operation should be applied repeatedly over what Conj has thrown to unify the variables appropriately. Finally, these variables are bounded by quantifiers through the operation of ‘existentialization’ (“Existsx”).9 There is an immediate reason why immanent universals should be disqualified to construct possible worlds according to this theory. In effect, a possible world—with the sole exception of the maximal structural universal actually instantiated—is a non-instantiated universal. The modal ontological space is constructed by an infinite plurality of maximal structural universals. If the universals are limited to those that are actually instantiated, this modal space disappears. Only the actual world would be metaphysically possible. Perhaps a conception of this kind would be acceptable to a Spinozist or a Hegelian, but it is not for anyone who is not inclined to think that everything that happens, happens necessarily. The modal theory based on universals, then, seems from the beginning a theory that can only work with transcendent universals. This conclusion, however, might be precipitated. Advocates of immanent universals have usually argued that a conjunctive property—for example, the property of being F and being G—only exists if it is instantiated, that is, if there is something that is F and is G (see Armstrong 1978b, 30–36). A structural universal must include a conjunctive universal so, according to the same principle, it would only exist if something instantiates it. Perhaps, however, one could relax this requirement, granting that the existence of complex universals is admitted, provided that there are instantiated ‘simple’ universals that constitute them, according to the standards of the defender of immanent universals. Thus, if the universals U1 and U2 are instantiated, although nothing instantiates at the same time U1 and U2, the universal of being U1 and being U2 exists. The same instantiated universals could have a different instantiation distribution. Each of these different instantiation distributions could be reflected in a maximal structural universal. For example, suppose that there is actually something that instantiates universals U1 and U2, and something that

A construction with ‘negation’ and ‘universalization’ could also be used. In effect, the property of there being a Ui should be constructed as Neg (Univx (Neg (Uix))) where Neg (Uix) is the property of not being Ui and Univx (Uix) is the property of all being Ui. It is better to use the operation Existsx because it allows one to evade the difficult issues associated with the negation operation Neg. The remaining operations are: (1) Plugx (Ux, a) is the operation by which the particular object a is ‘plugged-in’ to the variable x in universal U. What results from the Plug operation is a not purely qualitative property; and (2) Convx1, x2 (Ux1, x2, . . ., xn) ¼ Ux2, x1, . . ., xn. It is the ‘conversion’ operation that inverts the order of the variables x1 and x2. With the operations Existx and Plugx propositions can be generated from universals. 9

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instantiates U3 and U4. According to what the defenders of immanent universals have traditionally held, there would also be the conjunctive universals of being U1 and U2, and of being U3 and U4 and—if the remaining required operations of properties on properties other than the conjunction are granted—the structural universal of, for example, having something with a part that is U1 and U2, and another part that is U3 and U4, etcetera. There are several additional disadvantages for an Aristotelian to admit this intermediate position. As indicated, to constitute a structural universal, it is necessary to admit operations of conjunction, reflection and existentialization.10 The operation of ‘reflection’ has not been subject to explicit consideration, but it should also generate problems. Armstrong rejects the existence of reflexive relationships, since—prima facie—they would not be authentic relationships (see Armstrong 1978b, 91–93). If one admits a reflection operation, however, one should admit these relations, since [Refx1, x2 (x1 loves x2)] ¼ (x1 loves x1)]. Admitting loving as an authentic relationship and admitting the operation of reflection involves accepting the reflective relationship of loving oneself. If one wants to maintain the prohibition of reflexive relations in ontology, the operation of reflection cannot be accepted, but without reflection, there are no structural universals. There are several reasons for rejecting the theory of possible worlds as maximal structural universals constructed with instantiated ‘basic’ universals. Consider that it would be compatible with what we know that all universals were structural or complex (see Armstrong 1978b, 32–36, 1997, 31–32). In a scenario in which all universals are structural, there would be no “basic” universals to be instantiated. If all universals were complex and complex universals could exist without being instantiated in something, then all universals could exist without being instantiated. Admitting this hypothesis would be for the defender of immanent universals to simply abandon his position. The defender of immanent universals should, therefore, accept the existence of complex universes not instantiated only with the additional condition that there are ‘simple’ or ‘basic’ universals not constructed by other universals. But it would not be prudent to admit such a thing. There are neither empirical nor purely philosophical reasons for sustaining the actual existence of ‘basic’ universals. A second reason for rejecting maximal structural universals constructed from instantiated ‘basic’ universals is that in such a position there would be no way to accommodate intuitions about Alien properties. It would only be metaphysically

If one appeals to operations of ‘universalization’ and ‘negation’ there is an additional problem. Advocates of immanent universal have generally accepted the existence of conjunctive universals with the precautions listed above, but have not accepted the existence of disjunctive and negative universals (see Armstrong 1978b, 19–29, 1989a, 82–84). The defenders of immanent universals have rejected the existence of negative universals because they would generate spurious similarities between objects. The fact that two objects share the same property grounds an objective similarity between them. However, it seems unacceptable to admit that a cat and a quark are similar because they both instantiate the property of not being a galaxy. Something similar has been maintained for disjunctive properties. It does not seem reasonable to think that there is an objective similarity between cats and quarks because they both have the property of being a cat or being a quark. 10

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possible the instantiation of a different distribution of universals already instantiated, but no other. This restriction seems, of course, arbitrary. It would result that if a universal has not been actually instantiated, then it would be metaphysically impossible for it to exist. The defender of immanent universals could here reply that for cases like the property of having mass, masses actually instantiated could be sufficient to represent the instantiation of any other.11 For example, let n be the mass of the mereological fusion of everything. The property of having a mass of n + 1 gr is not instantiated, but the property of having 1 gr of mass is also instantiated. Then, the situation in which something instates the property of having n + 1 gr of mass could be represented as the instantiation of the structural property of having a part that has a mass of n gr and having another part that has a mass of 1 gr. For the case of the property of having a form, arbitrary parts of any object having exactly any of the infinite shapes could be selected. Thus, even if a stone does not have a perfectly spherical shape, there would be a part of that stone with a perfectly spherical shape. Something similar could be maintained, perhaps, concerning color, if it were a property reducible to others that could be treated as mass or form. These explanations would allow us to accommodate these cases, although they are not enough to construct alien properties that cannot be ‘constructed’ from properties actually instantiated. Nevertheless, even for these restricted cases, there are serious problems for a strategy of this type. Indeed, arbitrary parts for any object would be required. In any object with a spatial extension, there should be a part that has exactly a cube shape, for example. But these are assumptions for which we have no independent evidence. For all we know, the mereological structure of physical objects could include mereological atoms or—even if there were no atoms—there could be physical parts not constituted by arbitrary parts. For all we know, for example, a proton is made up of quarks, not ‘quark parts’. There are also no known parts of an electron. We have no reason, then, to think that forms like those of a cube are instantiated. We have no reason to think, either, that all the colors of the chromatic space are structural properties constructed by instantiated properties. We do not even know that any of the infinite determinate masses—whose magnitudes have always been assumed to be isomorphic to the real numbers—can be treated as a structural property constituted by masses of smaller magnitudes actually instantiated. To have masses of arbitrary quantities, arbitrary parts are required, and there is no guarantee that there are arbitrary parts, as explained above.12 Then, the problem remains: if

11

The defender of Aristotelian universals could also appeal to a conception of functional laws as relationships of direct necessitation between determinable universals. This alternative will be discussed at length later (see § 47). It would not be acceptable, in the end, for an Aristotelian. 12 For example, suppose the mass of n + i gr in which n gr is the mass of the mereological fusion of all physical objects and i is a very small quantity, smaller than the mass of the particle with the smallest mass according to the Standard Model of Particle Physics. It is not at all clear that there is something that actually has a mass of i gr, so it is not at all clear that the property of having a mass of i gr is instantiated. Then, it is not at all clear that the property of having a mass of n + i gr can be considered as the structural property of having a part that has a mass of n gr and having another part that has a mass of i gr.

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only instantiated universals are admitted, the metaphysical modal space would be restricted in a way that seems arbitrary. Another reason for rejecting maximal structural universals constructed only from instantiated ‘basic’ universals is that it would have theoretically inconvenient consequences if some forms of metaphysics of time were accepted. Suppose that everything that exists is present that is, that there is nothing past or future. This philosophical position has been called “presentism” in the literature (see Crisp 2003). Every moment of time, there are universals that cease to be instantiated and others that become instantiated. Since the modal ontological space is grounded on which universals are instantiated, then the extension of such a space of objective possibilities will vary in time. Suppose that at time t1 it is metaphysically possible that there is an F because something instantiates the universal F, if in time t2 later nothing instantiates the universal F, then being F will turn to be impossible. In the same way, it could happen that at time t1 it is not metaphysically possible for something to be F, since nothing instantiates the universal of being F, and at time t2 later it is metaphysically possible because something has come to instantiate the universal of being F. Not only would a modal ontological space suffer very strange ‘systoles’ and ‘diastoles’ in time. It would be especially strange if something happened that was metaphysically impossible at the immediate previous moment of time. If in t1 it is metaphysically impossible for something to have a certain determinate tone of color C—since nothing instantiates color C—, how is it that in t2 there can be something with the color C? If having the color C was really impossible, how is it that something can then have that color? Perhaps it should be sustained at this point that a fact can be actual at a time t only if that fact was metaphysically possible at times before t. The consequence that this would have, however, is that new universals could not ‘appear’ in time. They could only ‘disappear’. The course of history would have to be a process of progressive narrowing of reality because it would be a process of losing universals that were instantiated and that are no longer—so they will not be instantiated in the future either. The reality would be destined to a sort of ‘metaphysical entropy’. Many defenders of immanent universals are free of these problems because they defend an eternalist ontology concerning time (see Armstrong 2004, 145–150) in which past, present and future exist equally, without ontological privileges for the present. In these views, a universal exists in a possible world if and only if it exists at some time in that world. It does not matter if—in relation to the present—the universal was instantiated, although it is not instantiated, or if it is instantiated, but it will not be, or if it will be instantiated, although it was not before. It is clear, however, that only adopting very substantive ontological commitments in metaphysics of time the postulation of immanent universals would be relatively plausible. The theory is not neutral in metaphysics of time, and this is a theoretical cost that must be weighed. More serious than the above are analogous problems, but of modal nature. It is only metaphysically possible with respect to the world w1 a possible world w2 if each of the universals instantiated in w2 is also instantiated in w1. In effect, the possible world w2 is a maximal structural universal that must have as constituents only universals existing in w1 which implies that such universals must be instantiated in

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w1. From the perspective of the defender of immanent universals, there are only instantiated universals and no other. At least, this should be true for ‘basic’ universals if one is going to admit structural universals freely constructed from such universals. Suppose now that there is a universal U that is instantiated in w1, but is not instantiated in w2. The maximal structural universal that is w2 can be ‘constructed’ with the universals existing in w1. The point is that w1 cannot be ‘constructed’ inversely with the universals existing in w2, because U does not exist in w2. Thus, although w2 is accessible from w1, w1 is not accessible from w2. Therefore, accessibility relationships between possible worlds would not be symmetric, if one adopts immanent universals together with a theory of possible worlds as maximal structural universals. This non-symmetry brings counter-intuitive consequences. For example, it seems reasonable to think that, if the fact that p obtains, then it should not be possible that p is impossible. Also, it seems reasonable to think that, if it is possibly necessary p, then it should not be the case that the fact that no-p. The assumption that accessibility relationships between possible worlds are not symmetric, however, implies that we should admit that something is the case, even if it is possibly impossible, or that something possibly necessary is not the case.13 Although what may be called “possibly necessary” or “possibly impossible” may be farther from our ordinary understanding than what “necessary” or “impossible” expressions seem to designate, there are certain intuitions that can be developed regarding their content when what is at issue is metaphysical modality. A state of affairs s is metaphysically possible if and only if there is a way in which everything might be in which s obtains. The existence of this possibility has nothing to do with what one can imagine or conceive, nor with what can be justified by empirical or a priori means. The metaphysical possibilities have to do with how natural dynamisms can develop with complete independence that we are (or not) aware of such dynamisms (see, for this approach, Alvarado 2009). Some processes are, by their intrinsic nature, stochastic. Whatever the background of these processes, there is an open space of different ways in which they can develop. If one also posits the existence of free agents, whose freedom is incompatible with determinism, then the extension of metaphysical possibilities would also be grounded on the space of open alternatives for the freedom of these agents. It is very difficult to understand how

13

It is characteristic of modal systems in which the accessibility relations are symmetrical the modal principles [p ! □◊p] and [◊□p ! p]. Consider, for example, the principle [p ! □◊p]. [□p] is true in possible world w if and only if in all possible worlds accessible from w is true p. [◊p] is true in a possible world w if and only if in some possible world accessible from w is true p. For [p ! □◊p] to be false, it would take a world w1 in which p is true, but [Ø□◊p]. [Ø□◊p] is equivalent to [◊□Øp]. And [◊□Øp] would be true in w1 if and only if there were at least one possible world, be w2, accessible from w1 where [□Øp]. For [□Øp] to be true in w2 it is required that in all worlds accessible from w2 be true Øp. If the accessibility relationships are symmetric, we must necessarily find w1 among the worlds accessible from w2, because —by hypothesis— w2 is accessible from w1. But in w1 it is true that p. Then, it is not true [□Øp] in w2. Then, if the accessibility relationships between possible worlds are symmetric, there would be no counter-examples to [p ! □◊p]. The assumption that relations of accessibility are not symmetrical brings, therefore, that scenarios in which [p ^ ◊□Øp] and scenarios in which [◊□p ^ Øp] should be admitted.

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accessibility relationships might not be symmetrical from this perspective. Indeed, if from the perspective of the world w1 there are ways in which natural processes or free decisions of agents could develop, let them be possible world w2, then w2 should also include those same processes and alternatives open to free agents that could lead to what happens in w1. The world w1 should also be accessible with respect to w2. Consider this example: in the world w1 a die is thrown, and a one comes out, and it is possible with respect to w1 that a six had come out instead of a one when throwing this same die. Let w2 be a possible world in which the die is thrown, and a six comes out. It seems obvious that, now from the perspective of w2, it is possible that there was a one instead of a six, which is what happens in w1. If one adopts an epistemic conception of the space of metaphysical possibilities, on the other hand, it is perfectly reasonable to admit breaks in the symmetry of accessibility relations. For example, one could argue that the state of affairs s is possible if and only if one can ‘conceive coherently’ that s obtains (see Chalmers 2002; Alvarado 2007c). Be that there is a scenario w1 with rational thinkers that have developed computers with great capacity of information processing using which they can calculate arithmetic operations of high complexity in seconds. From the perspective of this scenario, given the conceptual resources available to rational thinkers there, it is conceivable that there are no computers and that rational thinkers have very limited capacities to perform basic arithmetic operations. Be this scenario w2. From the perspective of w2, however, it may not be coherently conceivable that there are computers that can solve complex arithmetic problems in seconds, simply because in that scenario one does not even have the concept of ‘computer’. The symmetry in the accessibility relationships is characteristic of metaphysical modality. Asymmetry is characteristic of epistemic modality. If the accessibility relationships are symmetric—as they should be, if metaphysical possibilities are involved, independent of our ability to conceive or justify them—then it cannot be that something obtains and is possibly impossible, or that something that is not the case and is possibly necessary. If the state of affairs s is possibly necessary from the perspective of w1 then there is a possible world w2 accessible from the perspective of w1 in which the state of affairs s is necessary. But if s is necessary from the perspective of w2 then it cannot be that s does not obtain in w1, since w1 is one of the worlds accessible from the perspective of w2. It happens, moreover, that in a scheme such as the one indicated, a sort of ‘metaphysical entropy’ occurs as in the case of time. The farther away a possible world is in the chain of accessibility relationships, the fewer universals can be instantiated in that world. The difference between possible worlds is given, in effect, by having a different distribution of actual universals and by having less instantiated universals than those instantiated in the actual world. Suppose a world w1 accessible from wA—the actual world—in which there are fewer instantiated universals than those that are actually instantiated. Suppose then a world w2 accessible from w1 in which there are fewer universals instantiated than those that are instantiated in w1. The world w2 will be accessible from wA because it cannot have instantiated universals that are not also actually instantiated. The issue is that the ‘distance’ from the actual world is linked to a smaller diversity of universals instantiated. If a

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metric is established between possible worlds given by the accessibility relationships between these worlds, it turns out that the more ‘distant’ a world is from the actual world, the fewer universals will be instantiated in those worlds. The ontological space of the metaphysical modality then seems to be arranged in such a way that the possible worlds become more and more ‘uniform’ up to the limit of totally uniform or maximally uniform worlds that will have to be the most distant from the actual world. This peculiarity highlights the arbitrary nature of the restrictions imposed by the requirement that only instantiated universals should be admitted. What results is that it is only metaphysically possible an ontological ‘impoverishment’ with respect to the actual world. Reality could only be ‘poorer’, more ‘uniform’, less heterogeneous than what it is actually. It seems obvious, however, that not only could the reality be more ‘uniform’, with less instantiated universals, but that it could also be more ‘heterogeneous’, with more universals instantiated. After all this examination, it appears, then, that a conception of metaphysical possibilities in which there are maximal structural universals, but where only ‘basic’ universals are admitted would be theoretically inconvenient. Not only would our intuitions about Alien properties be incoherent with this conception, but it would require very controversial assumptions about the existence of ‘basic’ universals and would bring about unacceptable consequences in the metaphysics of time and for the construction of the complete modal ontological space. It should also be remembered that the existence of maximal structural universals if one admits only instantiated universals, requires several hypotheses that have not been traditionally accepted by the friends of Aristotelian universals. To construct structural universals, operations on universals are required that have been seen with suspicion by Aristotelians, such as reflection. If the postulation of immanent universals were ultimately incompatible with the conception of possible worlds as maximal structural universals, this would be a serious reason to reject Aristotelian universals. The costs of its acceptance would be too high.

6.4

Combinatorial Modal Theories

§ 43. The combinatorial modal theories postulate that possible worlds are ‘combinations’ of actually existing objects and properties (see Cresswell 1972; Skyrms 1981; Bigelow 1988; Armstrong 1989b, among others; Divers 2002, 174–177). In an ontology of universals and particular objects, a ‘possible state of affairs’ can be represented as an n-tuple that has as elements an n-adic universal, n particular objects and a time. I will ignore the time in what follows to simplify the discussion. The possible state of affairs of x1, x2, . . ., xn instantiating the universal U is represented by the n-tuple , which is a set-theoretic construction.14 For the

14

The ordered pair is defined as {{x}, {x, y}} according to Kuratowski, as indicated above. An n-tuple of the elements z1, z2, . . ., zn can be defined as an ordered pair whose elements are z1 and

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existence of possible states of affairs, then, only the existence of its constituents and the axioms of set theory are required. Although there are some philosophers who have understood by a ‘state of affairs’ an abstract entity (see Chisholm 1976, 114–137, 1989, 141–155; Plantinga 1974, 44–46), as explained above (see § 24), it will be designated here by “state of affairs” a concrete entity constructed, for example, from objects and properties and consisting in the fact that such objects—or object—effectively instantiate such property (see Armstrong 1978a, 113–116, 1997, 113–138; Alvarado 2012a, 2013a). There are only actual states of affairs. A ‘possible state of affairs’, therefore, must be the representation of a state of affairs that, although it does not actually exist, could exist. A set-theoretic construction is that with which such representation can be made. A possible world is a ‘maximal’ set of possible states of affairs,15 because it is the representation of all the states of affairs that would exist in the contemplated possibility—understanding “all” unrestrictedly.16 In a combinatorial conception, the modal ontological space is grounded on: (i) the particular objects and universals actually existing, and (ii) the mutual independence of these objects and universals. Actually existing objects are instantiating certain universals, but they could not instantiate them and instantiate others, instead. Actually existing universals are being instantiated by certain objects, but they may not be instantiated by those objects, but by others instead. Combinatorics of objects and universals are grounded on the fact that objects and universals do not mutually depend on each other,17 so they can occur jointly or separately. Of course, combinations will be restricted by the essences of the respective objects —if one admits essential properties for those objects that they could not but instantiate A defender of immanent universals will argue that only instantiated universals ground the modal ontological space (see, especially, Armstrong 1989b, 37–86). It is perfectly coherent also to postulate a combinatorial theory of modality with transcendent universals. It

the n-1-tuple of z2, . . ., zn. The n-1-tuple, in turn, can be defined as the ordered pair of z2 and the n-2tuple z3, . . ., zn, and so on. 15 This formulation has not been uniform. Cresswell treats possible worlds as sets of ‘particular basic situations’ (see Cresswell 1972, 6–7), where ‘situations’ are states of affairs as they have been characterized. Skyrms treats possible worlds as ‘collections of facts’ (see Skyrms 1981, 199–200), and a fact is an n-tuple of objects and a property. Armstrong characterizes possible worlds as ‘conjunctions of atomic states of affairs’ (see Armstrong 1989b, 45–49). It seems more reasonable to treat possible states of affairs and possible worlds as set-theoretic entities to capture what is common in all these proposals. A ‘conjunction’, on the other hand, is an operation defined only over propositions or sentences. 16 If one admits ‘negative states of affairs’, a possible world could be defined as a maximally consistent set of possible states of affairs. That is, it would be the set of possible states of affairs such that for all possible state of affairs s, either the possible world has as element s, or it has as element not-s. If s ¼ then not-s ¼ According to the way of defining a possible world indicated above, a possible world should be understood as the set of all the positive states of affairs that would obtain if that possible world were actual. 17 In abstract, the basic elements of the combinations could be mutually independent tropes. Recall, however, that a combinatorial theory with tropes would have serious problems (see § 30).

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will be shown that only in this second way can the combinatorial theory function in a plausible way. A first difficulty that has the combinatorial conception with immanent universals is that it does not seem to be able to accommodate our intuitions about Alien properties. As has been indicated, it seems obvious that properties that are not actually instantiated could be instantiated: there could be objects with masses not instantiated in anything; there could be objects of perfectly spherical shape, although nothing is in fact perfectly spherical; there could be surfaces with a certain shade of green that does not, in fact, have any surface. In a combinatorial theory with immanent universals, it is not just that in fact such universals are not instantiated. Its existence is metaphysically impossible. When this same problem has been considered for the case of the theories of possible worlds as maximal structural universals, the Aristotelian has been able to appeal to the resource of postulating structural universals formed by ‘basic’ universals actually instantiated. That is, although actually nothing has a mass of n + 1 gr —where n gr is the mass of the mereological fusion of everything— there are objects with a mass of n gr and objects with a mass of 1 gr, respectively. Then the structural universal of having two parts, one of which has a mass of n gr and the other one has a mass of 1 gr can be built. There were specific difficulties for this strategy, as has been explained since arbitrary parts for an object that are instantiating any mass or any form cannot be assumed. In this case, there is an additional difficulty. It is not clear how structural properties should be treated from the perspective of a combinatorial theory. It has been explained above that the operations on universals required to generate a structural universal have usually been viewed with suspicion by the Aristotelians. It is also required that structural non-instantiated universals are admitted —although constructed from “basic” universals that are instantiated. For the case of the modal conception based on universals, it is necessary from the outset to accept structural universals because otherwise, the position would not make any sense. The combinatorial theory, on the other hand, is in itself neutral regarding these issues.18 It is instructive to consider how Armstrong has sought to resolve this issue over time. In the work A Combinatorial Theory of Possibility (Armstrong 1989b) he argued that alien properties are ‘conceivable’ but not metaphysically possible (see 18

And a structural universal cannot be treated as a combinatorics of its constituent universals. For example, consider a molecule of water. From a combinatorial point of view, one could think of the possible states of affairs of: (i) being a hydrogen atom x, (ii) being a different hydrogen atom y, (iii) being an oxygen atom z, (iv) being a chemical bond between x and z, and (v) being a chemical bond between y and z. These possible states of affairs, however, do not ground the existence of a water molecule—and, with it, the existence of the structural property of being a water molecule. Without the additional postulation of a mereological sum of x, y, and z, and without the specification of a property that has to be instantiated by such a mereological sum different from the constituent properties, there is no water molecule. The specification of such property is precisely the construction of a structural property that requires, as already indicated, operations on universals. Here, too, one cannot think of the structural universal as a ‘type of state of affairs’ (Armstrong 1997, 34–38), since the whole whose elements are (i)–(v) is not a state of affairs and, even if it were, it is not the object that instantiates such property.

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1989b, 54–57). The restrictions in the modal ontological space are simply accepted as a necessary cost to maintain the general position according to which only instantiated universals exist. In A World of States of Affairs (1997), however, Armstrong offers a different treatment in which two ‘spheres’ of possibility are distinguished. Certain possibilities belong to an ‘internal’ sphere and others, on the other hand, belong to an ‘external’, ‘more distant’ sphere in relation to the actual world. The ‘internal’ sphere of possibilities is configured in the standard way in the combinatorial conception. These possibilities are grounded on existing objects and universals and their mutual independence. The ‘external’ sphere, however, includes possibilities concerning alien properties, as well as merely possible objects (see Armstrong 1997, 165–167). The central idea of Armstrong is to suppose that these should not be seen as, for example, the possibility that a specific universal is instantiated in a specific object, but as the possibility that some or other universal is instantiated in an object. The same treatment is proposed for possibilities involving merely possible objects.19 The possibilities of the ‘external’ sphere seem to be constituted by existential quantifications or something that takes its place. A state of affairs in the ‘internal’ sphere is simply the n-tuple where U is a universal actually instantiated and x1, x2, . . ., xn are actual objects. For the ‘external’ sphere, however, one should select a ‘representative entity’ that fulfills the functions of a universal without being one —be it X— to integrate an n-tuple that represents the possibility that X is instantiated by x1, x2, . . ., xn. There is an obvious problem, however, with this alternative. Whatever X is, it is not universal. If one adopts a combinatorial conception of the modality, one is arguing that the possibilities are grounded on the mutual independence of the entities that actually exist. There are no possibilities generated by nonexistent entities because there are no such. In this case, there is no universal X. There is also no metaphysical possibility to represent with respect to such universal.20 Perhaps one could maintain that, although a universal X does not exist, yet it is possible that such universal exists. But for a philosopher who coherently defends a combinatorial theory of metaphysical modality, the space of possibilities arises from combinatorics of independent entities. If the authentic metaphysical possibility of the existence of a universal X is to be admitted, this possibility must be grounded on the combinatorics of objects and universals actually instantiated. Which recombination would result in the putative universal X? If it is not a structural universal constructed from operations on other universals, it is not possible to see how such a possibility could be generated. Armstrong again modified his position in Truth and Truthmakers (2004). He maintains, first of all, that alien properties are metaphysically possible, but the 19 Armstrong follows here Skyrms (1981), who deals with the possibilities of merely possible objects and properties not actually instantiated by ‘analogy’. Those possibilities concern ‘something’ that fulfills the same roles that actual objects and properties fulfill. 20 Armstrong, on the other hand, seems to perceive these difficulties. He points out that: “There are reasons to think that the solution is unsatisfactory. Some violence is done to combinatorial intuitions because the existence of an alien [property] is not understood in terms of the recombination of existing items”.(1997, 167).

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truthmaker of the proposition it is possible that there are properties that do not actually exist is a fact of ‘totality’ concerning the actually existing properties (see Armstrong 2004, 86–89). Armstrong argues that necessary propositions —and impossible ones, for the same reasons— do not have truthmakers. When it comes to contingent propositions, on the other hand, its truthmaker, whatever it may be, is the same as the truthmaker of the statement of the possibility of the negation of the proposition declared possible (see Armstrong 2004, 83–86).21 Suppose that e is the truthmaker of the proposition: It is possible that there is a rabbit. Then, e must also be the truthmaker of: It is possible that there is no rabbit. If this is so, then, the truthmaker of a proposition such as: It is possible that there is a universal different from all universals actually existing must be the same truthmaker of the proposition: It is possible that there is no universal different from all actually existing universals. It happens here that the proposition under the scope of the possibility operator is: There is no universal different from all actually existing universals. This is a proposition that establishes a general fact. If U1, U2, . . ., Un are all actually existing universals, what is being stated is that for all universal X: X ¼ U1 or X ¼ U2 or . . . or X ¼ Un.22 Armstrong has a theory about the truthmakers of propositions that state generality (see Armstrong 2004, 68–76). A proposition of the form All F is G has as truthmakers the facts that x1 is F and G, x2 is F and G, . . ., xn is F and G, and the fact that x1, x2, . . ., xn are all the Fs. For the case of interest here, the truthmaker of the proposition: All universal is actually identical to some universal should be the plurality of facts that U1 is actually identical to some universal, U2 is actually identical to some universal, . . ., Un is actually identical to some universal, and the fact that U1, U2, . . ., Un are all actually existing universals. As it is contingent that there could be more universals than actually exist, then, the proposition: It is possible that there are universals that do not actually exist must have the same truthmaker that: There are no universals that do not actually exist. And the truthmaker of: There are no universals that do not actually exist is the plurality of facts already indicated. There are serious problems with this proposal, however. For many cases, the idea that the truthmaker of possibly p is the same as the truthmaker of possibly not p seems plausible from the perspective of a combinatorial theory of modality, but it is not acceptable as a general principle. Suppose the truthmaker for it is possible that x is F. It is reasonable to think that the same entities that are its truthmaker —i.e., the object x, the property F, and their mutual independence— are also the truthmaker of

21

The argument for this thesis is as follows. Let p be a contingent proposition. Let e be the truthmaker of p. If p is contingent, then it is possible also that Øp. Then by definition of contingency, p implies [◊p ^ ◊Øp]. But, if e is the truthmaker of a proposition q, and [□(q ! r)], then e is a truthmaker of r. Since the truthmaker of [◊p ^ ◊Øp] is also the truthmaker of [◊Øp], e truthmakes [◊Øp]. 22 That is, the proposition There is no universal different from all actually existing universals can be formulated as [Ø∃XA8Y (X 6¼ Y )]. Here the variables ‘X’ and ‘Y’ have universals as range, and ‘A’ is the modal operator ‘actually’. If U1, U2, . . ., Un are all actually instantiated universals, it can also be formulated as [Ø∃X ((X 6¼ U1) ^ (X 6¼ U2) ^ . . . ^ (X 6¼ Un))]. [Ø∃XA8Y (X 6¼ Y )] is equivalent to [8XA∃Y (X ¼ Y )].

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it is possible that x is not F. In effect, the possibility that x is not F is grounded on the same mutual independence of x and F. The situation is very different, however, in the case of the truthmaker of a proposition such as: It is possible that there are universals that do not actually exist. Whatever may be adduced as a truthmaker of this proposition, it cannot simply be the possibility of recombination of universals and existing objects. In particular, the truthmaker of the proposition: There are no universals that do not actually exist, cannot serve as a truthmaker of: It is possible that there are universals that do not actually exist. The universals that actually exist —from the Aristotelian perspective— do not ground the existence of other universals. Moreover, the mutual independence of the entities that serves to ground modality in the combinatorial perspective makes it impossible for the totality of universals actually instantiated to ground the possible existence of universals not actually instantiated. These are different entities so that the existence of one of them does not make the existence of others necessary. The existence of actual universals does not ground the possible existence of other different universals. All the strategies have failed. This returns us to the initial problem with respect to Alien properties. If a universal is not instantiated, it is not metaphysically possible that it is instantiated. To this difficulty, it is added that, as for the conceptions of maximal structural universals with Aristotelian universals, it turns out that the relations of accessibility between possible worlds are not symmetrical. Indeed, if the universal U is actually instantiated in the actual world wA, then, it can be represented from the perspective of wA a possible world in which U is not instantiated, be w1. If one adopts the perspective of w1, however, it is not possible to represent the possibility realized in wA, since there is no U in w1, so there could not be n-tuples that include U as an element. Thus, although w1 is accessible from wA, wA is not accessible from w1. For the reasons indicated in the previous section, this implies that scenarios should be admitted in which something is the case, although it is possibly impossible and cases in which something is not the case, although it is possibly necessary. The reasons already explained to argue that accessibility relationships should be symmetric —if they concern metaphysical modality— are also applicable here. Similarly, in a combinatorial conception of modality with Aristotelian universals, a sort of ‘metaphysical entropy’, such as the one explained in the previous section, would result. As possible worlds are ‘moving away’ from the actual world through chains of accessibility, they are becoming increasingly ‘uniform’ with less instantiated universals. In a conception such as this, only worlds with less heterogeneity are metaphysically possible, with fewer universals instantiated. Only greater ‘ontological poverty’ is possible. Our intuitions, however, point to the fact that not only ‘ontological poverty’ is possible but also less ‘ontological poverty’, with more universal instantiated. These same considerations would lead to very unlikely results if one accepted a presentist ontology of time. They will not be repeated here again. It turns out, then, that the combinatorial conceptions of modality with Aristotelian universals are not in better conditions than the conceptions of the possible worlds as maximal structural universals with the same assumption. A reasonable actualist modal theory seems to require the postulation of transcendent universals.

6.5 Linguistic Theories of Modality

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§ 44. It remains to consider the situation of Aristotelian universals in modal linguistic theories. In these conceptions, possible worlds are ‘maximally consistent stories’ in some language, as already explained (see §§ 24, 30). Let L be a language in which infinitely meaningful sentences can be constructed. A possible world w is a set of sentences of L such that, for every sentence s of L, either s belongs to w, or not-s belongs to w. By construction, w is free of contradiction and is also a ‘complete’ description of how things would be if all the sentences of w were true, according to the expressive resources of the language L. Theories of this type have been quite popular despite their obvious weaknesses (see Divers 2002, 178–180). Natural languages —and any artificial language created for special purposes— are human artifacts. Although one cannot assign a single creator for each language, they are the result of the deliberate action of different people in time and space. Languages have a history and could have developed in very different ways from the way they have. As has been pointed out previously, creatures with different forms of perception and different interests would have developed —perhaps— also very different languages to talk about the features of reality that would have resulted interesting for them. It seems very arbitrary that the extension of the modal ontological space is fixed by the features of some language or another. If there were no expressions to designate electrons, for example, that is, if the term “electron” did not exist, then there would be no possibilities in which electrons were involved. If we had never had a term like “galaxy”, then the modal space would lack possibilities relative to galaxies. All this is absurd. Moreover, whatever terms a natural language possesses, a denumerably infinite number of different sentences can be generated at most, but some possibilities involve —or seem to involve— a continuum of spacetime points. Each one of those indenumerably infinite points could be occupied or empty. Then, no natural language could generate enough sentences to express all these possibilities (see Lewis 1973, 90, 1986a, 142–144). One resource for resolving these difficulties has been to propose what has been called a “Lagadonian language” (see Lewis 1986a, 145). The idea is that each property and each object are names of themselves. Actually existing objects and properties can form sets —which is guaranteed by the axioms of set theory. The sentences of the Lagadonian language can be identified with n-tuples of objects and properties that are being ‘attributed’ of them. Possible worlds are maximally consistent sets of sentences of this ‘language’. It is, however, difficult to admit that a ‘Lagadonian’ language really counts as a language. It would be a language that would not be learned even by angels —no angel could learn indenumerably infinite different names— so it could not fulfill communicative functions of any kind. However, even accepting a ‘language’ like this there are serious problems for a linguistic theory of modality if only immanent, and not transcendent universals are accepted, so it is to be assumed in what follows that the language is ‘Lagadonian’. In a Lagadonian language there are, by definition, terms to designate all actually existing universals. If one admits only immanent universals, there are only terms to

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designate instantiated universals. How would it be metaphysically possible that there were universals actually not instantiated? A possible world is a maximally consistent story that completely describes everything that would happen if all the sentences in the story were true. For it to be metaphysically possible that there are universals that do not actually exist, it would be required, therefore, that at least one of those stories contains a sentence “--- U ---” in which “U” is the expression that designates a property that does not actually exist. But there are no expressions to designate alien properties, for there are no such properties in a conception that admits only immanent universals. There are no universals not actually instantiated that may be names of themselves. A similar problem arises about merely possible individuals. Therefore, it would not be metaphysically possible that there were instantiated universals that are not actually instantiated. Various ways of solving the problem have been proposed, both concerning alien properties and merely possible individuals (see Roy 1995; Melia 2001; Sider 2002; but, Alvarado 2008c, 2010d). A first way of solving it is to suppose that alien properties could be included in the sentences in question, but in a ‘descriptive’ way. Analogously an individual can be designated by a proper name that is his ‘rigid designator’ but can also be designated by a definite description. One can designate Pablo Neruda, for example, using the proper name “Pablo Neruda”, but also using the description “the author of Veinte poemas de amor y una canción desesperada”. Similarly, one might think that the possibilities involving alien properties are grounded on sentences that describe such properties. Although no property can be designated by a name, we can speak of “the universal that is in the relation R with the universals U1, U2, . . ., Un”. A procedure of this style is one that has been used when the ‘Ramsey sentence’ of a theory has been postulated (see Lewis 1970).23 This proposal has been criticized, first, because it assumes that a universal has to fulfill the same theoretical role in all possible worlds in which it exists (see Lewis 1986a, 161–165), but many have argued that the causal powers that a property confers to its bearers are contingent. The property that in the world w1 is in the relationship R with U1, U2, . . ., Un is perhaps not the same as that which fulfills the same role in another world w2. But this criticism can be left aside, as there are many reasons to think otherwise, as will be argued below (see § 51). A second, much more serious criticism is that the semantic value of a term can be fixed by this procedure only if there is a semantic value already defined for the remaining terms that appear in the description. A description in which many terms lack semantic value will be inept at fixing their meaning implicitly. But there is no reason to think that all alien universals will have sufficient relations with instantiated universals so that one can appeal to this proce-

Let a theory be a set of sentences in which ‘theoretical’ terms and ‘non-theoretical’ terms appear. A proposed way to implicitly define the meaning of a theoretical term is by the procedure of substituting that term in all its occurrences in the conjunction of the sentences of the theory by a higher-order variable. The meaning of the term is whatever it is that exactly satisfies the open sentence that results from such substitution.

23

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dure of implicit definition reasonably. We do not know which relations have transcendent, non-instantiated universals among themselves. It has also been argued that, whatever the language L that is used as a basis for the construction of the maximally consistent sets of sentences, this could be supplemented with additional names corresponding to the properties that could exist, although they do not exist. The possibilities that involve such properties would be based on sentences containing these new expressions (see Roy 1995, 225–233). There is an obvious problem with Roy’s proposal, and that is that there is no reason to attribute the character of ‘names’ to the new expressions that have been introduced. A name, be proper or common, acquires a semantic value by its connection with really existing entities to which it is referring. For the case of proper names, as is widely known, this connection is given by a chain of transmission of uses that goes back to the moment of ‘initial baptism’. Something analogous has been assumed for common names that should designate properties. Thus, Roy’s ‘names’ do not seem to be names. If there were transcendent universals one, perhaps, could postulate some mechanism of access to them that makes possible an ‘initial baptism’ and then chains of transmission of use that try to preserve the same reference. But if there are no transcendent universals, there is nothing of this kind that can be proposed. There is no way to access what does not exist because there is nothing to access. According to the linguistic theory, a maximally consistent and complete story grounds metaphysical possibility because the names and terms that make up the story have a semantic value. The sentences constructed by such names and terms have therefore truth conditions, and there is something determined, too, that would be the case if such sentences were true. But, if ‘sentences’ have names or terms without semantic value, there will be no truth conditions for such sentences, and there will be nothing determinate that would be the case if those sentences were true. It has also been argued that the possibility of universals that do not actually exist could be based on ‘pixels’ (see Melia 2001, 2003, 166–168). A ‘pixel’ is not a name, but something that performs functions analogous to those of shades of gray in a black and white image of something colored. A catalog, for example, may contain only black and white photographs of the paintings in an exhibition. There is a multitude of shades that are being omitted in this representation, but the catalog has informative value to the extent that exactly the same shade of gray is correlated with the same color throughout the catalog. Thus, although one cannot know if a certain pixel of gray represents a yellow or light blue tone, one can know that that tone in a certain painting is the same as in another painting. ‘Pixels’ in this type of linguistic theory of modality are not shades of color, but some kind of linguistic item —something that can be put into composition with names and predicates to form sentences— that represents a property that would have something of what it is predicated. Although you cannot know which property is representing the pixel, you can know that it is the same property that is used in other sentences. Thus, maximally consistent and complete stories that include such pixels have informative value about the possibilities represented. But it is difficult to see what advantages Melia’s proposal has over Roy’s previous proposal. Even if a ‘pixel’ is presented as

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something that is not a name, it is supposedly something with certain semantic value if it is going to be able to concur to form sentences endowed with truth conditions. They have to function as names, they occupy the position of names, but they are not supposed to be. What semantic value do these ‘pixels’ have? It does not help to say that they are designating universals that, although they do not actually exist they could exist, because there is no such metaphysical possibility in the first place. It is assumed that pixels ground such possibilities. It is, on the other hand, very counterintuitive to suppose that the introduction or not of ‘pixels’ grounds facts about what is metaphysically possible. After all, the introduction of ‘pixels’ into a language is something we can do arbitrarily. It seems that we could expand or contract the modal space at our will. Another proposal to solve the problem of the possibility of alien properties has appealed what has been called a “pluriverse” (see Sider 2002; Alvarado 2008c, 2010d). The ‘pluriverse’ is the full extension of all possible worlds. In the linguistic perspective of modality, each possible world is a maximally consistent story, as it has been explained. There are no names to designate properties not actually instantiated that can ground possibilities concerning such properties. It has already been indicated that it would not be good to use the Ramsey procedure. First, there is no guarantee that non-instantiated universals can be defined implicitly by their systematic connections with other universals. Secondly, there is no way to specify whether the universal that is instantiated in a possible world is identical or different from some universal in another possible world. Assume that one assigns higher-order variables to take the place of the —eventual— alien properties for which no names exist. The maximally consistent story that is a possible world will be a set of open sentences in which higher-order variables X1, X2, . . ., Xn occur. Take the conjunction of all these sentences and bound the variables with existential quantifications. Each possible world would be represented by a sentence of this form: (1)

∃X1 ∃X2 . . . ∃Xn [(‐‐‐X1‐‐‐) ^ (‐‐‐X2‐‐‐) ^ . . . ^ (‐‐‐Xn‐‐‐) ^ ((X1 6¼ U1) ^ (X1 6¼ U2) ^ . . . ^ (X1 6¼ Un)) ^ ((X2 6¼ U1) ^ (X2 6¼ U2) ^ . . . ^ (X2 6¼ Un)) ^ . . . ^ ((Xn 6¼ U1) ^ (Xn 6¼ U2) ^ . . . ^ (Xn 6¼ Un))]

This world-sentence (1) should be infinitary to contain, eventually, infinite conjunctions. The universals U1, U2, . . ., Un are all universals actually instantiated. It seemed at first glance that all that would be needed to ground the metaphysical possibility that there are actually non-instantiated universals are sentences like (1) for every possible world. The problem is that the metaphysical possibilities that must be grounded include facts about whether, for example, the same universal that could be instantiated in x1 and not instantiated in x2, could be instantiated in x2 and not in x1. Naturally, in the same world it cannot be that the universal is and is not instantiated in an object. What is at issue is the distribution of instantiations of the same universal in different possible worlds. For this it is indispensable to determine facts about the identity of universals in these worlds. Such a thing would be achieved if one had names to rigidly designate such universals in the worlds in which they are instantiated, but such names are not possessed, as has already been explained several times. Theodore Sider’s proposal consists of replacing a plurality of sentences with

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the form of (1) by a single infinitary sentence preceded by a sequence of existential quantifications. When you have several different sentences of the form of (1) there is no way to determine if, for example, the universal designated by the bound variable X1 of one of them is identical or not to the universal designated by the bound variable X2 of another. A sentence of pluriverse, however, would have the following form: (2)

∃X1 ∃X2 . . . ∃Xn [((X1 6¼ U1) ^ (X1 6¼ U2) ^ . . . ^ (X1 6¼ Un)) ^ ((X2 6¼ U1) ^ (X2 6¼ U2) ^ . . . ^ (X2 6¼ Un)) ^ . . . ^ ((Xn 6¼ U1) ^ (Xn 6¼ U2) ^ . . . ^ (Xn 6¼ Un)) ^ (((‐‐‐X1‐‐‐) ^ (‐‐‐X2‐‐‐) ^ . . . ^ (‐‐‐Xn‐‐‐)) _ ((‐‐‐X2‐‐‐) ^ (‐‐‐X1‐‐‐) ^ . . . ^ (‐‐‐Xn‐‐‐)) _ . . . _ ((‐‐‐Xn‐‐‐) ^ (‐‐‐Xn-1‐‐‐) ^ . . . ^ (‐‐‐X1‐‐‐)))]

The length of (2) may be confusing -it is an infinitary sentence scheme. The central aspects that should be considered concerning (2) is that it is an existential quantification of the higher-order variables X1, X2, . . ., Xn that have universals as range. Each of these universals X1, X2, . . ., Xn is different from all universals actually instantiated U1, U2, . . ., Un. This is what is stated in the clause [((X1 6¼ U1) ^ (X1 6¼ U2) ^ . . . ^ (X1 6¼ Un)) ^ ((X2 6¼ U1) ^ (X2 6¼ U2) ^ . . . ^ (X2 6¼ Un)) ^ . . . ^ ((Xn 6¼ U1) ^ (Xn 6¼ U2) ^ . . . ^ (Xn 6¼ Un))]. What follows is a disjunction that is constituted by clauses of the form [(‐‐‐X1‐‐‐) ^ (‐‐‐X2‐‐‐) ^ . . . ^ (‐‐‐Xn‐‐‐)] each of which is a complete description of how things would be. These clauses are those that fulfill the role of being ‘maximally consistent stories’ —the possible worlds in the traditional linguistic theories of modality. The difference is that all these ‘maximally consistent stories’ are within the scope of the initial existential quantifiers. It is then perfectly determined whether the universal designated by, for example, the variable X1 that appears in one of the disjunctive clauses is identical or not to the universal designated by the variable X1 that appears in another disjunctive clause. If they are bound by the same existential quantifier, they are the same universal. If they are not, they are not identical. The problem here is that sentence (2) is supposedly true since one of the disjunctive clauses included must be true. The pluriverse sentence grounds the complete modal ontological space in which only one of the possibilities represented by the disjunctive clauses is realized. The sentences of form (1) are, however, all of them false with the sole exception of a true sentence that describes exactly how things are actually. This contrast brings with it a very serious problem, because if the pluriverse sentence is true then there must exist not instantiated universals X1, X2, . . ., Xn, otherwise, it would not be true that there are universals X1, X2, . . ., Xn such that . . . etc. What are such universals, however? It is assumed that there are only universals actually instantiated and no other. The sentence (2) cannot be true, therefore, because it precisely states that there are universals not actually instantiated. If the sentence is false, however, why should we consider it as representing the modal ontological space? It seems, in summary, that modal linguistic theories do not leave space for the metaphysical possibility of alien properties, if it is assumed that there are only instanced universals. This concludes the revision of the different actualist theories of the modality. In all cases, it has been found that the restrictions imposed by the Aristotelian bring

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about an arbitrary impoverishment of the modal ontological space. If it is possible for universals that are not instantiated to be instantiated, then there are reasons to reject such restrictions. It is reasonable, then, to postulate transcendent universals whose existence is independent of having or not instantiations. Transcendent universals are those that fit our intuitions about the extension of metaphysical possibilities in any reasonable modal theory.

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Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Armstrong, D. M. (2004). Truth and truthmakers. Cambridge: Cambridge University Press. Armstrong, D. M. (2010). Sketch for a systematic metaphysics. Oxford: Clarendon Press. Bealer, G. (1982). Quality and concept. Oxford: Clarendon Press. Bealer, G. (1993). Universals. The Journal of Philosophy, 90, 5–32. Bigelow, J. (1988). Real possibilities. Philosophical Studies, 53, 37–64. Bigelow, J., & Pargetter, R. (1990). Science and necessity. Cambridge: Cambridge University Press. Carmichael, C. (2010). Universals. Philosophical Studies, 150, 373–389. Chalmers, D. J. (2002). Does conceivability entail possibility? In T. S. Gendler & J. Hawthorne (Eds.), Conceivability and possibility (pp. 145–200). Oxford: Clarendon Press. Chierchia, G., & Turner, R. (1988). Semantics and property theory. Linguistics and Philosophy, 11, 261–302. Chisholm, R. M. (1976). Person and object. A metaphysical study. Chicago: Open Court. Chisholm, R. M. (1989). On metaphysics. Minneapolis: University of Minnesota Press. Church, A. (1950). On Carnap’s analysis of statements of assertion and belief. Analysis, 10, 97–99. Cresswell, M. J. (1972). The world is everything that is the case. Australasian Journal of Philosophy, 50, 1–13. Crisp, T. M. (2003). Presentism. In M. J. Loux & D. W. Zimmerman (Eds.), The Oxford handbook of metaphysics (pp. 211–245). Oxford: Oxford University Press. Divers, J. (2002). Possible worlds. London: Routledge. Fine, K. (1985). Plantinga on the Reduction of Possibilist Discourse. In J. E. Tomberlin & P. van Inwagen (Eds.), Alvin Plantinga (pp. 145–186). Dordrecht: Reidel. Reprinted in Kit Fine, Modality and Tense. Philosophical Papers. Oxford: Clarendon Press, 2005, pp. 176–213. Forrest, P. (1986). Ways worlds could be. Australasian Journal of Philosophy, 64(1), 15–24. Ingram, D. (2015). Platonism, alienation, and negativity. Erkenntnis, published online 14 December 2015. https://doi.org/10.1007/s10670-015-9794-2. Jubien, M. (1989). On properties and property theory. In G. Chierchia, B. H. Partee, & R. Turner (Eds.), Properties, types, and meaning, Volume I: Foundational issues (pp. 159–175). Dordrecht: Kluwer. Lewis, D. K. (1970). How to define theoretical terms. The Journal of Philosophy, 67, 427–446. Reprinted in David Lewis, Philosophical Papers. Volume I. New York: Oxford University Press, 1983, 78–95. Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Lewis, D. K. (1986a). On the plurality of worlds. Oxford: Blackwell. Lewis, D. K. (1986b). Against structural universals. Australasian Journal of Philosophy, 64, 25–46. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 78–107. Melia, J. (2001). Reducing possibilities to language. Analysis, 61, 19–29. Melia, J. (2003). Modality. Chesham: Acumen. Plantinga, A. (1974). The nature of necessity. Oxford: Clarendon Press. Roy, T. (1995). In defense of linguistic Ersatzism. Philosophical Studies, 80, 217–242. Sider, T. (2002). The Ersatz Pluriverse. The Journal of Philosophy, 99, 279–315. Skyrms, B. (1981). Tractarian nominalism. Philosophical Studies, 40, 199–206. Swoyer, C. (1998). Complex predicates and logics for properties and relations. Journal of Philosophical Logic, 27, 295–325. Tugby, M. (2015). The alien paradox. Analysis, 75, 28–37. Van Inwagen, P. (2004). A theory of properties. In D. W. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 1, pp. 107–138). Oxford: Clarendon Press. Williamson, T. (2013). Modal logic as metaphysics. Oxford: Oxford University Press.

Chapter 7

Transcendent Universals and Natural Laws

Abstract One of the theoretical functions attributed to universals is to work as natural laws – or, at least, to be something on which natural laws are dependent. It is argued in this chapter that only transcendent universals can satisfy this theoretical function. Transcendent universals have been postulated as constituents of natural laws since 1977 by Michael Tooley. The cases presented by Tooley are discussed and generalized. Functional laws are not of use for Aristotelians to explain those cases. On the contrary, functional laws are an additional reason to postulate Platonic universals. Finally, it is considered whether there is a unique nomic structure invariant through all possible worlds. § 45. It has been explained above (see §§ 14 and 21) that the most reasonable conception of natural laws ground them on universals. Some theories have postulated that natural laws are ‘nomological’ or ‘necessitation’ relationships (see Armstrong 1983; Dretske 1977; Tooley 1977, 1987, 37–169). The nomological or necessitation relationship is a higher-order relation between universals. Natural laws are in these views states of affairs of higher-order, therefore. Other philosophers—within the same views critical of Humean ontologies—have argued that natural laws should not be seen as ‘relationships’ between universals. Every universal confers, by itself, by its very intrinsic nature, causal powers to its possessors. The deployment of such powers is what we ordinarily identify with the operation of a natural law. Natural laws are simply universals with essential causal powers (see, among others, Molnar 2003; with preventions, Mumford 2004, 65–123; Bird 2007, 43–65). Although the second position will be defended in this work (see § 51), it will be assumed that natural laws are universals or are grounded on universals. Natural laws consisting of two universals will be considered to be in a nomological relationship of necessitation (N ) to facilitate the discussion, but all considerations can apply to the second alternative, mutatis mutandis. If U1 and U2 are universals, a natural law is the state of affairs [N(U1, U2)]. Since the conception of natural laws as higher-order relations among universals was proposed, it was postulated that there should be transcendent universals. This thesis that was defended by Michael Tooley (see 1977; 1987, 113–120; also, Swoyer © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_7

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1982) was, however, immediately resisted by David Armstrong (see 1983, 117–127). In this chapter, in the first place, the reasons that led Tooley to defend non-instantiated universals (§ 46) will be examined in detail, and the strategies that have been proposed by the Aristotelians to neutralize these motives will be considered (§§ 47–48). Secondly, the presentation of a generalization of the Tooley-type cases will be made, which will make more credible the thesis that the most reasonable form of metaphysics of natural laws requires the postulation of transcendent universals (§ 49). Third, another line of argument is going to be examined for the same conclusion proposed years later by Simon Bostock (2003), which is perfectly acceptable (§§ 50–52). The argumentation that is going to be presented here also has a closeness to that proposed by Matthew Tugby (2013) on the nature of the dispositions.

7.1

Tooley-Type Cases

§ 46. As indicated, from the first proposals of theories of non-Humean natural laws, some scenarios were proposed, one of which is this: Imagine a world containing ten different types of fundamental particles. Suppose further that the behavior of particles in interactions depends upon the types of the interacting particles. Considering only interactions involving two particles, there are 55 possibilities with respect to the types of the two particles. Suppose that 54 of these possible interactions have been carefully studied, with the result that 54 laws have been discovered, one for each case, which are not interrelated in any way. Suppose finally that the world is sufficiently deterministic that, given how particles of types X and Y are currently distributed, it is impossible for them ever to interact at any time, past, present or future. In such a situation it would seem very reasonable to believe that there is some underived law dealing with the interaction of particles of type X and Y. (Tooley 1977, 669; see 1987, 47-48).

In the scenario proposed by Michael Tooley, one could not admit a natural law concerning the interactions between particles of type X and particles of type Y—from the regularist perspective—since there are no regularities of events involving these particles together. It can only be accepted as a natural law that every F is followed by a G in the possible world w if every event of having an F in w is followed by the event of having a G in that world. But here there are no events in which a particle of type X interacts with a particle of type Y.1 In such a scenario, however, it would seem

In fact, one could admit regularities in a similar scenario, but it would be ‘empty’ regularities. Indeed, it would be true in this scenario a universally quantified conditional as follows: [8x8y (((x is a particle of type X) ^ (y is a particle of type Y ) ^ Rxy) ! Sxy)], in which R is an interaction and S is anything. This quantified conditional is trivially true because its antecedent is false for any objects x and y. But this is no advantage for the defender of the regularist theories since it is very counterintuitive to admit natural laws for empty regularities. It would be a natural law, for example, that every diplomatic pig feeds on caviar, for the simple reason that there are no diplomatic pigs. The existence of these vacuous regularities is one of the arguments that have been adduced against the regularist views (see Armstrong 1983, 19–23).

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reasonable to assume that there must be a natural law, even if there are no regularities of events. Then, there would be justification to maintain that laws are not mere regularities of events. The consideration of this scenario is not yet a reason to maintain that natural laws are higher-order nomological relations between universals. An examination of the various ways in which the nature of a law could be understood from non-regularist perspectives has to be done. It is not necessary to enter now in such a comparative examination. The point is that, if there are reasons to postulate natural laws as higherorder nomological relations between universals due to scenarios such as that described by Tooley, then there are also reasons to postulate transcendent universals. Indeed, the interaction of particles of type X with particles of type Y is the instantiation of a relational universal or another, be it UR. But what is stated is that nothing instantiates UR. Then, if there were a natural law of the type [N(UR, US)]—where US is the relational universal that must be instantiated because particles of type X and type Y instantiate UR—then there must also exist the universals on which such a law depends ontologically. Then there must be, at least, the non-instantiated universal U R. This scenario presented by Tooley considers a case in which the ‘first’ universal of a natural law was not instantiated—which corresponds to a case in which the antecedent of the respective conditional was false if it were a universally quantified conditional stating a regularity. There are also cases in which the ‘first’ universal connected by a natural law would be—by hypothesis—instantiated, but it might not (see Tooley 1977, 685, 1987, 49–50). Suppose there were psycho-physical laws that correlate physical states with mental states. These laws are not derivable from other laws that have a more basic character. One can assume a regularist perspective to understand the nature of such laws. It would be an actual regularity of events according to which every physical event of type F is followed by a mental event of type M. For example, certain sets of synaptic events in an area of the brain are followed by the phenomenal appearance of something red to the subject in whose brain such synaptic events occur. But things could be different from how they actually are. The Earth, for example, could have been farther or closer to the Sun, so that the proper conditions for the appearance of conscious life would never have occurred either on Earth or in any other region of the universe. Let this possibility be possible world w1—by hypothesis—accessible from the actual world wA. It seems reasonable to assume that in w1 it is also a natural law that, if something is F, then there will be an M. It is also reasonable to maintain that if something were F in w1, it would also be M. But there is no regularity by which every F is followed by an M in w1 –there is a counterfactual fact in w1 about what would happen with an F, but nothing is F in w1. There would be a natural law without any regularity. Natural laws, therefore, do not seem to be regularities of events. Again, if there is a reason to accept that natural laws are higher-order nomological relations between universals, then there are also reasons to accept the non-instantiated universals F and M. Armstrong, who also defends the same kind of ontology of natural laws as relations of necessitation between universals, does not want to accept non-instantiated universals. He has offered, therefore, several strategies to neutralize

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the Tooley scenarios that will be examined next (see Armstrong 1983, 111–127, 1997, 242–248).

7.2

Functional Laws

§ 47. A first strategy for dealing with cases of laws with non-instantiated universals is to subsume such cases—or most of them—under ‘functional laws’. In the standard formulation, a natural law is the fact that two—or more—universals are in a nomological or necessitation relationship. When formulating natural laws, however, what is typically done is to enunciate equations between physical quantities. It is said, for example, that the force of gravity obeys an equation whereby [F ¼ Gm1m2/ d2]. In this equation what is stated is that the magnitude of the force of attraction between two objects, respectively, with the masses m1 and m2 and located at a distance d, is directly proportional to the product of the masses—weighted by the gravitational constant G—and inversely proportional to the square of the distance d between those objects. The expression of the law does not state anything about the masses and distances of two or more objects being ‘nomologically’ connected with a force of attraction between those objects, or ‘make necessary’ the existence of a force of attraction between those objects, although what is supposed is that precisely the fact that there is such a nomological relation is what grounds the truth of the equation [F ¼ Gm1 m2/d2]. For what interests here, moreover, neither “F” nor “m1” nor “m2” nor “d” designate properties, universal or not. It is assumed that the infinite determinate properties of mass and the infinite determinate relations of distance are isomorphic to the set of real numbers to obtain a value from the equation for the variable F. What enters the equation [F ¼ Gm1m2/d2] is not the properties in question but real numbers that ‘represent’ them. It happens, then, that the equation [F ¼ Gm1m2/d2], with all these preventions, is not the expression of one natural law, but indenumerably infinite natural laws corresponding to the indenumerably infinite properties of force, mass, and distance. Therefore, when it is argued that there is a law of universal gravitation, it is being argued that there is a nomological or necessitation relation between each of these indenumerably infinite forces, masses, and distances. Such infinite natural laws are also that which ground infinite counterfactual conditionals that seem true. It is evident, however, that the indenumerably infinite universal determinations of force, mass, and distance required for the existence of such natural laws are not all instantiated. Throughout the history of humankind, only a finite number of these determinate properties of force, mass, and distance have been observed. However vast the universe, there is also a finite number of those magnitudes realized. At least, there will be infinite of such unrealized magnitudes: all infinite masses greater than the largest mass that is instantiated. If natural laws seem to exist—which is indispensable if the counterfactual conditionals grounded on such laws are true—then their constituents must exist. Then, there must exist non-instantiated universals.

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Being things like that, almost any natural law under the standard form in which they are normally enunciated would be a reason to postulate transcendent universals. Armstrong believes, however, that these cases can be solved satisfactorily for an Aristotelian with the postulation of “functional laws” (see Armstrong 1983, 111–116). If, for example, it is a law that every F is a G, where “F” and “G” have as range determinate universals under the same determinable, then a functional law would be that which would establish a regularity of this general form: (3)

For all x, for all F: if Fx, then there is a G such that: G ¼ f(F) and Gx.2

Here in (3) the variable ‘x’ has objects as range, variables ‘F’ and ‘G’ have determinate universals as range and ‘f’ is a function that maps each property under a certain determinable F to a property under a determinable G. It is assumed that the function f is what is stated in the equations normally used to express a law. What would be the nature of a functional law, then? Armstrong postulates that it must be a state of affairs of ‘second-order’ that grounds the nomological or necessitation relations of ‘first-order’ between universals in which the ‘usual’ natural laws consist. It would be a natural law of natural laws: It is a law [second-order] concerning P-type properties, that, if a particular [first-order] has one of these properties, then it is a law [first-order] both that this particular has a Q-type property, and that a certain relation [the function: Q ¼ f(P)] holds between this P-type property and this Q-type property. (Armstrong 1983, 113).

Just as a natural law of the form [N(U1, U2)] grounds the regularity by which everything that is U1 is also U2, here it would be that a law of higher-order grounds laws of first-order. Armstrong’s idea is that a higher-order law of this kind could ground counterfactual laws that, although they do not actually exist, could exist. These same merely possible laws could then ground the correlative counterfactual conditionals. Thus, even if nothing instantiates a given property U1, if such a universal were instantiated, then there would be a natural law [N(U1, U2)]. Since there would be such a natural law in these counterfactual circumstances, then if something were instantiating the universal U1, it would instantiate the universal U2. It seems, then, that neither the actual existence of the universals U1 and U2 nor the existence of the natural law [N(U1, U2)] are required since an appropriate functional law could ground their metaphysical possibility. For a functional law to fulfill these theoretical functions, it could not be based on natural laws of ‘first-order’ between determinate properties. Indeed, a very simple way of understanding the nature of a functional law by which every universal of type F is nomologically related to a universal of type G is as the fact that [N (F1, G1)], [N (F2, G2)], . . ., [N(Fn, Gn)] and the fact that F1, F2, . . ., Fn are all the determinate universals under the determinable universal F. Understood in this way, the functional law is grounded on each of these infinite natural laws of first-order and is More precisely: [8x8F (Fx ! ∃G ((G ¼ f(F)) ^ Gx))]. Higher-order quantification is required here on the variables ‘F’ and ‘G’ that have as range determinate universals, that are supposedly falling under the corresponding determinables. 2

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ontologically dependent on each of the determinate universals F1, F2, . . ., Fn and G1, G2, . . ., Gn. The functional law could not exist without the existence of these universals, which brings back the initial problem. It would not be useful to appeal to functional laws to solve the question of natural laws that seem to include non-instantiated universals, because such laws would depend on the previous existence of these universals. The idea of a functional law, then, must be that of a nomological or necessitation relation but between determinable properties and not directly between the determinate properties under those determinables. The nomological relation between determinable properties then grounds nomological relations between the determinate properties. The idea of a functional law in these terms seems plausible, on the other hand, if one considers what is normally stated by a natural law. When it comes, for example, to the law of universal gravitation, we supposed to have justified a general nomological relationship between masses, forces, and distances. This law seems to be a relationship directly between the determinable properties and not between the determinate properties. Otherwise, no one would have justifiably supposed that the empirical evidence collected about finite determinate masses, distances and forces could be sufficient to justify a law that—by its standard formulation—involves all determinate masses, distances, and forces. In other words, if the functional laws were based on the ‘first-order’ laws between determinate universals, the evidence we possess to postulate the existence of the functional law would be completely insufficient. One difficulty with this conception of functional laws, however, for the particular case of Armstrong, is that he does not accept the existence of determinable properties (see Armstrong 1978, 117–120, 1997, 50–51). How could ‘first-order’ laws be grounded in ‘second-order’ laws between determinable universals, when there are no such determinable universals? It seems that Armstrong, in particular, could not postulate functional laws of the required form. The matter presents additional complexities, however. Although Armstrong does not admit the existence of determinable properties, as indicated, he does have a general conception of how certain properties can fulfill such functions (see Armstrong 1978, p. 120–131, 1997, 51–63). Perhaps this same conception can serve here to integrate the theory of functional laws. Armstrong’s idea is that all determinate properties under the same determinable have a ‘partial identity’ between them. If x, for example, has a mass of 1 gr and y has a mass of 10 gr there is something in common between x and y because y must have a proper part, let it be z, such that z also has 1 gr. The only way in which something can have a mass of 10 gr is by having proper parts, each of which has a mass of 1 gr. All the infinite determinate properties of mass result, therefore, from the multiple instantiations of the same determinate universal. The application of this scheme in a general way shows that all the supposedly determinable properties are simply the families of structural properties that arise from the multiple instantiations of a certain “minimal” universal for each of the determinable properties. The functional laws would, therefore, be the generalization of a single ‘basic’ natural law integrated by the ‘minimal’ determinate universals that must be ‘part’ of all other determinate property under the same determinable. For example, concerning the law of universal gravitation, one should

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maintain that there is something like a ‘minimal mass’ that has to be ‘part’ of all other mass and a ‘minimal distance’ between objects that has to be part of all other distance. Objects with these minimal masses at that minimal distance are nomologically connected with a ‘minimal attractive force’ between them. When dealing with objects with larger masses or other distances, the masses involved will be multiples of the minimal mass, just as the other distances will be multiples of the minimal distance. The resultant force of attraction will then also be a multiple of the minimal, attractive force. This conception of determinable properties and the correlative conception of functional laws, however, is fraught with problems. First, it is not admissible as a reasonable theory about the systematic relations between determinate and determinable properties, as was explained above (see § 12). Nor is the theory of functional laws acceptable if it depends on a misconception of determinable properties. The scheme proposed by Armstrong may perhaps seem plausible concerning determinable properties with a single dimension of determination, but it does not fit determinable properties with more than one dimension of determination. In effect, in what sense would an isosceles triangle be ‘part’ of another scalene triangle? In what way would a particular color—with a certain hue, brightness, and saturation—be ‘part’ of another color of the chromatic space—with a different hue, brightness, and saturation? Talking about “parts” or “partial identity” in these cases does not seem to make sense. Even in the case of determinable properties with a single dimension of determination, however, Armstrong’s proposal seems implausible. For it to work, as it has been explained, minimal determinate properties would be required, but we have no reason to think that there are such. Perhaps there is no ‘smaller’ mass as there is no ‘smaller’ distance. The usual assumption when considering, for example, the law of universal gravitation is that masses and distances can be put into bijection with real numbers. This is why masses and distances can be ‘represented’ by such numbers, and that is why the equation [F ¼ Gm1m2/d2] can be given a physical meaning. Perhaps this assumption is wrong because not for any real number there is a correlative property, but—anyway—it is not obvious that no mass or distance is the smallest. As can be seen, therefore, the idea that functional laws are relations of necessitation of second-order between determinable universals that ground the relations of necessitation between determinate universals simply is not consistent with the way Armstrong deals with determinable properties. Armstrong could not, then, pretend to solve the problems that come with natural laws between non-instantiated universals by appealing to functional laws. However, this is an ad hominem difficulty that specifically affects someone, like Armstrong, that simultaneously holds that there are only instantiated universals, that natural laws are relations between universals and that determinable universals are grounded on the respective determinate universals. Maybe these problems could be solved if any of these assumptions were abandoned. In principle, it seems that a coherent Aristotelian position could be formulated if it is rejected that determinable universals are grounded on the determinate universals that fall under them. What seems to be sustained by those who do not want to postulate transcendent universals is that functional laws are nomological relations that occur

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directly between determinable universals. These functional nomological relations are those that later ground the natural laws between determinate universals. It is essential for this that determinable universals are not, in turn, grounded on the determinate universals that fall under them. It must be held that determinable universals only exist if they have instantiations to respect the Aristotelian conception of universals. But the instantiation of some determinate universal under a determinable is also the instantiation of the determinable. If x instantiates the universal of having n gr of mass, for example, then x instantiates the determinable universal of having mass. It is sufficient then that only one of the determinates under a determinable is instantiated for the determinable to be also instantiated.3 Although a theory of functional laws and determinable universals of these characteristics seems very attractive, it brings difficulties that make it inappropriate for a general conception in which there are only immanent universals. The problem, in summary, is the following: one should postulate the ontological priority of the determinable universal over the determinate universals so that the functional laws have enough ontological ‘autonomy’ to ground laws that involve non-instantiated universals, but this is incompatible with the ontological priority that must be given to determinate instantiated universals, of which the determinable universals depend for their existence. It is typical of an Aristotelian theory of universals to postulate a direction of ontological priority that is the inverse of what is required for the conception of the functional laws. Indeed, the advantage of functional laws for an Aristotelian is that they would ground natural laws between determinate universals or they would ground, at least, the metaphysical possibility of such laws, although those universals are not instantiated. As already explained, the grounding relationship is irreflexive and asymmetric (§ 4). The functional laws that are required for the defender of Aristotelianism are functional laws ontologically prior to the laws between determinate universals. The question here, however, is that the Aristotelian conception can only be respected if the determinable universal is—in turn— grounded on the instantiation of the determinate universals that fall under the determinable. Then, in an Aristotelian conception, there is the determinable universal of, for example, having mass because there is the determinate universal of having exactly n gr of mass—the existence of the determinate universal grounds the existence of the determinable universal. But, then, there seem to be cross-grounding relationships that go from the determinate universal to the determinable universal and then from the determinable universal to the determinate universals under it. A ‘circle’ of grounding of this kind, however, is unacceptable due to the asymmetry of grounding relationships. One way that could be attempted to overcome this problem is to introduce a distinction between the functional natural law and its constituents. As indicated, the

3 It can be seen, furthermore, that a theory of functional laws of this kind would also allow solving many of the counter-intuitive consequences of Aristotelianism when it comes to integrating ‘possible worlds’ as maximal structural universals (see § 42). The metaphysical possibilities relative to certain non-instantiated universals would be grounded on instantiated determinable universals.

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determinable universal must be grounded on the instantiation of some determinate universal determined under it. As it has also been indicated, laws between determinate universals must be grounded on the functional law between determinable universals. What could be argued here is that the functional law is not grounded, however, on the determinable universals that make it up. One could admit—prima facie—that the functional law is ontologically dependent on those determinable universals without being grounded on them. Recall that the ontological grounding between entities x and y is a relation by which x is ‘constitutively sufficient’ for the existence of y, whereas ontological dependence between entities x and y is a relation by which x is ‘constitutively necessary’ for the existence of y. What one would be maintaining is that, although a functional law [N(U1, U2)] is dependent on the universals U1 and U2, because without the existence of such universals the law would not exist, the law would not be grounded on universals U1 and U2, because the existence of these universals would not be sufficient for the existence of the functional law. That is, the natural law should be understood as a state of affairs that implies a genuine ontological ‘novelty’ regarding its constituents. In this way, it would seem to avoid the circle of grounding to which mention has been made. The determinable universal is grounded on the existence of a determinate universal. Then, the functional law depends ontologically on that determinable universal. That same functional law grounds the natural laws between determinate universals. It does not follow from this that the determinate universal—or the determinable, for the same reasons—is grounded on itself. That would happen if the functional law were also grounded on the universals that constitute it, given the transitivity of the grounding relationship, but by replacing grounding with dependence, this vicious circle is broken, or so it seems. Unfortunately, this strategy has the problem that it rests on one of the less acceptable features of the theory of natural laws advocated by Armstrong, Tooley, and Dretske. It is an abandoned trait—for good reasons—by the philosophers of the next generation who have worked on the development of non-Humean ontologies of natural laws. In the initial formulations of Armstrong, Tooley, and Dretske, the second-order nomological relations of necessitation are contingent for these universals. Natural laws are, therefore, contingent. This brings with it several counterintuitive consequences. This question will be discussed with detention below (see § 51). For example, although in the actual world certain natural laws explain observable regularities, there are possible worlds that present exactly the same regularities, but without such laws. If in the actual world it is a law that [N(U1, U2)], it follows that there is a regularity by which all instantiation of U1 is followed by the instantiation of U2. There are possible worlds, however, in which empirically everything happens just like in the actual world. Any instantiation of U1 is followed by the instantiation of U2, but there is no natural law [N(U1, U2)]. These are possible worlds in which natural laws are replaced by huge coincidences. The nomological relation N connects certain universals in some possible worlds, but not in others. From a Humean perspective, this makes the N relationship especially disgusting. It is a way of connecting ‘different existents’—universals—without explanation. It must simply be accepted as a brute ontologically basic fact that in some possible worlds

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N does connect universals and in others, it does not (see for criticism in these lines, Lewis 1983, 39–43; Van Fraassen 1989, 94–128). It is not strange that then several philosophers have simply rejected the contingency of laws. Some have argued that necessitation or nomological relationships are essential to the universals connected by them (see Swoyer 1982; Ellis 2001, 203–228, 2009, 51–72). Others have gone further and argued that no higher-order relationship of necessitation is required between universals to generate a natural law. It is enough to postulate a universal endowed with essential causal powers. What we call a “natural law” is simply such causal powers (see Bird 2007, 43–65, 169–203; with preventions, Mumford 2004, 65–159). Moreover, for these philosophers to think of natural laws as structured by ‘first-order’ universals and a ‘second-order’ relationship between them forces us to consider the problem about why it is that such a connection must necessarily hold. If you are dealing with numerically different entities, you might be inclined to think that they are independent of each other. On the contrary, if laws are simply the causal powers of the universals and such causal powers are essential to such universals, no further explanation is required as to why there are natural laws. What we call a “natural law” simply is what is given by the intrinsic nature of one or several universals. As can be appreciated, therefore, the strategy that could be tried to solve the circle of grounding between determinate universals and functional laws depend on an unlikely assumption. If natural laws are necessary for the universals that integrate them, they must be grounded on them. But this brings us back to the initial problem. Functional laws are supposed to ground ‘first-order’ laws into which determinate universals enter, but such laws are, in turn, grounded on such universals. It turns out, then, that the only alternatives open to the defender of immanent universals in regard to functional laws are: (i) to argue that functional laws are relations directly between determinable universals that must not be grounded on the instantiation of determinate universals; or (ii) to argue that functional laws must be grounded on determinate universals and their respective nomological relationships with other determinate universals. None of these alternatives would be acceptable to the Aristotelian. Under alternative (i), which seems the most reasonable independently, it should not be required that determinable universals are instantiated for the existence of functional laws—for these universals must be ontologically prior over determinate universals under them. But this would be to admit the possibility of non-instantiated universals. Under alternative (ii), functional laws should be grounded on first-order laws between determinate universals. There would be no way to argue, then, that the existence of the functional law allows us to explain the existence of natural laws between certain non-instantiated universals. The Aristotelian should, rather, reject the existence of such laws. What results from this examination, to conclude, is that taking functional laws seriously, far from being a relief for the Aristotelian, obliges us to accept non-instantiated universals.

7.3 Powers to the Rescue?

7.3

169

Powers to the Rescue?

§ 48. Functional laws are a way of solving the problem of how there seem to be natural laws between non-instantiated universals, but the Tooley-like cases that have been indicated do not include determinate properties. Even if one accepted the idea that functional laws were sufficient to cover certain cases—and it has already been found that they are not acceptable—this would not be adequate to cover the scenarios presented by Tooley. What has been proposed in such cases is that there could be laws non-derivable from others that involve non-instantiated universals. It will be instructive to consider the additional strategies proposed by Armstrong to address these scenarios. The failure of these strategies will serve to show how the same motives that are making reasonable to think of natural laws as relations of higher-order between universals or simply as universals make it reasonable to postulate non-instantiated universals. Recall that the first case proposed by Tooley is that of a possible world in which there are ten different types of fundamental particles, in which the interactions between the particles of each of these types obey “basic” natural laws, not derivable from other natural laws, and in which these laws have been discovered, except for the case of the interaction between particles of the type X and Y. In the case of these particles, moreover, it is not that their interaction has not yet been experienced, but that they cannot interact in that world, for example, because they are too far apart from each other. Tooley argues that in such a case, it would be reasonable to assume that there is a natural law that governs the interactions between particles of type X and type Y. Armstrong argued that in these cases one could assume that the respective natural law would be grounded on the universals X, Y and the relation R of interacting such particles (see Armstrong 1978, 156–157). Although there are no particles of type X in the relation R with particles of type Y, the universal X, the universal Y, and the relation R—by hypothesis—are instantiated. The existence of these universals should be sufficient for the existence of the respective natural law.4 There are two problems with this response. First, a natural law must connect universals so that the instantiation of one of them makes the instantiation of another necessary. Here, it would be the instantiation of the relational universal R between particles of type X and type Y that necessitates the instantiation of another relational universal S. All that has been held is that X, Y, and R would be instantiated, but this is not enough. For the existence of the law, the existence of the relational universal S is also required. Second, a solution of this kind does not apply to the second type of case proposed by Tooley. If actually it is a law that the instantiation of the universal 4

With some preventions. For Armstrong, natural laws are contingent. The existence of the constituent universals of a natural law is not a guarantee of its existence. Although there are universals X, Y, and R, the existence of the natural law that includes them is not necessary. The cases that are being considered, however, are cases in which it seems reasonable to postulate a natural law from the outset, although their constituent universals are not instantiated, so it is not relevant for this discussion to question the existence of the natural law. Below, it will be seen that Armstrong does such a questioning as a last resort to accommodate Tooley-type cases.

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F makes the occurrence of the universal M necessary, and universals F and M are actually instantiated, it seems reasonable to think that, even if F were not instantiated, it would still be a natural law that the instantiation of F makes the instantiation of M necessary. In a possible world where the law exists, and the universals that make it up are not instantiated, it cannot be that those same universals are the ground of the law since from an Aristotelian perspective there would not be such universals. A second way of dealing with the problem of Tooley-type cases would be the postulation of ‘causal powers’ (see Armstrong 1983, 121–123). A ‘causal power’ is the disposition to produce an effect if certain circumstances occur. Analogously to an intentional state, a power is, in some way, ‘intended’ towards the production of an effect of a certain type, even when such an effect does not exist. In the case of intentional states, one can think of a unicorn, even though there are no unicorns. The idea here would be to argue about the first Tooley case that it is inscribed in the intrinsic nature of the properties X, Y, and R that whatever it is that instants these universals will have a causal power to produce the occurrence of an S relation. Although from the perspective of an Aristotelian, the relation S does not exist when it is not instantiated, it is ‘potentially’ in the universals that are instantiated. This would be enough to postulate the existence of a natural law by which the instantiation of an R necessitates the instantiation of an S, even when S does not exist. This strategy, unfortunately, clashes with the position that Armstrong himself has defended regarding dispositional properties or powers (see Armstrong 1997, 69–84). What Armstrong has argued is that a ‘disposition’ to make φ must be understood as a ‘categorical’ property P together with a natural law or a series of natural laws according to which the instantiation of P in the circumstances C necessitates the occurrence of a φ. Dispositions are reducible to categorical properties and natural laws. This is what one must admit, in effect, if one generally holds that natural laws are contingent for the universals that they are connecting, as indicated above. In effect, if it is contingent for a universal U1 to be or not integrating a law [N(U1, U2)], then it is not by its intrinsic nature that the universal U1 is ‘intended’ to generate the occurrence of an U2. Considered by itself, taking into consideration only the intrinsic nature of U1, it is ‘inert’. What causal relations provokes the instantiation of the universal will be a function of facts ‘external’ to the universal itself that must be grounded, among other things, on ‘extrinsic’ natural laws. In a theory of this kind the same universal will ground different causal powers in different possible worlds because in different possible worlds it will be integrating different natural laws.5 It is evident, then, that Armstrong could hardly appeal to this strategy since the only

This is a conception generally known as “functionalist”, because the same disposition can be constructed in different possible worlds by different categorical properties, according to which are the natural laws in each world. The categorical property that, in our world, for example, is a disposition to attract objects with opposite electric charges, can be in another world a disposition to repel such objects. The disposition to ‘attract objects with opposite electromagnetic charges’ selects different categorical bases in different possible worlds. See Mumford 1998, 192–215; Prior et al. 1982.

5

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way in which a ‘causal power’ could ground a natural law in its ontology is if a natural law obtains already together with the respective categorical property. This is, however, an ad hominem difficulty which affects someone like Armstrong who holds at the same time: (i) that natural laws between non-instantiated universals can be accommodated by causal powers, and (ii) that causal powers are reduced to a base constituted by a categorical properties and natural laws—or several natural laws. Perhaps the relaxation of the requirement (ii) would allow treating adequately natural laws that have as constituents non-instantiated universals. It has been anticipated that in this work we are going to prefer a theory in which the causal powers are ontologically fundamental. Why not use those same powers to ground natural laws that seem to be connecting non-instantiated universals? This would be adequate, perhaps, to accommodate the first type of case proposed by Tooley, but it would not be adequate, in any case, for the cases of the second type. In the cases of the second type are considered possible worlds in which it is not instantiated any of the universals that make up a natural law that actually exists. It seems reasonable to assume that in such possible worlds it would still be a natural law that, for example, [N(F, M )], although there neither F nor M are instantiated, since it seems reasonable to suppose that in such worlds it would still be the case that if there were an F, there would be an M. At least it would require that F be instantiated to ground the causal powers that this universal confers. As there is no such universal for an Aristotelian— because it is not instantiated—hardly can it attribute causal powers. Armstrong, finally, proposes a skeptical attitude about the Tooley-type cases (see Armstrong 1983, 123–126, 1997, 243–248). One is inclined to admit that in the scenarios presented, there should be a natural law, although the universals that make up such laws are not instantiated. There seems to be no way to accommodate these intuitions in a general Aristotelian conception in which there are only instantiated universals, as has been shown. So, what a consistent Aristotelian must do is to reject what these intuitions seem to show. An Aristotelian could admit in these scenarios the counterfactual conditional that there would be a natural law if the universals in question were instantiated would be true. This is not to admit a natural law in these scenarios, but something much weaker. The counterfactual conditional should be grounded on some ‘meta-law’, as a functional law does concerning the laws between determinate universals. But lacking an adequate ‘meta-law’, one should not even admit the counterfactual. In principle, adopting a skeptical position of this kind is an admissible theoretical option. A good theory with great explanatory power can have ‘costs’, that is, counter-intuitive consequences that we must accept. What an Aristotelian could argue is that this is what happens here. Although we have the intuition that in the scenarios presented by Tooley, there would be natural laws that include universals not instantiated, this is an illusion. We cannot accept such possible laws for the simple reason that it is not metaphysically possible for there to be non-instantiated universals. It is the entire Aristotelian theory of immanent universals that recommends this skeptical position. A skeptical attitude in these lines would win in plausibility if the Aristotelian had an acceptable explanation of functional laws. The functional laws would be the ground of laws between non-instantiated universals. Most cases of natural laws—or

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apparent natural laws—connecting non-instantiated universals would be explicable by such functional laws. Then, the Tooley-type cases would be marginal, so the ‘cost’ of the skeptical attitude would be very low. The problem is that, as explained in the previous section, the Aristotelian does not have an acceptable theory of functional laws. Discarding what Tooley-like cases seem to show in these circumstances is not a marginal ‘cost’. It is an explanatory failure that reaffirms the explanatory failure concerning functional laws. Whatever the degree of plausibility that Aristotelianism has had up to this moment, this plausibility has diminished with the skeptical attitude. But it also happens that the Tooley-type cases admit a generalization, as will be shown in the next section, which makes the ‘cost’ of the skeptical attitude much higher.

7.4

Generalized Tooley-Type Cases

§ 49. As has been explained, Tooley has presented two types of scenario in which one would be inclined to postulate the existence of a natural law, although the universals that makeup such a law are not instantiated. These scenarios are a reason both to reject the regularist conceptions of natural laws and to reject Aristotelianism over universals. It is a problem of these cases, however, that they depend on a very dubious modal metaphysics. Of course, if it is metaphysically possible that there is a natural law constituted by non-instantiated universals, then it would be metaphysically possible for there to be non-instantiated universals. And this implies the falsity of the principle of Instantiation and, with it, of Aristotelianism. But this reasoning requires that, in effect, it is metaphysically possible that there is a natural law connecting non-instantiated universals. The thought experiments proposed by Tooley make certain intuitions plausible, but they do not seem sufficient in themselves to justify the much stronger thesis of metaphysical possibility. It is not enough to justify a thesis of metaphysical possibility simply to describe a scenario in which there is no apparent incoherence, according to what one can notice by its examination. What makes such scenarios especially dubious as a justification for metaphysical possibility is that they seem to assume that natural laws are contingent upon their constituents and that, then, there are possible worlds for all the ‘laws’ that one can conceive coherently. There are ‘closer’ possible worlds with the same laws as our world, but there are other ‘farther’ worlds in which those laws do not exist and are replaced by others. If laws are contingent, then what should be postulated is that, for example, there are possible worlds for the whole combinatorial of nomological relations between different universals, in the same way, that one could postulate all the combinatorial possible states of affairs constructed by objects and universals. As explained, however, in the previous section, the idea that natural laws are contingent is probably the weakest flank of the conception of natural laws as relations of higher-order among universals. A reasonable conception of natural laws should reject this idea (see § 51).

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The issue is that there are not ten types of fundamental physical particles. According to the standard model, there are seventeen—six types of quark, six types of lepton, and five types of boson. There are no ‘basic’ laws for each of the interactions between these particles. We have no reason to think, then, that a scenario as described by Tooley is metaphysically possible. It seems coherent, of course, but this is insufficient for a thesis of metaphysical possibility. Something similar happens with the second type of case proposed by Tooley. According to everything we know, there are no “basic” laws that are connecting physical properties with mental properties, whether epiphenomenal or not. We do not have, therefore, sufficient reasons to think that the scenario proposed by Tooley is metaphysically possible. What we can or cannot conceive about those scenarios is irrelevant to justify the thesis of possibility.6 The Tooley-type cases, however, admit being generalized in a way that makes the conclusion to which Tooley wanted to reach perspicuous (see Alvarado 2008, 2010). Take any existing natural law, according to all the Aristotelian parameters. It is a natural law, for example, that objects with opposite electromagnetic charges are attracted with a force directly proportional to the charges—weighted by a constant— and inversely proportional to the square of the distance between those objects (Coulomb’s law). Suppose a possible world w1 in which only objects with negative electromagnetic charge exist. This nomologically possible world is accessible from the actual world wA. In effect, in w1, there are no existing objects that behave differently to the way they behave in wA. All that is being postulated is that certain properties actually instantiated—like the property of having some positive electromagnetic charge—are not instantiated. The instantiated properties in w1 obey exactly the same laws as those that they obey in wA. The issue here, on the other hand, concerns the determinable properties of having some negative electromagnetic charge and having some positive electromagnetic charge. This allows us to separate this type of cases from the questions concerning the functional laws already explained. This is not a problem related to the non-instantiation of certain determinate properties under a determinable that has some other determinate properties instantiated. These are cases in which there is no determinate property instantiated under a determinable. It seems reasonable to postulate that in w1 the following counterfactual conditional is true: if something had a negative electromagnetic charge of –q, it would be attracted with an object with an electromagnetic charge of + q with a force directly proportional to the charges and inversely proportional to

6

Understandably, Armstrong did not raise objections of this kind to the cases raised by Tooley, since Armstrong defended a combinatorial theory of metaphysical modality (see Armstrong 1989) and defended—as it has been explained—the contingency of natural laws. In general, the space of metaphysical possibilities is generated by the mutual independence of universal properties and objects to construct possible states of affairs (§ 43). Natural laws are states of affairs of ‘higherorder’ constructed from universals and the relation of necessitation. Again, the universals constituent of a law and the nomological relationship are independent of each other and can exist ‘jointly’, or ‘separated’. In principle, any universal can be nomologically related to any other. Then, the scenarios described by Tooley are—from this perspective—metaphysically possible.

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the distances between those objects. The counterfactual conditional must have some ground to be its truthmaker in w1. If Coulomb’s law does not exist in w1—and it should not exist in w1 according to the Aristotelian, since there are no negative charges in w1—there should not be any truthmaker of the counterfactual conditional. But, by hypothesis, the counterfactual conditional is true in w1. Even if nothing has a negative charge on w1, if there were something that has it, it would obey the law of Coulomb. Then there must be Coulomb’s law in w1. Then, it is possible that there is a natural law, even if one of its constituent universals is not instantiated. Then, it is possible that there are non-instantiated universals, and Aristotelianism is false. This case can be generalized. Just as you can contemplate nomologically possible worlds with respect to the actual world without negative electromagnetic charges, you can also contemplate possible worlds without quarks, or without neutrinos, or leptons, or without photons, and so on. A world without photons could include photons. A world that only had photons and gluons, for example, would be a world without masses and electromagnetic charges. It is not necessary to postulate these possible worlds anything other than existing natural laws. This makes it necessary to postulate certain counterfactual conditionals that require truthmakers. One might be inclined to think at this point that all that would be required for the truth of the counterfactual conditional in w1 is that the existence of Coulomb’s law is possible. This possibility is guaranteed because Coulomb’s law actually exists. If the relations of accessibility between possible worlds are symmetrical—something that has already been argued above (see § 42)—the fact that w1 is accessible from wA makes wA accessible from w1. In wA Coulomb’s law exists, then in w1, it is possible that Coulomb’s law exists. The cases presented here systematically exploit the fact that the possible worlds contemplated are nomologically possible with respect to the actual world. This same trait, then, could be used to neutralize the Platonic consequences that are intended to be derived from these cases. This strategy, however, is not sufficient to guarantee a truthmaker for the counterfactual conditional in w1. A possible truthmaker is not a truthmaker. What is required is something that acts as truthmaker in w1 and not something that would be a truthmaker, if it existed in w1. It will be necessary to consider the situation again in more detail to clarify this point. In w1 it is true that if something had a negative electromagnetic charge –q it could be attracted to an object with an electromagnetic charge of + q with a force directly proportional to the charges and inversely proportional to the square of the distances between those objects. What is sought is a truthmaker of this counterfactual conditional. Counterfactual facts require, in effect, something that grounds them. They are not fundamental facts. The most reasonable ground for this fact is a natural law that must have a ‘regulative’ character with respect to what may happen in w1, as well as what might happen in wA. What is being argued in the disputed argument is that the possibility of a natural law in w1 and not the existence of a natural law in w1 would suffice as a basis for the counterfactual fact. That is, if in w1 there were an object with a negative electromagnetic charge, then Coulomb’s law would exist. If Coulomb’s law existed, then that object would be attracted to others of opposite charge. Then— so the objection reads—if there were an object with negative electromagnetic charge in w1, it would be attracted with others of opposite charge. But the possibility of a

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natural law in w1 is not enough to postulate the regulatory force of such a law in w1. Indeed, it is true that in all possible worlds ‘closest’ to w1 objects with opposite charges are attracted, because there must be facts that ‘regulate’ what is possible in w1. If in w1 a natural law with such a ‘regulative effect’ is only possible, all that is being postulated is that there is a possible world, w2, such that in all the ‘closest’ worlds to w2—worlds regulated by the existing Coulomb’s law in w2—objects with opposite charges are attracted. The class of worlds closest to w2 does not have to coincide with the class of worlds closest to w1. The possibility of a natural law in w1 is not a guarantee that in all the worlds ‘closest’ to w1 things happen as established by the law. The most that would follow from such a situation is that in w1 if there were an object with negative charge, it might be attracted with objects of opposite charge.7 Notoriously in the reasoning used above has been appealed to the transitivity of the conditional mentioned, but it is known that the counterfactual conditional is not transitive (see Lewis 1973, 32–35). It has been crucial to block this objection that the class of worlds ‘closest’ to any given world is variable. However, if natural laws are necessary entities, invariant between possible worlds, there is no difference to be considered between the worlds ‘closest’ to w1 and the worlds ‘closest’ to w2. The worlds ‘closest’ to w2, that is, the worlds in which what happens is regulated by Coulomb’s law are simply all the worlds, without exception. And the worlds ‘closest’ to w1 will be a subset of all the worlds. But if natural laws are necessary entities, what necessity would there be for all this argumentation? To sustain that natural laws are necessary—whether or not their universals constituent are instantiated—is to admit universals that are not instantiated. It can be seen, therefore, that the mere possibility of a natural law in w1 is not sufficient to ground the counterfactual fact in w1 that if there were an object with a negative charge, it would be attracted with objects of opposite charge. On the contrary, the counterfactual fact requires a natural law. And if there is a natural law, there are the universals from which it is constructed.8 7

This is what has been called a might counterfactual conditional as opposed to a could counterfactual conditional (see Lewis, 1973, 1–43). The could conditional if p were the case, then q could be the case [p ⃞! q] is true in w if and only if in all possible worlds closest to w in which p is true, q is true. The might conditional if p were the case, then q might be the case [p ⋄! q] is true in w if and only if in at least one possible world closest to w, p and q are true. For what interests us here, Coulomb’s law ¼ LC; there is some object with negative charge ¼ CN; and objects with opposite charges are attracted ¼ CO. In w1 it is true that [CN ⃞! LC], then in the class of the worlds closest to w1 in which CN, it also happens that LC. Be one of these worlds w2. As in w2 LC is true, it follows without problems in w2 that [CN ⃞! CO], because in all the worlds closest to w2 in which CN it also CO, given the regulative function of Coulomb’s law. This is not a guarantee that all the worlds closest to w1 are also worlds in which LC is true, that is, worlds in which Coulomb’s law is regulating what happens. All that could be said is that one of the closest worlds to w1 is regulated by Coulomb’s law, and this guarantees that [CN ⋄! CO] is true in w1. These same reasons make counterfactual conditionals—at least under the semantics used here—not transitive. 8 As explained above, how these cases would be treated in a counterfactual conception of natural laws is not being considered here (see Lange 2009). In Lange’s theory, natural laws are identified with classes of fundamental counterfactual facts. It is not required in this approach a ground of them in natural laws. Although this point of view seems unlikely, it can easily be seen that the same

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The Necessary ‘Nomic Network’

§ 50. An additional justification for transcendent universals comes from certain considerations about the nature of natural laws that must be essential for the universals that integrate them (see Bostock 2003; also, Bird 2007, 50–59). As explained above (see § 45), although the initial proponents of the conception of natural laws as constituted by universals had held that laws are contingent entities for their constituents, this feature is one of the weakest aspects of the theory. The most reasonable position is to maintain that it is essential for a universal to integrate the natural laws that it integrates, whether one conceives such natural laws as nomological relations with other universals or as the causal powers that universals confer on its instantiations. Of course, there is a difference between postulating natural laws essential to universals and postulating necessary natural laws. In principle, one could defend the existence of immanent universals that exist contingently along with natural laws that are essential to them. In the possible worlds in which these universals exist, there will also exist the natural laws associated with such universals, but in the possible worlds in which such universals do not exist, because they are not instantiated in those worlds, the natural laws associated with them will not exist either. If universals are immanent, natural laws will be contingent entities, even if they are essential for their constituent universals. A much stronger thesis is to postulate that natural laws are necessary entities that exist invariably in all possible worlds. Bostock has argued, however, that the essential character of the natural laws for universals that are being connected in such laws is a reason to hold the stronger thesis of the necessity of natural laws. It is this derivation that is interesting to examine with detainment in this section. For this, a review of the reasons for postulating essential natural laws will be made first. Given the systematic importance of this thesis, it will be convenient to show in more detail why it is the most acceptable. Then Bostock’s argument will be discussed directly.

7.5.1

Natural Laws Essential for Universals

§ 51. To suppose that natural laws were contingent for the universals that make up such laws implies what has been called “quidditism”. Traditionally, the essence of a substance has been called its quidditas, that is, the properties that make it ‘what’ it is

conclusions that have been raised here would also follow from a position like Lange’s. The law of Coulomb exists in w1 because in w1 there are all the primitive counterfactual facts of the type if there were an object with a negative charge –q it could be attracted with objects with a positive charge of + q. The possibility of a natural law in w1 would be, instead, the fact that in a possible world w2 accessible to w1 those same facts are given. But, the counterfactual facts of w2 are not eo ipso counterfactual facts of w1.

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(in Latin: quid)—and that, therefore, are properties possessed by the substance in question in all possible worlds in which the substance exists. The quidditas is what is answered to the question about what (quid) is a substance.9 The expression “quidditas” as it has been used since Black’s work (see Black 2000) designates something very different. It is the character of a property P—not a substance—by which P is that property and not another. It is a qualitative character that is not reducible to the causal powers that such property confers, nor is it grounded on such powers.10 A property, in effect, has to confer certain causal powers on the objects that instantiate it. That is, because an object instantiates some properties, it can enter into certain causal connections. If natural laws are contingent to such universals, then in different possible worlds the same property will confer different causal powers. Negative electromagnetic charge, for example, is actually a property that grounds attraction with objects that have a positive electromagnetic charge. In possible worlds with different natural laws, the same property would be that which grounds the repulsion with objects that have a positive electromagnetic charge. In other possible worlds, all the functions actually associated with negative electromagnetic charge are fulfilled by positive electromagnetic charge, and all the functions actually associated with positive electromagnetic charge are fulfilled by negative electromagnetic charge. What makes the property of negative electromagnetic charge in the actual world identical or different from the property of negative electromagnetic charge in another possible world cannot be determined by the specification of what powers such property confers or what natural laws govern its behavior. The identity or difference of properties in different possible worlds must be determined simply by an irreducible and fundamental qualitative component: its quidditas. It is called “quidditism” the thesis according to which the identity of the properties depends on such quidditates. To postulate contingent natural laws for the universals that are connected by them requires postulating quidditates for such universals. Their identity and difference between possible worlds are completely disconnected from the causal relations in which the objects that instantiate such universals can enter. Moreover, as the way in which a property comes to be known is through the causal relationships that the

9

For example, St. Thomas Aquinas points out that: illud quod res constituitur in proprio genere vel specie est hoc quod significatur per diffinitionem indicantem quid est res, inde est quod nomen essentiae a philosophis in nomen quiditatis mutatur; et hoc est etiam quod Philosophus frequenter nominat quod quid erat esse, id est hoc per quod aliquid habet esse quid. (De ente et essentia, I, 5). The Greek expression tò tí hên eînai of Aristotle (see, for example, Aristotle (1998) VII, 3–6) which was translated by William of Moerbecke as quod quid erat esse has been notoriously difficult to translate, but it seems to designate what is being X for an entity that is, indeed, an X. The essence expressed by its proper definition is what manifests, for example, what it is to be a man for a man. 10 When it comes to the character by which an individual object is this individual object and not another, the term haecceitas has been used, which could be translated as “thisness”. Those who have postulated an haecceitas have argued that the essence of an individual object cannot be determined by universal properties of a purely qualitative nature (see Adams 1979; Alvarado 2006, 2007a). Analogously, the character by which one property is this and not another has been called its quidditas.

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instantiations of such property come to have with, for example, our measuring devices—or our senses that, for these purposes, count as ‘mechanisms to detect properties’—it happens that there would be no way of knowing if what is detected with a voltmeter in the northern hemisphere is the same property or properties that are detected in the southern hemisphere. They could be, according to all we know, different properties, with different quidditates, but this difference would be undetectable for us.11 A first reason for rejecting quidditism has to do with epistemological problems (see Shoemaker 1980). How the existence of a property is known has to do with the causal connections in which an object that instantiates it must enter. Even when it comes to properties that should be ‘phenomenal’, such as the one that exhibits something that has a certain color of the chromatic space, ‘seeing’ that color is entering in a certain causal interaction with the surface that possesses it. If the ‘color’ is to be admitted as an authentic property, then there will be causal powers that the objects will have by the instantiation of such property.12 If the differences in quidditates are undetectable, what reasons would we have to postulate them? The only criteria we have to select properties and to distinguish properties are causal powers. A difference of quidditas does not seem to be a difference. This type of objection seems, however, to depend on verificationist premises that are already highly discredited. The fact that a difference is undetectable to us does not make it metaphysically impossible.13 Not everything that exists must be 11

It should be noted that the postulation of natural laws essential to universals does not exclude the possibility of additional quidditates for these universals. If natural laws are essential for a universal, all that is required is that in no possible world would that universal be identical to a property that confers different causal powers, but this still does not prevent that different properties confer exactly the same causal powers if they have different quidditates. It is not necessary to enter now directly into the question about the conditions of identity for universals (see § 75–77), but it can be appreciated why quidditism is deeply problematic. 12 Of course, it is a moot question of whether colors are authentic properties or not. Eliminativist positions on color have been advocated according to which there are no colors as surface properties independent of the minds that perceive them, but only certain phenomenal features of the experiences of such minds. For each maximally determinate hue of the chromatic space, there is no single property that is that which is instantiated on all the surfaces that exhibit that color. If, on the other hand, colors are authentic properties, then perceiving a color must be a certain causal connection with its instantiation. Colors look the way they look, from this perspective, because they confer specific causal powers on surfaces. 13 It has also been argued that the epistemological motives adduced here are skeptical motives that we already know how to resist in a general way (see Schaffer 2005). If we do not have to worry about skeptical scenarios in which we are deceived by an evil genius, or we are brains in a vat, we should not worry about the skeptical scenarios in which we would be detecting different properties in the northern hemisphere and the southern hemisphere. But there is a very substantive difference between our epistemic situation in those skeptical scenarios and our epistemic situation with the postulation of quidditates. In traditional skeptical scenarios, there is sufficient evidence to justify our ordinary beliefs, and that is a reason for neglecting them. If one postulates a quidditas, on the other hand, what we are doing is precisely discarding the epistemological relevance of the mechanisms that we ordinarily use to identify properties. It is not a question—pace Schaffer— that the same attitude we adopted in the face of skeptical challenges should lead us to accept

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empirically verifiable in one way or another. At least, the metaphysical possibility of empirically undetectable differences should not be prohibited a priori. For example, when it comes to specifying the individual essence of a particular object, the metaphysical possibility of two or more objects that have exactly the same purely qualitative universal properties has seemed perfectly acceptable.14 This has led to postulate an haecceitas for objects, which is neither reducible to other facts, nor is it grounded on other facts (see Adams 1979; Alvarado 2006, 2007b). Although actually the individuals that are known to us ordinarily always possess qualitative differences among themselves, the metaphysical possibility of qualitatively indiscernible replicas makes it reasonable to postulate an individual constituent that is not reducible to these same universal properties. Why, then, do not postulate for the properties a quidditas that fulfills, mutatis mutandis, the same theoretical functions? There are, however, very substantive differences between the theoretical situation that recommends postulating haecceitates and the theoretical situation that would recommend postulating quidditates. Particular objects are known by us by qualitative aspects, but also very noticeably as something determinate that we can normally indicate by an indexical gesture like this or that. The haecceitas of a particular object seems to have to do precisely with that particular incommunicable character of a particular. When it comes to a universal property, on the other hand, all that is offered to us is the qualitative character. Although a particular instantiation of a universal is, in effect, a particular—even if the universal is not such instantiation— the universal is the essentially communicable qualitative character that appears there exemplified. One who defends the existence of a quidditas is assuming that analogously to what happens with individual objects concerning their individuality, there is a character by which that quality is that quality and not another, not reducible to other facts, nor based on other facts. And it is assumed that this irreducible qualitative character can be separated from the causal relationships in which objects possessing such character can enter. But there is no such. If an authentic property appears phenomenally to one as a quality, it is because there has been a causal transaction with the instantiation of that property. Properties appear to us phenomenologically in a certain way because they confer certain causal powers rather than others. What is designated by the quidditas—as opposed to an haecceitas—exactly is what is shown in the manifestation of causal powers. It happens, therefore, that the postulation of an haecceitas for individual objects does not require to suppose exorbitant cognitive capacities. That haecceitas is already something with which we have ordinary epistemic contact when we have

quidditates without epistemological qualms. On the contrary, what this attitude recommends is trusting in our ordinary epistemological resources, which is achieved by discarding the quidditates (see Bird 2007, 78–79). 14 Recall that a ‘purely qualitative’ property is a property that does not include any individual. For example, the property of being five meters from Napoleon Bonaparte is a property that includes Napoleon Bonaparte as a constituent and, therefore, ontologically depends on Napoleon Bonaparte. In contrast, the relational property of being x five meters from y does not include any individual. The latter is ‘purely qualitative’.

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epistemic contact with a particular. When, on the other hand, a quidditas is postulated in addition to the causal powers conferred by a universal, something is being postulated that must be epistemologically disconnected from those ordinary cognitive capacities. Our natural science claims to know or, at least, to have well-justified beliefs about different authentic properties of the world. This knowledge is what explains the empirical predictive capacity of our best theories. If one supposes that there are quidditates as constituents of the identity of the properties, it would be necessary to attribute all these successes to good fortune, because we would not have any way to justify our identification of properties. A metaphysics of properties that conceive them epistemologically disconnected from us, not only makes its object doubtful but also—with it—makes all our natural science dubious. As has been explained on several occasions, we must respect a requirement of ‘integration’ between the ontology and our epistemological access to the world. Dubious epistemological requirements should not impose extravagant ontologies, but neither a proposed ontology should turn postulated entities unknowable. What is appreciated here is just a violation of this requirement. Quidditism would make properties epistemologically unattainable. Another reason why quidditism must be rejected arises from considerations analogous to what has been made for the case of the haecceitates of individual objects without essential non-trivial properties.15 If there are no limitations for the vicissitudes that are metaphysically possible for an object, every object will be identified with any other (see Chisholm 1967). The same reasons that make a position like this unacceptable are reasons to reject quidditism (see Mumford 2004, 103–104, 151–152; Bird 2007, 73–76). Assume possible world w1 in which are Genghis Khan in the thirteenth century and Napoleon Bonaparte in the nineteenth century, with the same characteristics as these historical figures have in the actual world. It seems—prima facie—perfectly possible that the circumstances of Genghis Khan’s life would have been a little different from how they actually were. It seems that Genghis Khan could have been conceived a little after when he was conceived, and in a place a little west of where he was conceived. He could have had a life a little different from the one he had in fact. The same could be said of Napoleon Bonaparte, because—prima facie—it seems that he could have been conceived a little before when it was conceived and a little further east. There seem to be no problems, then, with postulating a possible world w2 in which are also Genghis Khan and Napoleon Bonaparte with these small variations with respect to w1. Genghis Khan-@-w1 ¼ Genghis Khan-@-w2 and Napoleon Bonaparte-@w1 ¼ Napoleon Bonaparte-@-w2. As these small variations between w1 and w2 seem 15

It has been common to understand the essential properties of x as those possessed by x in all possible worlds where x exists. This way of specifying the essential properties makes trivially essential for any object properties such as being a cat if it is a cat or being such that: if something is a cat and it is fat, then it is a cat. Any object will have all these properties. As these are properties that necessarily everything has, they do not allow discriminating between different objects. An essential ‘non-trivial’ property is an essential property that it is possible for at least one object not to possess it.

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to be admissible, there seems to be no difficulty in postulating small variations with respect to what happens in w2. Genghis Khan could have had a life that takes place a bit after when it takes place, and Napoleon could have had a life that takes place a little before when it takes place, etc. Let w3 be a possible world with these small differences. Then, Genghis Khan-@-w2 ¼ Genghis Khan-@-w3 and Napoleon Bonaparte-@-w2 ¼ Napoleon Bonaparte-@-w3. This exercise of introducing small modifications can then be repeated as many times as you want. In each case, the differences will be so minor that it will not seem reasonable to think that they must be impossible for Genghis Khan or Napoleon, respectively. It happens, however, that identity is transitive and that the accumulation of many small differences makes a big difference. If there is a chain of identities between Genghis Khan-@-w1 and Genghis Khan-@-wn, then Genghis Khan-@-w1 ¼ Genghis Khan-@-wn. The same goes for Napoleon. It would be, then, that upon reaching a possible world wn—for an n large enough—Genghis Khan will be a great French general of Corsican origin who develops his career in the XIXth century, triumphs in the battle of Austerlitz, is defeated in Waterloo and is called “Napoleon Bonaparte”. Napoleon Bonaparte, on the other hand, is a leader of the Mongol tribes of central Asia who conquered an empire in the thirteenth century and is called “Genghis Khan”. One can suppose that, moreover, everything happens in wn as it happens in w1 as regards to the distribution of universal properties instantiated there. These are qualitatively indiscernible worlds with each other and whose only difference is that everything that Genghis Khan instantiates in w1 is instanced in wn by Napoleon Bonaparte, and everything that Napoleon Bonaparte instantiates in w1 is instantiated in wn by Genghis Khan. These identifications are absurd. In the same way, any object could be identified with any other. What seems advisable from the consideration of this type of cases is that there must be essential properties for an object that set limits for the vicissitudes that are admissible modally. This is what has been done when, for example, the necessity of origin for objects has been defended (see Mackie 2006, 47–69, 93–117; Alvarado 2005, 2007b). If there is a necessity of origin, Genghis Khan could not have had different parents than he had and could not have been born in Corsica in the eighteenth century. Something similar happens with Napoleon Bonaparte. It is reasonable here to think that the facts about the identity and difference of objects in different possible worlds cannot be completely disconnected from the properties that those objects are instantiating in those worlds. This is not to suppose that such facts of identity and difference must be grounded on the distributions of purely qualitative properties—this would oblige the acceptance of the principle of identity of the indiscernibles, that is, two objects that have exactly the same purely qualitative properties should be the same object. One can admit that there is an haecceitas for each individual object, but there must also be essential properties of a purely qualitative nature that make impossible that any vicissitude can happen to any object. ‘Radical haecceitism’, on the other hand, is the assumption that the only thing that determines identity and difference of objects in different possible worlds are their respective haecceitates. If it is reasonable to reject ‘radical haecceitism’ by which any object could be any other, it is also reasonable to reject radical quidditism that

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emerges from the postulation of contingent natural laws for the universals that constitute them. Indeed, if any universal could be nomologically connected to any other, then the mass property could be that which works as the spin actually works, or it could be what works as the electromagnetic charge actually works, or anything else. If these assumptions are too drastic, to begin with, one can try an exercise similar to that proposed for Genghis Khan and Napoleon Bonaparte. No matter how dissimilar two properties seem, with a sequence of many small modifications such properties can be completely inverted. We will have possible worlds indiscernible from the actual world about what things ‘seem to happen there’ and with exactly the same individual objects—each with its own haecceitas—but with a difference of quidditas so that, for example, the determinate properties of mass are there fulfilling the role that actually fulfill the determinate properties of charge and the determinate properties of charge are there fulfilling the role that actually meet the determinate properties of mass. But these identifications are absurd. It is reasonable, therefore, to suppose that the identity or difference of properties in different possible worlds must have to do with the distribution of causal powers that these universals confer. To these reasons against the quidditism—reasons that already have been exposed in several places, as it has been explained—a third one can be added. Those who have defended the contingency of natural laws for the universals that constitute them have been guided by the intuition that the same universal can be fulfilling different ‘nomic functions’ in different possible worlds, just as the same ‘nomic function’ can be satisfied by different universals in different worlds. In the traditional formulations of functionalism in the philosophy of mind,16 a ‘mental state of belief’—for example—could be realized in bases of very different nature provided that any of them is an instantiation of the same network of connections with other ‘states’. The ‘same’ belief could be realized in a computer, or a plurality of neurons connected synaptically to each other, or the population of China. Similarly, it has been thought that a disposition to do φ or the power to do φ must be understood as consisting of a categorical base and a natural law. The disposition will be the same if what works as ‘stimulus’ and what works as ‘manifestation’ are the same, although the categorical basis that is fulfilling such functions is variable. Thus, there is a property that is actually designated with the common name “mass”, but that property might not have been the property of mass if the natural laws were different. Expressions such as “mass”, “electromagnetic charge” or “spin” designate whatever it is that satisfies a specific nomic function—which depends on what natural laws exist in a world—but they are not expressions that rigidly designate the same basic categorical property.17 It is indispensable for this ‘functionalist’ conception of dispositions or ‘powers’ to This ‘functionalism’ should not be confused with the postulation of functional natural laws that has been made above (see § 47). 17 The analogy is obvious with what has been put forward for the contrast between proper names and definite descriptions. Although Aristotle is the tutor of Alexander the Great, the expression “the tutor of Alexander the Great” does not necessarily designate Aristotle, since Aristotle may not have been the tutor of Alexander the Great, but Aristotle could not have been different from Aristotle. The name “Aristotle” rigidly designates Aristotle in all possible worlds. The description “the tutor 16

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make sense that there is a clear difference between the ‘manifestation’ and the ‘stimulus’ of a disposition, on the one hand, and the ‘categorical’ property that is going to fulfill a certain nomic role, on the other. The manifestations and stimuli are what determine such a nomic role, which can be satisfied by different categorical properties. What, however, is the ‘manifestation’ of a disposition? It must be to cause something to be in a certain way; that is, it must be to cause some property to be instantiated. But it is assumed that the properties have a primitive quidditas. For example, suppose that in the world w1 the categorical universal U1 is a disposition to make a U2 be instantiated because, in w1, there is a natural law [N(U1, U2)]. In another possible world w2, on the other hand, the same universal U1 is a disposition to make a U3 be instantiated and it is not a disposition to the instantiation of an U2 as in w1, since in w2 there exists the natural law [N(U1, U3)] and there is no natural law [N(U1, U2)]. Suppose that neither U2 in w1, nor U3 in w2 are integrating other natural laws. The same categorical universal U1 satisfies different nomic roles in w1 and w2, but this difference consists of nothing more than the difference of quidditas between U2 and U3. It turns out, therefore, that there is no way to make the idea of a nomic role intelligible if it is not appealing to the properties’ quidditates that integrate the natural laws in question. What has been normally assumed when a quidditas has been postulated for the properties is that they are accessible to us epistemologically by their nomic roles, whereas the quidditas is an epistemologically opaque ‘addition’. The situation, however, is much worse. If there are primitive quidditates, then there is no way to access the nomic roles epistemologically. In the case presented, all the difference between the nomic role of U1 in w1 and w2 is the difference in quidditas between U2 and U3. If there is no epistemological access to the quidditas, then, there is also no epistemological access to a supposed ‘nomic role’ that a property could be satisfying contingently. It turns out, therefore, that the distinction between a ‘categorical’ property and the nomic role that is at the base of the functionalist theory of the dispositions or causal powers is not intelligible. ‘Nomic roles’ would be as epistemologically opaque for us as quidditates are. Extreme quidditism is, therefore, an implausible position. Natural laws must be considered essential for the universals that integrate them.

7.5.2

Necessary Laws

§ 52. Since natural laws, however they may be conceived, must be essential for the universals that constitute them, there are solid reasons to think that those laws must be necessary entities. If they are necessary entities, ontologically dependent on the universals that constitute them, then those universals must also be necessary entities. Bostock’s argument (see Bostock 2003) can be summarized as follows: all the of Alexander the Great”, on the other hand, does not designate him rigidly. In some worlds it designates Aristotle, but in others, it designates Diogenes the Cynic, for example.

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universals of a possible world—assuming that they are universals instantiated in that world, according to the restrictions of the Aristotelian—must belong to a ‘nomic network’. There is a single nomic network for every possible world. If the instantiations of a universal can interact with instantiations of another universal, then those universals must belong to the same nomic network. But the instantiation of any universal can interact with the instantiation of any other. Then, all universals belong to the same nomic network. This nomic network, then, exists invariably in all possible worlds, which is what we wanted to show. It will now be necessary to examine this argument with more detention. In a general way, it must be understood by a “nomic network” the closure of all and only universals nomologically connected between themselves. For example, suppose that [N(U1, U2)], [N(U2, Conj(U3, U4))] and [N(U4, U1)]. Suppose, also, that these are all the natural laws that the universals U1, U2, U3, and U4 constitute. All of them are connected nomologically, and there is no other universal connected nomologically with any of them. Each of these universals depends ontologically on all the others. For example, U1 depends ontologically on U2, since [N(U1, U2)], and it is essential for U1 to integrate [N(U1, U2)], as explained in the previous section. Like the natural law [N(U1, U2)] depends on its constituents, it turns out that U1 depends on U2, on which depends the natural law essential to U1. But, in turn, U2 depends ontologically for the same reasons on the conjunction of U3 and U4, [Conj(U3, U4)]. As a conjunction of universals depends on the conjunctively linked universals, it follows that U2 depends ontologically on U3 and ontologically depends on U4. But the ontological dependence is transitive, so it follows that U1 depends on U3 and depends on U4, etc. A nomic network, then, is a structure of universals mutually dependent with each other.18 The existence of one of these universals in a possible world makes necessary the existence of all the rest of the same nomic network. The absence of one of the universals of a nomic network makes that none of the others exist. A premise of importance for Bostock’s argument is that every possible world must have only one nomic network for all universals existing in such a world. The main motivation for this premise is that all the instantiations of the universals in a possible world must be able to interact with each other. For it to be possible for the instantiations of two different universals to interact with each other, it is necessary a nomic connection between these universals. Each of these universals essentially integrates certain laws. Such laws ground what would happen with an instantiation of that universal if it were subject to certain interactions. If two universals can interact with each other, then there are natural laws of which those universals are constituents that must include them both. They must belong, therefore, to the same nomic network. Conversely, consider the hypothesis that there were universals in the same possible world that did not belong to the same nomic network. These universals, for the reasons already indicated, could not interact with each other. Sometimes

18

There are several questions associated with a structure based on ontological dependencies of this type that will be dealt with in §§ 75–77 below.

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scenarios have been proposed in which, for example, there are different disconnected regions of spacetime, so that there are no causal connections between events in those different regions. A scenario of disconnected nomic networks would be much stronger. In a world with disconnected regions, instantiations of the universals instantiated in those regions can interact with each other, although they do not do so in the contemplated scenario. If there were disconnected nomic networks, that is, if there were two different nomic networks in the same possible world, it should not be possible that instantiations of the universals of these networks interact with each other. The universals of any of these networks would be metaphysically inaccessible to the universals of the other. A hypothesis of this kind would be extremely extravagant. We can see now that these same reasons by which all universals of the same possible world must belong to the same nomic network, also make that all universals whose instantiations can interact with each other must belong to the same nomic network. If there is a possible world in which the instantiations of a universal interact with the instantiations of another it is because there are natural laws that must include both universals which ‘regulate’ what would happen in such interactions. In a possible world in which, for example, there were no particles with negative electromagnetic charge, it is possible for objects with a positive electromagnetic charge to interact with objects with negative electromagnetic charge, and in that interaction, there are certain effects that would occur. The universal having positive electromagnetic charge must, therefore, belong to the same nomic network as the universal of having negative electromagnetic charge. All forces could interact with each other. For the effects due to electromagnetic forces, the effects of strong gravitational or nuclear forces can be superimposed. There must be a single nomic network that includes the universals corresponding to such forces. Note that for this, it is not required that the interactions between instances of different universals be ‘deterministic’. The natural laws in question can be stochastic. One reason to doubt this idea is that there could be universals whose instantiations were non-physical. If, for example, there were angels or demons of a purely spiritual nature, they would not have physical properties, but they would be instantiating certain universals. Should these universals integrate the same nomic network? I cannot see why not. It is assumed that spiritual creatures can intervene in the physical world. An angel that can destroy Sodom, or that can kill all the firstborn of Egypt in one night is an entity that can affect physical reality. So, whatever universals are instantiated by an angel, they must belong to the same nomic network as all other universals. Above it has been pointed out that the hypothesis of different nomic networks is extravagant, for the universals of these different networks would be modally disconnected from each other. These would be universals whose instantiations cannot interact with each other. But there is a stronger reason to reject the hypothesis of different nomic networks from a theistic perspective. God is the causal

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principle of everything that exists in any possible world.19 No possible object is causally disconnected from divine action. No object could be if God exists. This connection implies that all instances of universals must be able to interact with each other or interact with the instantiation of universals that must be able to interact with each other. From a theistic perspective, the hypothesis of different nomic networks is not only extravagant but simply unintelligible. This is for me sufficient reason to declare the hypothesis of different nomic networks incoherent. It turns out, then, that there are very good reasons to argue that there is a single nomic network of all universals. This nomic network must invariably exist in all metaphysically possible worlds. If there is a natural law, there must also be the universals on which those laws are dependent. The existence of a nomic network is the existence of all the universals that make up such a network. And these are all universals, without exception. It turns out, therefore, that universals must invariably exist in all possible worlds, whether they are instantiated in these worlds or not. This concludes the examination of the reasons that lead to postulate transcendent universals from the functions that must be fulfilled by universals in natural laws. It has been seen that the existence of functional laws that directly link determinable universals, the scenarios of the type proposed by Tooley—although I am not inclined to accept such scenarios in the precise manner in which they were proposed by Tooley—and the essential nature of natural laws for the universals that integrate them, are reasons to maintain that the principle of Instantiation is false. There are, then, non-instantiated universals.

References Adams, R. M. (1979). Primitive thisness and primitive identity. The Journal of Philosophy, 76(1), 5–26. Alvarado, J. T. (2005). Necesidad de origen y metafísica modal. Diánoia, 50(54), 3–32. Alvarado, J. T. (2006). Esencias individuales e identidad primitiva (J. Ahumada, M. Pantaleone, & V. Rodríguez, Eds.), Epistemología e Historia de la Ciencia 12, (pp. 31–36). Alvarado, J. T. (2007a). Esencias individuales e identidad primitiva. Analytica, 11(2), 155–195. Alvarado, J. T. (2007b). Cómo podrían ser las cosas. Nota crítica. Philosophica, 32, 203–219. Alvarado, J. T. (2008). Leyes naturales y universales trascendentes. In G. Agüero, L. Urtubey, & D. Vera (Eds.), Conceptos, creencias y racionalidad (pp. 65–71). Córdoba: Brujas. Alvarado, J. T. (2010). Laws of nature, modality, and universals. Epistemologia, 33, 255–282. Aristotle, Metaphysica (1998). Metafísica. Introducción, traducción y notas de Tomas Calvo Martínez. Madrid: Gredos.

19

Some philosophers have argued that God would be an entity that exists in certain possible worlds and not in others. This is not the way God has been traditionally conceived for very good reasons. It is not possible to enter here into this question, but it is assumed that God is an entity a se—as it has been explained above—that is, that it is a concrete entity such that it is necessary that, if it exists, it necessarily exists, and the only entity with these characteristics. That is, it is: [(ιx)⃞((x exists) ! ⃞(x exists))].

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Armstrong, D. M. (1978). Universals and scientific realism, Volume II, A theory of universals. Cambridge: Cambridge University Press. Armstrong, D. M. (1983). What is a law of nature? Cambridge: Cambridge University Press. Armstrong, D. M. (1989). A combinatorial theory of possibility. Cambridge: Cambridge University Press. Armstrong, D. M. (1997). A world of states of affairs. Cambridge: Cambridge University Press. Bird, A. (2007). Nature’s metaphysics. laws and properties. Oxford: Clarendon Press. Black, R. (2000). Against Quidditism. Australasian Journal of Philosophy, 78, 87–104. Bostock, S. (2003). Are all possible laws actual laws? Australasian Journal of Philosophy, 81, 517–533. Chisholm, R. M. (1967). Identity through possible worlds: Some questions. Noûs, 1, 1–8. Dretske, F. (1977). Laws of nature. Philosophy of Science, 44, 248–268. Ellis, B. (2001). Scientific essentialism. Cambridge: Cambridge University Press. Ellis, B. (2009). The metaphysics of scientific realism. Durham: Acumen. Lange, M. (2009). Laws and lawmakers. Science, metaphysics, and the laws of nature. Oxford: Oxford University Press. Lewis, D. K. (1973). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Mackie, P. (2006). How things might have been. Individuals, kinds, and essential properties. Oxford: Clarendon Press. Molnar, G. (2003). Powers. A study in metaphysics. Oxford: Oxford University Press. Mumford, S. (1998). Dispositions. Oxford: Oxford University Press. Mumford, S. (2004). Laws in nature. London: Routledge. Prior, E. W., Pargetter, R., & Jackson, F. (1982). Three theses about dispositions. American Philosophical Quarterly, 19, 251–257. Schaffer, J. (2005). Quiddistic knowledge. Philosophical Studies, 123, 1–32. Shoemaker, S. (1980). Causality and properties. In P. van Inwagen (Ed.), Time and cause (pp. 109–135). Dordrecht: Reidel. Reprinted in Identity, cause, and mind. Philosophical essays (pp. 206–233). Oxford: Clarendon Press, 2003. Swoyer, C. (1982). The nature of natural laws. Australasian Journal of Philosophy, 60, 203–223. Thomas, Aquinas Saint, De ente et essentia (2002). L’ente e l’essenza. Introduzione, traduzione, note e apparati di Pasquale Porro. Milano: Bompiani. Tooley, M. (1977). The nature of laws. Canadian Journal of Philosophy, 7, 667–698. Tooley, M. (1987). Causation. A realist approach. Oxford: Clarendon Press. Tugby, M. (2013). Platonic dispositionalism. Mind, 122, 451–480. Van Fraassen, B. (1989). Laws and symmetry. Oxford: Clarendon Press.

Chapter 8

Transcendent Universals and Ontological Priority

Abstract Universals are postulated to satisfy certain explanatory work. They partially ground the nature that objects have. They partially ground objective resemblances between objects. They partially ground causal powers. In effect, objects are as they are, because they instantiate certain universals. At the same time, the Aristotelian maintains that universals require instantiations to exist, i.e., universals are grounded in their instantiations. It is argued in this chapter that the grounding profile attributed to universals by Aristotelians is incompatible with the general grounding profile that any universal –either Platonic or Aristotelian– should have. § 53. Two major lines of defense of transcendent universals have already been considered, which have to do with the functions that universals must satisfy in an actualist modal metaphysics and a non-Humean metaphysics of natural laws. In both cases, it is not a single argument for transcendent universals, but a set of considerations that make it more reasonable to postulate transcendent universals than to postulate immanent universals. In this third line of argument, we also face considerations that make it more reasonable to postulate transcendent universals, but in this case, concerning issues of ontological priority. The central idea is that Aristotelian universals invert the relationships of ontological priority between universals and particulars. There are certain theoretical functions that a universal must satisfy, and such theoretical functions are only compatible with a ‘profile’ of grounding and dependence that the immanent universals do not satisfy. In what follows in this chapter, therefore, it will be discussed: (i) the theoretical functions that a universal should satisfy to ground the characteristics of particular objects (§§ 54–57); (ii) the ontological priority relations that immanent universals have with their instantiations (§ 58), and (iii) why only transcendent universals satisfy the theoretical functions that universals are supposed to satisfy (§ 59).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_8

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8 Transcendent Universals and Ontological Priority

The Ontological Priority of Universals

§ 54. Universals are entities that have been postulated to satisfy a plurality of theoretical functions, as explained above (see §§ 10–15). They are what makes that a plurality of objects share the same nature. They are what determine ontologically that the same object has different properties. They are what determine the objective similarities between objects and the systematic relations between determinable and determinate properties. They are the constituents of natural laws and confer the causal powers that can be attributed to an object. When we talk about “what makes something” or “what determines something” we talk about certain relationships of ontological priority that must be specified. It will be convenient, then, to examine in detail what kind of priority is assigned to universals with respect to the particular objects in each of these theoretical functions. Recall that there are two types of priority—grounding and dependence—as indicated above (§ 4). Something can be grounding-fundamental, but it can also be dependence-fundamental. Priority in one of those senses does not have to imply priority in the other.

8.1.1

Priority to Natures

§ 55. Universals must fulfill an explanatory function with respect to the natures of objects. In general, if x is F, for any x and any F, it is because x instantiates a universal U. For example, if the character in question is to have a mass of n gr, then there is a universal of having a mass of n gr correlative that x must be instantiating. What is indicated here for the nature of an object is valid for the community of nature of a plurality of objects—the so-called problem of the ‘one over many’ (see § 10). The fact that the objects x1, x2, . . ., xn have the ‘same nature’ must be determined by the fact that there is a universal U such that: x1 instantiates U, x2 instantiates U, . . . and xn instantiates U. And what applies to a plurality of objects with the same nature also applies to a single object that possesses different natures—the problem of the ‘many over one’ (see § 11). The fact that x is, for example, a perfect sphere with a mass of 10 gr must be determined by the fact that x instantiates the universal of being a sphere and also instantiates the universal of having 10 gr of mass. The difference between the universals of being a sphere and having 10 gr of mass determines the plurality of natures in the object. When it comes to the ontological explanation of the natures of things, there is a grounding relation between the instantiations and the characteristics of objects. More precisely, the fact that x is F is grounded in the fact that x instantiates a universal U. The fact that x1, x2, . . ., xn are all F is grounded in the fact that each of x1, x2, . . ., xn instantiate a single universal U. The fact that x is F and is G is grounded in the fact that x instantiates a universal U1 and also instantiates a different universal U2. Note that the ground for each of these facts, that is, that which is ‘constitutively sufficient’ for the obtaining of each of those facts, is not simply the existence of the universal U,

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or the existence of the universals U1 and U2, as the case may be, but the instantiation of the universal or the universals in the object. For the time being, it will not be necessary to consider more carefully the nature of the ‘instantiation’ relation, but this issue will be discussed below (see §§ 69–70). It is clear, however, that universals are essential constituents of the facts of instantiation, for nothing can be an instantiation of the universal U, without the universal U. Universals, then, are a partial ground of the natures of objects, because the full ground of the fact that x is F is the plurality of facts that x exists, U exists, and x instantiates U. Of course, not in every case in which there is ontological dependence, there is also partial grounding. But there is for facts of instantiation. Universals not only partially ground the natures of things, but they are also something that these natures depend on ontologically. There is, therefore, an ontological priority of dependence of universals with respect to the natures of things.

8.1.2

Priority to Resemblances

§ 56. Universals must fulfill an explanatory function with respect to the objective similarities between objects and the systematic relations between determinate and determinable properties (see § 12). The fact that two or more objects are similar—or not—to each other is determined ontologically by the universals that such objects are instantiating. The instantiations of universals are the ground of similarities or dissimilarities. Just as for the explanation of the natures of things, universals partially ground the facts of resemblance or dissimilarity, since what is ‘constitutively sufficient’ for these facts are the instantiations of universals. Universals are essential members of the facts of instantiation. There is, then, a grounding-priority of universals regarding similarities and dissimilarities.

8.1.3

Priority to Natural Laws and Causality

§ 57. Another field in which explanatory functions have been attributed to universals has to do with natural laws and causal powers. As has already been explained at length, the best conception of natural laws is as relations of higher-order of necessitation between universals or as the essential causal powers that a universal confers on its instantiations. In the first case, it must also be assumed that it is essential for the universals that integrate these natural laws to be integrating them. There may be some doubt here as to whether universals should identify with natural laws—in which case they would weakly ground laws—or whether they strictly ground such laws. If natural laws are understood as higher-order nomological relations, then it is clear that universals would strictly ground such laws. If natural laws are simply universals that confer causal powers on their instantiations, then there would be weak grounding due to the identity of the ground and the grounded. In any case, it is clear that universals must be grounding-prior to natural laws. The same

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considerations serve to justify the priority of dependence of universals on natural laws. Just as the laws have to be grounded on the universals that integrate them, they must also depend ontologically on them. This dependence will be strict or weak, depending on how the natural laws are understood, but this does not change the point. The priority of universals to causal relations requires greater caution, since there are large differences according to which conception of causality is adopted. As has been done above, the question will be considered concerning three types of theories of causation (see § 13): regularity theories, counterfactual theories, and non-reductivist theories. In the theories of regularity, the causal connections between particular events are facts about the regularity with which types of events occur in the possible world in question. Event c of type C causes event e of type E if and only if every event of type C is followed by an event of type E with the other usual conditions.1 The nature of the event c by which it is of type C and the nature of the event e by which it is of type E are grounded partially on the universals instantiated in c and e.2 The regularity by which every event of type C is followed by an event of type E is also partially grounded by the universals in question. Recall that an ontology of universals is assumed from the outset, by which all nature of an object must be grounded on the universals that such object instantiates. It is not difficult to see that both regularities and singular causal relationships are also ontologically dependent on the universals instantiated by the objects involved in such causal relationships.3 In the counterfactual theories of causality, the causal relationship between the events c and e consists in the fact that the following counterfactual conditionals are true: if c did not exist, then e could not exist, and if c existed, then e could exist. A counterfactual conditional such as if p were the case then q could be the case is true in the possible world w if and only if in all possible worlds closest to w in which p is true, q is also true. The metric of ‘closeness’ between possible worlds is determined by the similarity or dissimilarity between these worlds. In this way, the truth of the counterfactual conditionals and correlative causal relations are grounded on (i) what properties are instantiated in the different possible worlds, and (ii) how similar or dissimilar these possible worlds are to each other. It is clear that in an ontology of universals (ii) is grounded on (i). It is decisive, then, which universals are

1

To these requirements it is added that the event cause and the event effect must be spatiotemporally contiguous and that the event cause must precede in time the effect event (see Psillos 2002, 19). For the discussion that is made here, these later requirements are not relevant. 2 An event, from the perspective of an ontology of universals, is essentially the instantiation of a universal or of several universals in one or several objects in a spacetime region. 3 In § 13 above, the explanatory requirements that the regularist conception of causality imposes on property metaphysics have been explained. What was argued there is that the regularist theory does not impose important restrictions. It is compatible with the most liberal forms of nominalism. What is considered here is the problem about what function universals would have for causal relationships understood in the way they are conceived under the regularist perspective, assuming already an ontology of universals.

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instantiated in the different possible worlds, since what makes the possible worlds more or less similar to each other is their ‘qualitative’ character. The instantiation of universals grounds these similarities, so that the instantiated universals partially ground such similarities. These similarities between possible worlds are also ontologically dependent on the universals instantiated in the different worlds.4 The third type of conception of causality is non-reductive. In these perspectives, causal facts are not reduced to other ontologically more basic facts. As explained above, it is these conceptions that impose more serious requirements on the metaphysics of properties. Curiously and, perhaps, due to the autonomy that is granted to causal facts in this perspective, the priority of universals over causality is, in this case, less marked. The fact that an object instantiates a universal, although it grounds in the object a set of causal powers, does not automatically determine the causal interactions in which that object will enter. Causal interactions will be determined by what other universals are instantiated in other objects in its ‘neighborhood’. One could argue that the causal interactions in which objects enter are grounded on the distribution of instantiations of universals in spacetime but in many cases everything that guarantees that distribution is only objective probabilities of interactions, i.e., a space of possibilities of causal interaction.5 There is no doubt, on the other hand, that the causal interactions depend ontologically on the distribution of instantiations of universals and, with it, on the instantiated universals. It turns out, therefore, that in some cases universals partially ground causal relationships—if the laws that these universals ground are deterministic—but in many other cases they do not. Universals are not always grounding-prior to causality, in this perspective. In all cases, however, universals are dependence-prior to causal relationships, since these relationships depend ontologically on universals. In conclusion, then, universals are both grounding-prior and dependence-prior to natural laws. When it comes to causality, there is also grounding-priority and dependence-priority of universals to causal connections, but only on the regularist and counterfactual conceptions of causality. When it comes to non-reductivist conceptions, although there is dependence-priority of universals, not in all cases there is grounding-priority.6

4 In § 13 above, it has been explained that this conception of causality only requires entities that can function as ‘authentic properties’, but does not seem to require the postulation of universals. It is more demanding as far as it concerns the metaphysics of properties than regularistic theories, but less so than non-reductivist theories. Here, on the other hand, what is at issue is the function that universals would have in counterfactual theories, already assumed an ontology of universals. 5 This also concerns the causal interactions caused by the exercise of the free will of rational agents, assuming that freedom is incompatible with determinism. Whatever the properties whose instantiation makes an agent free, what those properties do is to determine a space of open possibilities of deliberate action. 6 It turns out, then, that there are causal connections that are ‘emergent’ entities, being ungrounded, but ontologically dependent —according to the sense explained above. This happens whenever we are faced with stochastic causal processes and the decisions of free agents.

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After this examination, it can be said that there is a profile of ontological priority that a universal must satisfy. This profile does not have to do with the fact that universals are immanent or transcendent. If there are reasons to postulate the existence of universals of any kind, it is due to the theoretical functions that these universals must satisfy. Universals must ground the natures of things—and, with it, the ‘one over many’, and the ‘many over one’—the objective similarities, natural laws, and causal connections, at least if they are understood from the perspective of regularist or counterfactual views. Universals must also be dependence-prior to the natures of things, objective similarities, natural laws, and the causal connections since all these entities should be ontologically dependent on universals.

8.2

The Priority Profile of Immanent Universals

§ 58. It is important to consider here what priority profile an immanent universal should have and whether it would be compatible with the priority profile that any universal must have, as explained above. Since immanent universals only exist if they possess instantiations, although it is not necessary that they are instantiated in one object rather than in another, immanent universals are generically dependent on their instantiations and are grounded on their instantiations. As they are not ontologically dependent on specific instantiations, immanent universals would be a case of a ‘multiply realizable’ entity, that is, of independent but grounded entities. The grounding in question here is strict and total. It turns out, therefore, that immanent universals must be grounding-posterior to their instantiations, although generic dependence does not seem sufficient for dependence-posteriority. If this priority profile of immanent universals is conjoined to the general priority profile for all universals, several situations merit reflection. In the first place, universals partially ground the nature of things, which is fully grounded on the instantiation of that universal, but the universal, in turn, must be fully grounded on such instantiation. Secondly, the natures of things depend ontologically on the universals that are being instantiated by those things, but at the same time, those universals depend generically on such instantiations. It can be argued that the second ‘circle of priority’ is intelligible, but the first is not. The difficulty here is basically that circular structures are not acceptable either for grounding or for dependence. The dependence profile of immanent universals with respect to their instantiations does not amount to a vicious circle of dependence, but the grounding profile of immanent universals with respect to their instantiations is viciously circular and, therefore, incoherent. The situation of ontological dependence must first be considered. As has been explained, dependence has been taken as a strict order, that is, as an irreflexive, asymmetric, and transitive relation (see § 4). There are important reasons for attributing these characteristics to dependence since nothing depends on oneself. It is assumed that ontological dependence is a notion that describes a form of priority and, correlatively, of ontological posteriority. Dependence must also be transitive. In

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the conception that has been followed here, the fact that x is dependent on y consists of the fact that y is part of the essence of x. That is, x would not be the entity that it is without the existence of y. Suppose that x is dependent on y, and y is dependent on z; how could x exist without z also existing? If the dependence of y on z obtains, then y would not exist without z. But without y neither could exist x, if the dependence of x on y obtains also. Given irreflexivity and transitivity, asymmetry follows.7 From the Aristotelian perspective, the fact that objects possess the nature they possess is ontologically dependent on the universals they are instantiating. These objects would not have the nature that they have if the universals in question did not exist, but the universals—on their part—depend generically on having some instantiation or another. In principle, it seems that a situation of this kind should not be a case in which universals or natures of objects depend on themselves,8 but a closer examination shows that the issue is not so simple. Suppose in the first place a possible world w1 in which the universal U only has a unique instantiation in the object x. The nature of x depends on U, but—given the distribution of instantiations of w1—U seems to depend on being instantiated in x. Is not this a circle of dependence, even if it is contingent? The problem would be that, as regards the dependence of immanent universals, although these are not necessary dependency circles, they are metaphysically possible. But this possibility is as unacceptable as the necessary existence of circles of dependence because the irreflexivity of dependence is not a merely contingent feature of this ontological relation. If dependence is irreflexive, then it is necessarily irreflexive. This problem could be evaded, however, considering the modal nature of the generic dependence. In the possible world w1, the universal U would not exist if it were not instantiated in x but, also, if it were not instantiated in any other object. Thus, although in w1, in fact, U is only instantiated in x, its existence is not ‘connected’ essentially to x, as is the nature that actually has x, since U would exist, even if it were not instanced in x if it were instantiated in some other object. There is no such thing as a ‘circle of contingent dependence’ since the notion of dependence is already an irreducibly modal notion. Although in a specific possible world is a single object that guarantees the existence of a universal, this does not make that object of the essence of the universal. Suppose now the possible world w2 in which the universal U is instantiated in the objects x1 and x2, and no other. The natures of objects x1 and x2 depend on U. U depends generically on some instantiation. Given the distribution of instantiations in w2, it cannot be argued that U rigidly depends on x1, or rigidly on x2. If it were not instantiated in x1, it would still be instantiated in x2. If it were not instantiated in x2, it would still be instantiated in x1. But here seems to be generated another form of dependency cycle. U could be instantiated in x2 only and not in x1. Instantiation in x2 7

In effect, suppose an irreflexive and transitive relation R. Suppose Rxy and also Ryx, in violation of asymmetry. Then, by transitivity, it would follow that Rxx, which violates irreflexivity. 8 Indeed, if the universal U depends on the nature of x, and the nature of x depends on U, from transitivity it follows that U depends on itself, which conflicts with the irreflexivity of dependence. On the other hand, if the nature of x depends on the universal U, and the universal U depends on the nature of x, then by transitivity it follows that the nature of x would depend on itself.

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is that on which U is then dependent, and then, the nature of x1 is dependent on U. There is no reciprocal dependence on the same items. The problem is that the same could be said about x2. U does not depend on x2 because it suffices the instantiation in x1 and then, the nature of x2 is dependent on U. It turns out then that the nature of x1 seems to depend on the nature of x2, whereas the nature of x1 seems to depend on the nature of x2. In effect, it seems that the nature of x1 depends on U and depends on its instantiation in x2, and vice versa. It seems to be a circle of dependence. Note that this case is generalizable to any possible world in which a universal has n instances. All those n instantiations would be mutually dependent on each other. In this second type of case, however, the modal character of the generic dependence seems to offer an escape for the Aristotelian. When the existence of U is guaranteed by its instantiation in x1 but is not instantiated x2, a variation does not occur in the dependence profile of U. U does not change from being generically dependent on any instantiation to being rigidly dependent on its instantiation in x1. The existence of U would be guaranteed by its instantiation in some other object different from x1 and different from x2. This can be generalized to n objects, of course. As it has been possible to appreciate, therefore, the dependency profile of immanent universals seems intelligible. The situation radically varies when it comes to the grounding profile of immanent universals. Universals partially ground the natures of the objects, because they are a necessary constituent of their instantiations, which fully ground the natures of the objects. The fact that an object instantiates a universal, however, fully ground the existence of the universal instantiated. If the question is considered superficially, it would seem that the risk of a circle of grounding, in this case, can be avoided because the universals only partially ground the natures of things, while the instantiations fully ground universals. Does not this generate a situation similar to that which has been seen with respect to dependence due to the contrast between generic dependence and rigid dependence? Unfortunately for the Aristotelian, it is not. It will be necessary to examine this issue with detention. The existence of a universal, from the Aristotelian perspective, is grounded on the fact that this universal has at least one instantiation. Instantiations are ‘constitutively sufficient’ for the existence of universals and, therefore, are its ground. There is, then, a grounding-priority of the particular instances with respect to the universals of which these instantiations are instantiations. Here, the weirdness of the theoretical situation that occurs with the postulation of immanent universals must begin to become apparent. How can it be, in effect, that the instantiations of a universal are the ground of the universal of which they are instantiations? The natures of things are grounded on the instantiation of universals. It is obvious that the instantiation of a universal is not such without the universal of which it is an instantiation. But the universal that has to be that which is to be instantiated is grounded on that same instantiation. In other words, the instantiation of a universal U on the object x cannot exist without U, but it happens that the same instantiation fully grounds the existence of U. What ‘makes’ U to exist is something that U has contributed to ‘make’ to exist. This seems simply impossible. Perhaps this theoretically uncomfortable situation can be remedied using a strategy analogous to that considered with respect to the alleged circles of

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dependency indicated above. Suppose that in a possible world w there are two instances of U in the objects x1 and x2. Any of these instantiations would be sufficient to ground the universal U. That is, if U were not instantiated in x1, it could exist if it were instantiated in x2. If U were not instantiated in x2, it could exist if it were instantiated in x1. One could perhaps argue that in a case like this U is not ‘constitutively connected’ with the instantiation in x1—because the instantiation in x2 would suffice—nor with the instantiation in x2—because the instantiation in x1 would suffice. In possible worlds such as w, universals would not be really grounded on each particular instantiation. It is sufficient that there is more than one instantiation to generate a situation of this type—or so it seems. It happens, however, that the fact that in w the existence of the universal is—to put it in some way— ‘overdetermined’ by several instances does not prevent each of these instances from being a strict and total ground of the universal. For the cases of apparent circles of dependence, it was crucial that a universal does not become rigidly dependent on an instantiation in an object, even though that instantiation is the only one that it possesses in a possible world. What happens in a possible world does not change its essence and, with it, the entities with respect to which it has—or not—ontological dependence. When it comes to grounding relations, it also happens that the vicissitudes that occur in a possible world do not change the essence of the entities involved. The irreducibly modal nature of the grounding relation implies that the ground of something does not change its character as ground by the existence—or not—of other entities that are also a ground. Note that the priority circle that is generated in this case is not a simple circle of grounding. The problem is not that the instantiations of a universal U grounds U and the universal U grounds its instantiations, because no universal grounds its instantiations. A universal partially grounds the natures of things whose full ground is the instantiation of that universal. The question is that the instantiation of the universal U is ontologically dependent on the universal U. According to the Aristotelian point of view, the instantiations of the universal U are grounding-prior to U, but U is dependence-prior to its instantiation. And it is this structure that is impossible. It has been explained above how there may be cases in which there is grounding-priority without there being dependence-priority (see§ 4) or vice versa. Thus, an “emergent” entity is one which, although it is dependent, is not grounded on anything. One could also call an entity grounded, but independent of any of the entities on which it is grounded, as “realized” on them. These hypotheses are coherent. It is crucial, however, that in none of these situations do we have two items in which the priority of grounding and the priority of dependence are crossed in opposite directions. If, for example, x is emergent with respect to y, it is because x depends on y, but is not grounded on y. If, for example, x is ‘realized’ in y is because x is grounded on y, but does not depend on y. What happens in the case of Aristotelian universals is that they are grounded on their instantiations, but their instantiations depend, in turn, on them. This is an unacceptable priority circle: the instantiation of U in x depends on U, since without U it would not exist and is, therefore, derivative with respect to U, but then it turns out that the instantiation of U fully and strictly grounds the existence of U,

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since it ‘makes’ U to exist, so U is derivative with respect to its instantiation. It seems then that U is derivative with respect to itself, which does not make any sense.

8.3

Immanent Universals, Similarities, Laws, and Causality

§ 59. It will help to understand the theoretical situation that is generated by the postulation of immanent universals to consider the priority profile of these universals with regard to objective resemblances, natural laws, and causal relations. Some precise theoretical functions of universals are also expected in these fields that immanent universals should be able to satisfy. It is assumed that universals should have priority over facts of resemblance, natural laws, and causal connections—with the proper cautions when it comes to non-reductivist conceptions of causality. It has already been shown how the facts of similarity are grounding-derivative and dependence-derivative on universals. It seems that—at least, at first glance—the problems indicated in the previous section can be set aside when it comes to facts of similarity. After all, the fact that x resembles y must be grounded on the fact that, for example, both x and y instantiate the same universal U. What has been shown is that the priority profile that results for instantiations of Aristotelian universals is incoherent, which is prior to any fact of similarity. But suppose the hypothesis that, in some way—per impossibile—, the instantiation of immanent universals could be made coherent. What will be shown here is that problems would still be generated for the intelligibility of similarity facts. It is characteristic of an Aristotelian position that the order of ontological priority of universals with respect to their instantiations is inverted since universals must be grounded on their instantiations. If the coherence of the instantiation of immanent universals were to be saved somehow, at least this should be preserved: that instantiations are prior to universals. What would then be the priority profile of resemblance facts? What results is that universals would become ‘explanatorily inert’ with respect to the facts of resemblance. Indeed, similarities must be grounded on the instantiation of the same universal. Universals must also be grounded on their instantiations. What role would universals play for similarities? None. They would be perfectly idle.9 It is important to note that in such a scheme, universals would not be fulfilling the functions that, in principle, they are supposed to satisfy in an ontology with universals. They are not what determine objective similarities, not even partially. In what has to do with the facts of similarity, the situation is analogous to that of some ontologies of tropes—in which a primitive intrinsic qualitative character is attributed to them—than to an ontology of universals.

9 Recall, moreover, that the dependence of instantiations on universals of which they are instantiations is what generates the coherence problem explained in the previous section. Here this dependence is put “into parentheses” to consider—per impossibile—what profile of priority should be assigned to the facts of similarity from an Aristotelian perspective.

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If one considers now what happens with natural laws, it turns out that these should be grounded on particular instantiations. In effect, natural laws are grounded on the universals that integrate them—or on the universals in which they consist, as the case may be— but those universals must be grounded on their instantiations. By transitivity, it follows that natural laws must be grounded on such particular instantiations. There has already been an occasion to explain how this goes against much of what is expected from a reasonable ontology of natural laws (see § 51) and, especially, from an ontology of natural laws that is not Humean. The value of the theories of natural laws that understand them as universal or as structures of universals is that it makes explainable why there is a distribution of instantiations of these universals. These gains are lost here, for if natural laws are ultimately grounded on their instantiations, then they cannot be expected to have a ‘regulative’ function of such particular instances. Things do not improve when one considers the priority profile that results in causal relations. For the reasons that have already been exposed regarding facts of similarity, it is no longer universals that must be assigned priority with respect to regularities of events or counterfactual dependencies, but rather to particular instantiations. The universals are here again explanatory idle. In the case of non-reductivist theories of causality, more precautions must be taken, as has been explained, but to the extent that ontological priority should be assigned to universals, the same problems appear. What results, then, is that, at best, an ontology of immanent universals leads to the explanatory irrelevance of universals. A theory that, on the one hand, pretends to postulate entities that must satisfy certain explanatory functions while, on the other, makes such functions impossible, cannot be correct. Aristotelianism is shown as the pretension of, at the same time, adopting universals but, without assigning them the theoretical functions that they must possess. These would be ‘epiphenomenal’ universals, so to say. The instantiations of universals are in a theory of this type what grounds the natures of objects, objective similarities, natural laws, causal connections, and even universals themselves. If things were like that, it would be legitimate to ask whether there really are reasons to postulate universals in the first place. It is assumed that the justification for postulating universals is that they can explain better than all their alternatives the natures of objects, objective similarities, natural laws, and causal connections. If it turns out that particular instances satisfy these explanatory tasks, then there would be no need for universals.10 It turns out, then, that the ontological priority profile that should be assigned to an immanent universal is incompatible with the general priority profile that any universal must have. Even if the coherence problems were corrected, immanent 10

Remember that one is here examining the priority profile that should be assigned to Aristotelian universals by putting ‘into brackets’ the ontological dependence of instantiations of universals with respect to the universals of which those instantiations are instantiations. Although this type of ontological dependence is obvious, it must be placed here ‘in parentheses’ because otherwise, the coherence problem pointed out in the previous section arises (§ 58). What results is that instantiations would have grounding and dependence priority.

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universals would not be satisfying the theoretical functions they are supposed to fulfill for objective similarities, natural laws, and causal relations. If universals must be postulated, it is reasonable —therefore— to postulate universals transcendent and not immanent. The same reasons that have been adduced to postulate universals, in general, are reasons to postulate transcendent universals. This concludes the ‘positive’ arguments for transcendent universals. Three major lines of consideration have been presented for the acceptance of transcendent universals that have to do: (i) with the functions that transcendent universals must fulfill in an actualist modal metaphysics (Chap. 6), (ii) with the functions that transcendent universals must satisfy in a non-Humean metaphysics of natural laws (Chap. 7), and (iii) with the requirements that emanate from the profile of ontological priority that a universal must satisfy, that is incompatible with the priority profile of immanent universals (Chap. 8). It turns out, then, that, if there are reasons to admit the existence of universals, there are reasons to admit the existence of transcendent universals.

Reference Psillos, S. (2002). Causation and explanation. Montreal: McGill-Queen’s University Press.

Chapter 9

Objections Against Transcendent Universals

Abstract Many objections have been directed against Platonic universals. These objections have seemed so grave that many philosophers are simply closed to consider any positive argument in favor of transcendent universals. This chapter considers the most often referred to objections: (i) the problem of causal powers, (ii) the lack of economy of Platonic universals, and (iii) the epistemological problem about how can one access a transcendent realm of universals. According to the ‘Eleatic principle’ everything should modify the causal powers of something. Supposedly, transcendent universals don’t, so they cannot be admitted into existence. It has also been argued that transcendent universals are less economical than immanent universals because an ontology of transcendent universals requires additionally ‘instantiations’ of universals. In the third place, it has been said that transcendent universals are epistemologically transcendent to our cognitive powers. It is argued in this chapter, nevertheless, that none of these objections is compelling. All causal powers depend on universals. Platonic universals are not less economical than Aristotelian universals, because the introduction of ‘instantiations’, ‘tropes’, ‘modes’ or primitive ‘states of affairs’ is a general requirement for any ontology of universals and particulars. Platonic universals are, finally, not epistemologically transcendent. They are known by their instantiations. § 60. As indicated above (see § 40), the justification of transcendent universals would be incomplete if the most usual criticisms that have been made to them since Classical Antiquity are not examined. Since, at least, Aristotle, it has been thought that Platonic universals would be a kind of unnecessary ‘duplication’ of reality.1 They would be entities without clear theoretical functions for the explanation of the natures of particular objects and their causal powers. The disconnection of these universals from the particular objects with which we have ordinary cognitive contact would make it difficult to understand how we might come to know of their existence. 1 For example, Metaphysics A, 9; Z, 13. Apart from the well-known argument of the ‘third man’, it has been argued that Platonic universals do not fulfill any causal function, that they ‘duplicate’ reality and that they cannot be common to many objects in a coherent way.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_9

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For many philosophers, these objections are of such weight that the acceptance of transcendent universals would be implausible. Any argument in favor of such universals, if convincing, should be treated as a ‘paradox’, that is, as an apparently correct argument that leads to an inadmissible conclusion. The theoretical task would be to discover what is the problem with that argument to detect what is the source of the error and what makes it appear to be correct. In this section, it will be shown that the motives traditionally presented against transcendent universals are false. The theoretical ‘costs’ of admitting Platonic universals are much lower than it seems at first sight. Neither are they causally disconnected entities, nor are they unknowable entities, nor are they entities whose introduction results in a gross violation of theoretical economy with respect to the postulation of immanent universals. It will be noted that among the traditional motives against the universals that are going to be discussed here will not be the famous argument of the ‘third man’. The reason for this omission is that, despite being an argument of venerable antiquity, it can easily be blocked (see Armstrong 1978, 71–72). Indeed, the argument of the ‘third man’ depends on two crucial assumptions: (i) all that is F, is F because it participates in the form of F, and (ii) the form of F is an F. In traditional terminology, this is formulated saying that ‘the F itself’ (autón) is an F. Since all that is F, is F because it ‘participates’ of a form, it follows that ‘the F itself’ can only be F because it participates of a second form of being F. Besides the particular object that is F and ‘the F itself’, the introduction of a ‘third’ F is necessary. This third F must be an F for the same reasons, which generates a vicious infinite regress.2 No contemporary advocate of transcendent universals, however, is willing to admit that the universal F is an F.3 If, for example, there is the universal to have a spherical shape, it is not itself spherical. The premise (ii) must be rejected and, with it, the vicious regress does not arise in the first place. There has also been an argument known as “the restricted third man” that follows the general lines of the old ‘third man’ but considering only the universal to be universal (see Armstrong 1978, 72–75). This argument can also be easily blocked. Although all universal of being F is not an F, it is a universal. So, if all universals are universal because they instantiate being a universal, it turns out that the universal of being universal can only be universal if, in turn, there is a previous universal of being a universal. At this point, one could argue that the universal of being a universal

2

It is a vicious infinite regress because the fact that the particular object x is F ontologically depends on ‘the F itself’. This form of F must be an F so that it can be that on which depends the fact that x is F. But ‘the F itself’ depends, in turn, on a form of F of which it participates. There is a chain of dependent entities, without any of them being ontologically basic, independent. 3 One exception is the conception defended by philosophers like Zalta (1988) in which it is true that the Platonic universal ‘F-ness is F’. But this predication is true not because the Platonic universal instantiates the property of being F, but simply because that universal ‘codifies’ the property of being F, i. e., what the predication says is that everything that instantiates F-ness is an F.

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self-instantiates, but this would be a form of reflexive dependence.4 If, on the other hand, an infinite hierarchy of universals of being universal is postulated, each of which explains the universal character of the universals that are lower in the hierarchy, a vicious regress is generated.5 What a defender of universals must uphold, however, is that no universal is a universal because it instantiates a supposed universal of being a universal. Recall that universals are authentic properties. The existence of a predicate or a concept is not enough for the existence of a correlative universal. Thus, although it is true to say of all universal that “it is a universal”, universals do not instantiate a universal to be a universal. Every universal has an intrinsic character by which it is a universal, but this intrinsic character or nature is not grounded on the instantiation of universals. Compare what happens with a substratum. By definition, a substratum or ‘thin’ particular is that which instantiates the properties of an object and which cannot be instantiated in anything (§ 80). Every substratum must have a nature by which, in effect, it can instantiate properties and cannot be instantiated—because it is not a property. This intrinsic nature of a substratum, however, is not grounded on the fact that substrata require a universal to be substrata (see Sider 2006). If this were so, the mere idea of a substratum would be incoherent, because only something that instantiates the property of being a substratum can be something that, considered in itself, with abstraction of the properties of the object, does not possess any property. In the case of universals, we are faced with an analogous situation. The fact that a universal is a universal is an ontologically basic fact, which is neither grounded on anything nor depends on anything. In this way, there is no universal to be a universal, and the ‘restricted third man’ is blocked. The difficulties that will be addressed here are: (i) the problem that arises from the eventual lack of ‘causal powers’ of transcendent universals (§§ 61–66); (ii) the problem that arises because whoever proposes transcendent universals must also postulate tropes, which seems theoretically uneconomic (§§ 67–70); and (iii) the problem that arises from the epistemologically ‘transcendent’ character of the Platonic universals (§§ 71–74). Many, indeed, are inclined to think that we have no epistemological access to universals of this nature.

4

In effect, the universal of being a universal is a universal because it depends on itself because only because it is instantiated in itself, it is a universal. Dependence —if it is not ‘weak dependence’ that is anti-symmetric— cannot be reflexive. 5 It would be a vicious regress because each universal of the hierarchy is universal because it instantiates the immediately following universal. None of the infinite universals of the hierarchy is ontologically basic.

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9 Objections Against Transcendent Universals

The Eleatic Principle

§ 61. An important difficulty against the postulation of transcendent universals is what seems to arise from the so-called “Eleatic principle”. The name is inspired by one of the characters in the dialogue The Sophist of Plato, the Stranger of Elea6 (see Oddie 1982). The intuition that seems to be justifying this principle is that only what can causally interact with other entities is real. Real entities are those that, in some way, are incorporated into the ‘causal frame’ of the world. ‘To exist’ would be to be inserted in such a causal frame, causing and receiving causal activity. Along with this ontological intuition, the Eleatic principle also seems to have an epistemological justification. In effect, it seems that we have reasons to postulate as existing those entities with which we have a direct perceptual contact and those entities that adequately explain the existence of those entities with which we have a direct perceptual contact. Thus, although we do not have a direct perceptual contact with an electron, electrons allow us to explain observational appearances, and for that reason, it is reasonable to suppose their existence for empirical reasons. Usually, in these contexts ‘explaining’ an event is ‘causing it’ or, at least, it must include a causal element.7 It is necessary to enter into a causal transaction with a trait of the environment to perceive it. Then, it turns out that we could only know those entities with which we can interact causally. Note, moreover, that this epistemological justification requires that the postulated entities not only be able to enter the causal frame when being caused by something but that they must be capable of causing something themselves. An epiphenomenal entity, which has causes but is causally inefficient, would be as unknowable as an entity completely separate from the causal frame. Only something that can be deployed causally is something with which we can have perceptual contact. Only something that can be causally efficacious is something that can explain the occurrence of that with which we have perceptual contact. Armstrong formulates the principle in this way (see Armstrong 1997, 41):

6 In Plato (1982) 247d-e, Plato puts in the mouth of the Foreigner of Elea: “I say that there really exists everything that possesses a certain power (dynamis), whether acting on any other natural thing, whether suffering, even in minimum degree and because of something infinitely weak, even if this happens only once. I then maintain this formula to define the things that are: they are nothing other than power. “ 7 All the discussion developed during the last century about the nature of ‘explanation’ from Carl Hempel’s nomological-deductive theory is well known (see for an overview, Ruben 2012). Many views have insisted on the importance of a causal connection between the explanans and the explanandum. Many others have insisted on the importance of epistemic and pragmatic elements for a good explanation. A ‘good explanation’ of a fact s is not only the elucidation of the causal network in which s is inserted but also highlighting those elements within this network that are ‘relevant’ because they are less familiar to an audience. For example, it can be informative for us to know that there was a forest fire because someone left a campfire. For extraterrestrial intelligent creatures, however, it can be informative to know that a forest fire occurred because the Earth’s atmosphere is full of oxygen and living beings on Earth are made of carbon. The campfire, the oxygen, and the carbon are all causal factors, but what is relevant as an explanation will depend on the previous beliefs and expectations of the audience and their theoretical interests.

9.1 The Eleatic Principle

[Eleatic principle]

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Everything that exists produces a difference in the causal powers of something.8

The principle has not been formulated directly in terms of causal powers, that is, it is not that only exists what has causal powers since it could be argued that only particular objects ‘have’ causal powers, but not properties. Properties, however, must enter the causal fabric of the world in determining the causal powers of objects (see Armstrong 1997, 41–42; Oddie 1982). The point is that transcendent universals would violate this requirement since their existence would be causally inert. Their existence neither seems to make objects more ‘powerful’ nor does it seem to make objects less ‘powerful’. That is, one could consider a scenario in which everything that happens in the complete spacetime system happens as it happens actually, but without any transcendent universal. What difference would this lack of transcendent universals in such a scenario make? It seems that their absence is not noticed at all. The causal powers that the objects have, and with it, the causal connections in which they will enter, seem to depend only on their intrinsic nature and on the external relations in which they find with each other. Transcendent universals are not ‘constituents’ of these intrinsic natures. It seems, then, that they are perfectly dispensable for the causal web of the world. Given everything that has already been explained about the non-substitutable theoretical functions of transcendent universals with respect to metaphysical

8

In quantificational modal logic, the principle should be formulated in this way:

[Eleatic principle]

□8x□ ((x exists) ! ∃y (x produces a difference in the causal powers of y))

The quantification is unrestricted and the variables ‘x’ and ‘y’ have as range entities of any category. The initial modal operator of necessity must be assumed, since the Eleatic principle, if true, must be true necessarily. It is not an accidental feature of our world, but a feature of what it means to ‘exist’ for any possible entity. It should also be assumed that a modal operator of necessity, under the scope of the first quantifier, since it would be part of the essence of any possible entity to be linked to the ‘causal network of the world’. It will also be noted that, according to this formulation, it is necessary for all that exists to produce a difference in the causal powers of something that exists in the same possible world. If x exists in w1, then the existence of x must produce a difference in the causal powers of something also existing in w1. A relaxation of this requirement would be to replace the Eleatic Principle by: [Eleatic principle’]

□8x□ ((x exists) ! ◊∃y (x produces a difference in the causal powers of y))

That is, the possibility of making a difference in the causal powers of something is sufficient for existence. The first formulation seems more in keeping with the spirit of an Aristotelian metaphysics that only admits immanent universals. Given the ambiguity that exists regarding what a ‘causal power’ is, as it is going to be shown (§ 62), it will not be necessary to discuss this in detail.

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modality, natural laws and, finally, the expected priority profile of a universal, this supposedly inert character of transcendent universals appears intriguing. If universals are indispensable for the nature of things, natural laws and causal connections, how can they be, then, inert? And they are not. A close examination of the question shows that transcendent universals, under any reasonable interpretation of the Eleatic principle, produce a difference in the causal powers of the objects that instantiate them and, thus, are not in violation of the principle (see Alvarado 2010, 2011, 2014a). For this examination, it will be essential to make a careful reconstruction of it. It is crucial to have much greater clarity about what should be understood by a ‘causal power’ (see Alvarado 2014a). As you can already see, it is a notion that can be interpreted in a multitude of ways, and that depends, in turn, on how the nature of causality and the nature of modal facts are understood. Discussions about the Eleatic principle until now have passed over these ambiguities, which has prevented an adequate evaluation of its scope. In the following, therefore, I will consider what should be understood by a ‘causal power’ and, then, I will directly consider the question of whether transcendent universals ‘produce’ a modification in the causal powers of something.

9.1.1

What Is a Causal Power?

§ 62. The attribution of a ‘causal power’ to an object is the attribution to that object of the character of being able to cause something. Three notes follow immediately from this characterization. First, it is a modal attribution de re. That is, it is the attribution that is made to an object that is metaphysically possible for it to be involved in certain causal interactions. Even if that object is not involved in such connections in the possible world of evaluation—i. e., the possible world in which the modal attribution is true—it is sufficient that the object is involved in a causal connection in a possible world accessible to the world of evaluation. Second, it is the attribution of causal connections, even if they are merely possible causal connections. It is obvious that the way in which the nature of the causal relationship is understood will have a central importance for the notion of ‘causal power’. Third, the type of causal power that is attributed to an object is specified by the type of effect that the deployment of such power would cause. The relata of the causal relationship are events, so the type of effect that a causal power could cause is specified by the type of event that could be its effect. A causal power can also be specified with respect to a particular event, that is, with respect to the particular event of instantiating the object a the universal U at time t. When causal powers are attributed, however, what matters is the attribution of a power to cause a general type of event. Since there are three constituents in the identity conditions of an event,9

9

Recall that Jaegwon Kim’s conception of the identity conditions of an event is being assumed (see Kim 1976): an event is essentially the instantiation of a universal at a time and coincides with a

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there are also three variables that can be used to specify a type of event. One can specify, for example, the general type of event of the universal U being instantiated by some object at some time. It can be specified the general type of event of some universal being instantiated in object a at some time. It can be specified, also, the type of event of being instantiated some universal in some object at time t. There are, of course, several additional alternatives.10 When a causal power is attributed, what matters is the power to instantiate a universal. The power to do something is usually understood as the power to make something in a certain way, and the specification of this way of being is the specification of the universals that would be instantiated in such circumstances. In what follows, therefore, it is to be assumed that the causal powers are specified by the general type of event they would cause and this type of event is, in turn, specified by the universal or the universals that would be instantiated. A first analysis of a causal power could be done in these lines: [Causal Power I]

U1 confers to x at t the power to cause a U2 ¼ df the state of affairs of x instantiating U1 at t would cause something to instantiate U2 at some time.11

It can be seen in this first analysis of the notion of ‘causal power’ that the attribution of a power is a modal attribution consisting of a possible causal relationship between the state of affairs of instantiating the object—to which the power is attributed—the universal that confers the power in question, and a state of affairs of instantiating a universal—which is the universal that specifies the causal power. It is notorious here,

‘state of affairs’, such as has been understood in this work and by authors such as Armstrong (see, in particular, Armstrong 1997, 113–127; Alvarado 2013a). States of affairs satisfy these two principles: [Existence of states of affairs] (x1, x2, ..., xn instantiate X at ζ) $ ([Xx1x2 ... xnζ] exists) [Identity of states of affairs] ([Xx1x2 ... xnζ] ¼ [Yy1y2 ... ynξ]) ! ((X ¼ Y) ^ (x1 ¼ y1) ^ (x2 ¼ y2) ^ ... ^ (xn ¼ yn) ^ (ζ ¼ ξ)) The variables ‘X’ and ‘Y’ have as range universals. The variables ‘x1’, ‘x2’, ..., ‘xn’, ‘y1’, ‘y2’, ..., ‘yn’ have as range objects and the variables ‘ζ‘and ‘ξ‘have as range times. Times should be taken as including an extended period and, eventually, punctual moments. The expression “[Xx1x2 ... xnζ]” should be read as “the state of affairs of being x1, x2, ..., xn instantiating the universal X in time ζ“. It has been omitted in these principles of Existence and Identity of states of affairs universal quantifiers that have to be bounding the variables and modal operators of necessity because, in effect, these principles, if they are valid, have to be universal and necessary. 10 There are also general types of events of the instantiation of the universal U in the object a at some time, the instantiation of the universal U at time t in some object, the instantiation of some universal in object a at time t and the maximally general type of event of some universal being instantiated in some object at some time. 11 Using the notation of quantified modal logic and following the conventions already introduced above, the analysis Causal Power I would be formulated as follows: (X confers to x at ζ the causal power to produce a Y ) ¼ df ∃y∃ξ ([Xxζ] would cause [Yyξ]) Recall that “[Xxζ]” designates the state of affairs or event of the instantiation of x of the universal X at time ζ.

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however, that the expression “would cause” is ambiguous and could be interpreted in different ways. A first interpretation could be: [Causal Power II]

U1 confers to x at t the power to cause a U2 ¼ df it is necessary that, if there is the state of affairs of x instantiating the universal U1 at t, then such a state of affairs causes something to instantiate U2 at some time.12

What is stated in this formulation is that the possession of the power to cause a U2 consists in the fact that an instantiation of U2 is necessarily caused if U1 is instantiated at some time. It is obvious that this formulation is too strong. For example, the fact that an object has a negative electromagnetic charge –q confers to this object the causal power to attract objects with positive electromagnetic charge, but in possible worlds in which there are no objects positively charged, the instantiation of the universal possessing a negative electromagnetic charge of –q does not cause any attraction. In possible worlds where other forces are interacting with objects endowed with such charges, such attraction will not occur either. It cannot be understood, therefore, a causal power as something that makes necessary the causation of the instantiation of the universal that specifies such power. These problems make this third formulation a better alternative: [Causal Power III]

U1 confers to x at t the power to cause a U2 ¼ df it is possible that the state of affairs of x instantiating the universal U1 at t cause something to instantiate U2 at some time.13

In this second formulation, the modal requirement is much weaker. The power to cause the instantiation of U2 consists in the fact that it is possible that the instantiation of U1 causes an instantiation of U2. Therefore, it is compatible with the existence of a causal power conferred by the universal U1 that in some possible worlds, the instantiation of U2 is caused and in other possible worlds, it is no. This formulation not only leaves room for stochastic processes and for the type of causal deployment that should be attributed to the free will of persons,14 but also for the forms of causal interaction that are governed by perfectly deterministic laws. For example, although it is a natural law that objects endowed with opposite electromagnetic charges of +q and –q at a distance d from each other are attracted with a force that is directly proportional to those charges and inversely proportional to the

12

Using quantified modal logic and with the conventions introduced, it results: (X confers to x at ζ the causal power to produce a Y ) ¼ df □ ([Xxζ] exists) ! ∃y∃ξ ([Xxζ] causes [Yyξ]) 13 Using quantified modal logic and with the conventions introduced, it results: (X confers to x at ζ the causal power to produce a Y ) ¼ df ◊ ([Xxζ] exists) ^ ∃y∃ξ ([Xxζ] causes [Yyξ]) The clause ‘([Xxζ] exists) has been maintained here in the definiens to maintain the symmetry with the formulation II, but it is redundant because only an existing state of things can cause something. 14 Assuming, of course, a conception of freedom incompatible with determinism.

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square of the distance d, the force in question will not manifest directly in the movement of the bodies if it is not assuming that there are no other forces influencing their behavior. Although the universals are connected by perfectly deterministic laws, in some possible worlds a regularity will result according to the law and in others, it will not. What establishes the natural law is the existence of a force, not the final acceleration, nor the final movement resulting. Along with these alternatives to analyse a causal power, an appeal can be made to counterfactual conditionals. For this, we assume Lewis’s semantics (see 1973a) in which two types of counterfactual conditional can be differentiated, as explained above: conditionals of the could type and conditionals of the might type. The counterfactuals of the could type can be formulated as: if p were the case, then q could be the case. A conditional of this type, as already explained several times, is true in the possible world w1 if and only if in all the possible worlds closest to w1 the material implication if p then q is true.15 The counterfactuals of the might type, however, can be formulated as: if p were the case, then q might be the case. This conditional is true in the possible world w2 if and only if in at least one of the possible worlds closest to w2, the antecedent and the consequent are true.16 One can analyse, then, the concept of ‘causal power’, in the first place, using the counterfactual conditional could: [Causal Power IV]

U1 confers to x at t the power to cause a U2 ¼ df if there were the state of affairs of x instantiating the universal U1 at t, then such state of affairs could cause something to instantiate U2 at some time.17

According to this formulation, the causal power conferred by the universal U1 consists in the fact that the counterfactual conditional is true according to which, if the universal U1 were instantiated, then the universal U2 could be instantiated. Much of the difficulties that affect form II can be evaded here since it is relevant to the evaluation of the conditional only what happens in the ‘closest’ possible worlds and not in all the metaphysically possible worlds. One would be inclined to argue that the class of possible worlds ‘closest’ to the world in which the attribution of a causal power is being evaluated are exactly the possible worlds in which there is nothing to prevent the deployment of such power, no force additional to modify the resulting final system. Recall that for the formulation II there was the problem that,

15 That is, in all possible worlds closest to w1 either Øp or q, because[( p ! q) $ (Øp _ q)]. For this reason, the counterfactual conditional is vacuously true in the case that the antecedent is false in all the worlds closest to the possible world of evaluation. Recall that the metric of ‘closeness’ between possible worlds is fixed by their global similarities. This counterfactual conditional could is written as [p □! q]. 16 This counterfactual conditional might is written as [p ◊! q]. It is the dual of the counterfactual could, so [( p □! q) $ Ø( p◊! Øq)] and [( p ◊! q) $ Ø( p□! Øq)]. 17 Using quantified modal logic with the introduced conventions, it results: (X confers to x at ζ the causal power to produce a Y ) ¼ df ([Xxζ] exists) □! ∃y∃ξ ([Xxζ] causes [Yyξ])

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although the instantiation of a universal is connected deterministically with the instantiation of another universal, such result can be affected by innumerable additional factors that can prevent the deployment of the causal power. One would suppose that all those possible worlds in which there are such additional factors are in ‘farther’ worlds and would not be relevant for the evaluation of the counterfactual conditional. Even with these precautions, however, formulation IV is not acceptable. In the first place, this analysis would not allow attributing a causal power to an object when the property that confers such power is only connected in a probabilistic manner with the effect specified by the power. Suppose that it is a natural law that the instantiation of U1 causes with an objective probability of 0.7 the instantiation of U2 and with an objective probability of 0.3 the instantiation of U3. Not in all possible worlds ‘nearby’ the world of evaluation U2 is caused, and not in all possible worlds ‘nearby’ the world of evaluation U3 is caused. Then, if one were to follow this analysis IV of the notion of causal power, the universal U1 would not confer any causal power, when it clearly does. Objects that instantiate U1 would cause a U2 and would cause a U3. Second, this analysis would not allow the attribution of a causal power to a person endowed with free will.18 It is not at all clear that any distribution of objective probabilities of deciding an intentional action can be attributed to a person endowed with free will.19 It seems obvious, however, that the fact that someone has free will determines that he could intentionally decide to perform the action φ. But not in all possible worlds closest to the world of evaluation the person freely decides to do φ. Then, the causal power of doing φ could not be attributed to that person, which seems absurd. Third, also, this analysis IV of the notion of causal power would not be adequate to assign causal powers to objects that instantiate universals that are deterministically connected to cause the instantiation of other universals. It is crucial for this that the metric by which the possible worlds are ordered is grounded on their mutual global similarities. The merely local similarities are not enough. Suppose, in effect, possible world w1 in which the object a instantiates the universal U1 at time t. It is a deterministic natural law that the instantiation of U1 causes the instantiation of U2. For the attribution of the power to cause a U2 to object a in w1, it is required— according to analysis IV—that in all possible worlds closest to w1 the state of affairs of a instantiating the universal U1 at t cause a U2. But there is no guarantee that it is so. Suppose that w1 is a possible world such as the actual world until the year 1980, but at this year there is an extraterrestrial invasion that extinguishes human beings, takes all the resources of our planet and leaves it devastated. It is obvious that, if

18

Assuming, of course, a conception of the freedom of the will incompatible with determinism. In principle, one would be inclined to assume that character habits, prior beliefs, and prior preferences make it more or less objectively probable that a person will freely decide to perform an intentional action. If there are distributions of objective probabilities here, then the case of free will would be a case of a stochastic process. The indicated problems that affect the stochastic powers would also affect the ‘powers of freedom’. It is not clear, however, whether the freedom of the will determines the purely stochastic nature of these factors or, rather, the cancellation of such probability distributions. 19

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everything else is maintained the same worlds closest to w1 will be worlds in which an alien invasion with these characteristics takes place in 1980, whether or not the causal process of producing the instantiation of a U2 by a U1 is prevented. Although at first, it seemed that the worlds closest to w1 are the worlds in which the instantiation of U1 is not hindered to cause a U2, this is only one among many other factors—infinite other factors—that are relevant to determine the global similarity or dissimilarity between w1 and other worlds. What results, then, is that even for deterministic causal powers analysis IV would not be adequate. However, there is another form of counterfactual analysis using now the conditional might instead of the conditional could. This analysis would be as follows: [Causal power V]

U1 confers to x at t the power to cause a U2 ¼ df if there were the state of affairs of x instantiating the universal U1 at t, then such state of affairs might cause something to instantiate U2 at some time.20

Unlike as what happens in analysis IV, here the only thing that is required to attribute a causal power is that in at least one of the possible worlds closest to the world of evaluation the causal connection occurs. If one considers the scenario just explained, although not in all possible worlds closest to w1, the instantiation of U1 in x at t causes the instantiation of U2, it is enough for one of these worlds to contain such a causality for the attribution of causal power to be correct. It can be seen then that this analysis is better than IV. There is a similarity of spirit between this analysis V and III. In both cases, the possibility of causality for the attribution of causal power is sufficient. The analysis V is, comparatively, more demanding than III because it requires that the causality occurs in one of the ‘closest’ worlds to the world of evaluation, while in III it is enough to occur in any world, whether or not it is ‘close’.21 As can be seen, analyses III and V are more acceptable. In some cases, analysis III will suffice, but in others analysis, V is indispensable to make the notion intelligible. With these analyses, the elucidation of the notion of ‘causal power’ is still halfway. In previous analyses II to V what has been done is to specify the notion in terms of modality and causality—together with the logical apparatus required for the formulations—but assuming that the modal and causal notions involved are already understood. As has been seen several times throughout this work, there are many alternative ways in which such concepts can be understood. These variations

20

In quantified modal logic with the introduced conventions, it results: (X confers to x at ζ the causal power to produce a Y ) ¼ df ([Xxζ] exists) ◊! ∃y∃ξ ([Xxζ] causes [Yyξ]) 21 Correlatively, there is a similarity of spirit between analyses II and IV. In both cases, it is required that the causal relationship occurs in all possible worlds. The difference is that according to analysis II, it must occur in all possible worlds, without exception, while according to analysis IV, it must occur in all possible worlds closest to the world of evaluation. If with respect to analyses III and V, the restriction to nearby worlds makes analysis V more demanding than III, here that same restriction makes the analysis IV less demanding than II.

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bring, as expected, very different notions of ‘causal power’ and, with it, very different ways of evaluating the Eleatic principle. Of course, a comprehensive evaluation of all conceivable theoretical alternatives is beyond the scope of this work, but attention can be focused on some of them that are especially salient. Concerning causality, the orientation that has already been used in previous sections will be followed, concentrating the examination on the theories of regularity, counterfactual theories, and non-reductivist theories (§ 13). Concerning modality, however, it will be useful here to make a variation. Concerning the problems considered up to now (see §§ 24, 30, 41–44), it has been useful to divide modal theories between possibilists and actualists. The possibilists are those that suppose at the outset the same ontological status for possible and actual entities. The actualists are, on the other hand, those who suppose at the outset that only the actual world is the unrestricted totality of what exists. This distinction is, to a large extent, orthogonal to the distinction between combinatorial conceptions and non-combinatorial conceptions of the modality that is now relevant. The most characteristic examples of combinatorial modal theories are David Lewis’s (see Lewis 1986, 86–92) and David Armstrong’s (see Armstrong, 1989b). Armstrong’s theory and several other actualist theories have been called, in effect, as “combinatorial” (see Divers 2002, 174–177), but the type of resources used in these are also common to the Lewisian approach and can be applied to other actualist conceptions. The crucial difference has to do with the fact that in ‘combinatorial’ modal theories, the modal space of metaphysical possibilities is grounded on a plurality of entities that are mutually independent of each other. As these entities are mutually independent, they can exist together—that is, in combination—or they can exist ‘separate’—that is, one of them and not the others. There are differences between the different conceptions about which entities should enter (or not) in combination and also with respect to the ‘combination’ in question, but the central intuition is the same in all cases. The modal space is based on the reciprocal ontological independence between entities of a fundamental repertoire (see Alvarado 2009). For example, in Armstrong’s theory, the modal space is constituted by the fact that universals may or may not be instantiated in specific particular objects— although they must be instantiated in some particular—and particular objects may or may not instantiate specific universals—although they must instantiate some universal or another (see Armstrong 1989b, 37–53). In Lewis’s theory, the modal space of metaphysical possibilities is fixed by a ‘principle of recombination’ (see Lewis 1986, 87–88) according to which any type of object—fixed by the intrinsic properties it would possess—could (or not) exist under any external relationship with any other type of object—also fixed by the intrinsic properties it would possess. In the same way, if in a possible world two objects of certain types exist together, an object of any of these types could exist without being ‘accompanied’ by another object of the other type. The only limitation imposed by Lewis for this principle of recombination is that which could be imposed by the extension of spacetime (see Lewis 1986, 89–91), since—in principle—the same region of spacetime could not exist being occupied by different objects. This restriction seems doubtful, but it can

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be set aside for this examination. In Lewis’s metaphysics, no object can exist in more than one possible world (see Lewis 1986, 198–220, 1968), so recombination does not affect ‘objects’, but rather ‘duplicates’ of objects. A duplicate of x is an object that has exactly the same intrinsic properties as x. Nothing prevents to use this same ‘combinatorial’ idea but with a different repertoire of entities or with other forms of ‘combination’. For example, one could postulate a modal space constructed by the recombination of tropes independent with each other. Or, from the perspective of a modal theory based on universals, a modal space could be postulated constructed by the different instantiation distributions without contradiction of transcendent universals. Although combinatorial conceptions have been the most frequent in the last 50 years, it is not the only way to conceive the extension of the space of metaphysical possibilities. A well-known alternative is offered by the modal theories based on primitive causal powers (see Borghini and Williams 2008; Alvarado 2009; Jacobs 2010; Vetter 2015; with preventions, Molnar 2003, 200–223; Mumford 2004, 160–181). In these conceptions, the central idea is that something is metaphysically possible if and only if it can be caused by an actual state of affairs. Actual objects and their actual causal powers determine the modal space of metaphysical possibilities. In these conceptions of modality that could be called “causal”, the order of priority between modality and causal powers is inverted. Here, causal powers cannot be analysed from previous and better understood modal notions. The situation is exactly the reverse since modality must be understood from ‘causal powers’. When it comes to an understanding of the concept of ‘causal power’, it will be crucial, therefore, to complement analyses III and V explained above with some of these combinatorial or causal conceptions.

9.1.2

Causal Powers and Combinatorial Modality

§ 63. In the first place, it is convenient to consider what a causal power is from a combinatorial modal perspective. It will be useful to examine the conception that should be had of a causal power in Lewisian metaphysics, as a characteristic example of this type of theory. In the central points, it will be appreciated that the features of this conception also apply to conceptions such as Armstrong’s and under any theory of causality. As has been explained, for David Lewis, the totality of possible worlds obeys a principle of recombination (see Lewis 1986, 86–92). Let F and G be ‘complete’ intrinsic natures, i. e., sets of all the intrinsic properties possessed by an object. If there is a possible world in which there is an F and there is a G, then there is also a possible world in which there is an F and there is no G, as there is also a possible world in which there is a G and there is no F. This applies to any intrinsic nature. If in a possible world an F and a G are each other under an external relationship R—for example, at a distance of 10 km from each other—then there are also possible worlds in which they are not under the external relationship R, but under another relation R’—for example, at a distance of 10.01 km, etc. The

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totality of possible worlds is grounded on the totality of coherent combinations of these intrinsic natures and external relations. Of course, for Lewis, possible worlds are mereological fusions of objects that are all connected spatiotemporally (see Lewis 1986, 69–86, 92–96), but this notorious feature of the Lewisian possible worlds is not relevant here.22 When it comes to the causal relationship, on the other hand, this is understood as the ancestral of the relation of counterfactual dependence. The central lines of this conception, already treated several times, will no longer be explained (see Lewis 1973b, 2004, § 13). Causal relationships are reducible to counterfactual conditionals true of propositions that state the occurrence (or not) of cause and effect events. And such counterfactual truths are grounded on global similarities and dissimilarities between possible worlds. The fact that event c causes event e is reduced to the facts that if c existed, then e could exist and that if c did not exist, then e could not exist. And these facts are grounded on what happens in the possible worlds closest to the world of evaluation. If one adopts analysis III, a universal U1 confers the power to cause a U2 to the object x at t if and only if it is metaphysically possible that the state of affairs of x instantiating U1 at time t cause the instantiation of U2. Suppose that the causal power is conferred in the possible world w1. For this, it is sufficient that there exists a possible world accessible to w1 in which the state of affairs of being the object x instantiating U1 at time t cause a U2, let this world be w2. In the counterfactual conception of causality this causal relation exists in w2 if and only if in all the possible worlds closest to w2 in which the state of affairs in question exists U2 is instantiated and in those same worlds in which the state of affairs does not exist, U2 is not instantiated. In Lewisian metaphysics, then, analysis III does not impose any restriction on the causal powers that can confer a universal. The only restriction imaginable is that there is no causal power to produce a contradiction, but apart from this, it seems trivial that any universal confers any causal power. In fact, in a combinatorial modal conception, any type of object—or any object—can exist accompanied or not with any other object. It is trivial that, for example, the property of having 1.7 mts of height confers the causal power of making an elephant appear because there are possible worlds in which to instantiate the universal of being 1.7 ms of height causes that an elephant appears. These are very different worlds to the actual world, but this has no relevance under analysis III. In any of those worlds, let one of these worlds be w3, it causes an elephant to appear, because in all the worlds close to w3 it happens that if something is 1.7 ms high an elephant appears and if there is not something with 1.7 ms of height no elephant is appearing. The mere combination of qualitative characteristics ensures that there are possible worlds sufficient to guarantee these metaphysical possibilities and, with this, actual causal powers of making an elephant appear. This can be generalized to almost anything. 22

An actualist theory in which the identity of objects in different possible worlds is not admitted should also postulate that possible worlds have to be specified in a purely ‘qualitative’ way by the distribution of properties and relations in such worlds. There are actualist theories, in effect, that reject the identity of objects in different possible worlds (see Sider 2002) and in which de re modal attributions are made through the use of counterparts, as Lewis does (see Lewis 1968).

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It is not difficult to see that a conception of causal powers from a combinatorial perspective such as the Lewisian—under analysis III—trivializes those powers. Besides, as any universal confers the causal power for anything, it is very unreasonable to postulate the Eleatic principle. Why should conferring causal powers or making a difference in the causal powers of something have so much ontological importance? It is difficult to ‘make’ a difference in the causal powers of something, because anything, by the fact of instantiating any universal, has the power to cause (almost) any universal. Variations in the distribution of property instantiations do not vary these causal powers since all objects have all powers at all times. Nothing can ‘make’ such powers grow or diminish. The same result of trivialization is found in any combinatorial theory of modality. As any entity can exist together or separately from any other, the instantiation of any universal can cause the instantiation of any other. If one postulates a regularist conception of causality, all that is required for it to be possible for the instantiation of one universal to cause the instantiation of another is that there is a possible world in which the instantiation of one is followed regularly by the instantiation of the other.23 And clearly, there are these possible worlds from a combinatorial perspective. If one postulates a non-reductivist conception of causality (see Anscombe 1971; Tooley 1987), causal facts are ontologically basic, not reducible to others. From a combinatorial perspective, it must be assumed that the causal relationship is a relational universal like any other that can be instantiated or not between any states of affairs—perhaps with temporal restrictions concerning the anteriority of the cause to the effect. No matter what is the universal in question, the instantiation of a universal confers a causal power for the instantiation of any other, because it is metaphysically possible that the instantiation of one cause the instantiation of any other. It may appear strange at this point that the instantiation of any universal can cause the instantiation of any other, because—in principle—universals are connected by natural laws, as explained. And if there are natural laws essential for the universals that integrate them, the universals that are connected by them cannot cause the instantiation of any universal. A combinatorial modal metaphysics that, at the same time, postulates essential natural laws for universals would not be incoherent. It is, however, an implausible metaphysics, because the central intuition of a combinatorial modal metaphysics is that the ‘different existents’ are modally independent of each other. It is more reasonable from a combinatorial perspective of modality a regularist theory of natural laws or a theory such as the one advocated by Armstrong, Tooley, and Dretske according to which, although natural laws are higher-order nomological relations between universals, they are contingent for the connected universals. Combinatorial modal theories have a Humean inspiration. The central idea of these theories is that nothing—in a localized region of spacetime—makes

23

With the other usual conditions of a regularist conception of causality: that the event cause is contiguous spatiotemporally with the event effect, and that the event causes occurs earlier temporarily to the event effect (see Psillos 2002, 19; 2009, 140).

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something else necessary or more probable in a different region. Theories of natural laws that conceive them as essential to universals are foreign in a combinatorial modal metaphysics.24 The most natural solution to conceive reasonably causal powers from a combinatorial modal perspective would seem to be to strengthen analysis III by the counterfactual resource used in analysis V. Instead of assuming that a possible causal relation is sufficient for that power to be conferred, it must be assumed now that such a causal relation might occur. As explained, this is to assume that in at least one of the possible worlds ‘closest’ to the world of evaluation, such a causal relationship occurs. It would not be enough, then, that causality is merely possible, if the possible world in which such causation is effective is too ‘far’ from the world of evaluation. The restriction imposed by the counterfactual conditional might to ‘near’ possible worlds seems to be adequate to block cases of trivialization. For example, as explained, according to analysis III—and from a combinatorial modal perspective—the property of being 1.7 ms high confers the causal power of making an elephant appear because there are possible worlds in which there is a causal relationship between having 1.7 ms of height and appearing an elephant. It does not matter for this how the nature of the causal relationship is conceived. Now this counter-intuitive consequence could be blocked by maintaining that, even if it is metaphysically possible that having someone 1.7 ms high causes the appearance of an elephant, if someone had 1.7 ms of height it might not cause an elephant to appear, for in no world close to the actual world, the height of something causes the appearance of elephants. When attention is restricted to the worlds closest to the possible world of evaluation, no causal relationship between the instantiation of any universals can be adduced, only the causal connections that occur in the worlds that are “similar” to the world of evaluation. This way of conceiving causal powers from a combinatorial perspective, however, brings with it some problems that have to do, precisely, with the restriction that is made in the analysis V to the possible worlds closest to the world of evaluation. As highlighted above with respect to the counterfactual analysis IV, similarities and dissimilarities between possible worlds relevant to the metric between these worlds —by which it is determined which worlds are ‘closer’ or ‘more distant’ to each other—are global in nature. This restriction produces that possible worlds in which causal relationships that would not seem at all extravagant are, for other reasons, too 24

Without prejudice to this, conceptions of natural laws essential to universals have been proposed in a framework that is basically combinatorial. An example is the position proposed by Stephen Mumford (see Mumford 2004, 170–181; especially, 175–180). Mumford postulates natural laws that are reduced to (or eliminated by) causal powers essential to a property, but contingency is grounded not on the causal powers essential to the properties, but the combinatorics of mutually independent objects and properties. Although causal powers impose substantive restrictions on the combinatorics that should generate the modal ontological space, they are ‘restrictions’ in a general scheme that has a different grounding. Mumford cannot be accused of inconsistency at this point, but there is an explanatory obligation that must be satisfied because it is not obvious why these restrictions should exist if, fundamentally, modality is grounded on the mutual independence of different existing entities.

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distant. In this way, it would result that certain properties would not confer causal powers to the objects that instantiate them, even when intuitively they should do it. It seems obvious, for example, that a negative electromagnetic charge of –q confers to the objects that possess it the causal power of attracting with objects that have a positive electromagnetic charge. Suppose the possible world w1 in which there are only objects with negative electromagnetic charge and nothing has a positive electromagnetic charge. This possible world is very different from the actual world and very different from any possible world moderately similar to the actual world. It would seem perfectly reasonable to maintain that all possible worlds closest to w1 are also worlds in which nothing has a positive electromagnetic charge because that is determinant for the global similarity between those worlds. But, if so, then, in none of the worlds closest to w1 objects with a negative electromagnetic charge of –q are attracted to other objects with positive electromagnetic charge. Then, it would result that in w1 the property of having a negative electromagnetic charge of –q would not confer the causal power of attracting objects with positive electromagnetic charge. But it seems obvious that the negative charge confers the power to attract positive charges. The restriction to nearby possible worlds, then, together with allowing ‘extravagant’ causal connections to be dismissed as irrelevant, leaves out in some cases causal connections that do seem relevant to a causal power. It is not clear, therefore, how causal powers should be properly understood from a combinatorial modal perspective. It is more reasonable to understand these causal powers by the counterfactual analysis V, but this analysis, as has been shown, has counter-intuitive consequences also. Analysis III seems too promiscuous. So promiscuous, that it would make all causal powers trivial. Analysis V, on the other hand, seems too demanding. So much so, that it would leave out causal powers that seem perfectly reasonable. The cost of analysis V, however, seems less than the trivialization that follows from III and, therefore, here it is going to be assumed that V is the way to understand a causal power from these combinatorial perspectives.

9.1.3

Primitive Causal Powers

§ 64. A second large family of conceptions about the modal ontological space of metaphysical possibilities is the conception that grounds it in primitive causal powers (see Borghini and Williams 2008; Alvarado 2009; Jacobs 2010; Vetter 2015). This type of modal conception has been gaining followers more recently since it seems an alternative much more coherent with a non-Humean metaphysics of natural laws and causality. As in a combinatorial conception, the fundamental thesis is that metaphysical possibilities are grounded on the mutual independence between the entities of a basic repertoire, here the fundamental thesis is that the metaphysically possible is what can be caused by actual states of affairs. In a more precise way:

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s is metaphysically possible ¼ df there is an object x, a universal X and a time t such that: the state of affairs of x instantiating X in t can cause s.25

It will be immediately noticed that the metaphysical possibility is analysed through the modal notion of ‘being able to cause’. As has happened in so many other modal conceptions, the principle of causal possibility does not pretend to be a reductive analysis of the metaphysical modality. Neither are reductive analyses those offered by combinatorial theories, since the combinatorial principle—also called of “modal plenitude”—is a modal principle about what may or may not exist jointly or separately.26 When it is said in the analysis of this principle of Causal Possibility that “there is an object, a universal and a time ..”. it is spoken of what there is actually, without restriction. The metaphysical possibility is, then, restricted according to this principle to what actual objects can cause at some time or another, given the universals that they actually instantiate. This principle of Causal Possibility allows us to analyse, using the usual resources, metaphysical necessity: [Causal Necessity]

s is metaphysically necessary ¼ df there is no object x, universal X, and time t such that: the state of affairs of x instantiating X at t can prevent s.27

Here ‘to prevent s’ is to make that s does not obtain. It could not be a causal relationship with not-s since there is only causality with what does exist.28 ‘Prevention’ should be understood here as the causal deployment that hinders or impedes the causal deployment of the state of affairs s. In a causal conception of modality, the causal relation cannot be reduced to other facts that are more basic or more fundamental. Causality cannot be reduced to counterfactual dependence. Causal facts in a counterfactual conception are grounded on the previous distribution of ‘local’ facts in the totality of possible worlds. Basic 25

In a more precise way: ◊ (s exists) ¼ df ∃x∃X∃ξ ([Xxξ] can cause s) The quantifiers of the definiens are actualist. Otherwise, the conventions already introduced are followed. The analysis can be generalized to n-adic relational universals. 26 It is well known that Lewis’s possibilist theory has been presented as a reductive theory by which modal propositions would be true (or false) about the domain of possible worlds, as Lewis understands them (see Divers 2002, 106–121, for a discussion). The problem is that this domain of possible worlds must satisfy the principle of ‘plenitude’ that operates in a combinatorial way (see Lewis 1986, 86–92). Lewis’s theory achieves certain adequacy with respect to our modal intuitions precisely because it rests on principles that are modal from the outset. 27 More precisely and following the above stipulations: □ (s exists) ¼ df Ø∃x∃X∃ξ ([Xxξ] can prevent s) 28 More precisely, there is no causality of ‘absences’ if causality is an ontologically basic, fundamental fact, not reducible to something else. In reductionist conceptions, such as the counterfactual or the regularist, there are no disadvantages of principle in admitting the causality of ‘omissions’ and the causality by ‘omissions’. As will be explained, however, only a non-reductivist conception of causality is compatible with the causal conception of modality.

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modal facts determine causal facts and not the other way around. Something similar happens with the regularist conception of causality. Causal facts are, from this perspective, those that result from the distribution of regularities in the different possible worlds. The theory assumes that there are already determined modal facts that ground causal facts. From a causal conception of modality, on the other hand, the relation of ontological priority is the inverse one. Causal facts or, more precisely, facts about the space generated by actual causal powers determine the totality of metaphysical possibilities. It is interesting to note that there is no problem, from a causal perspective of modality, with analysis III. Perhaps the only drawback here is that it cannot be assumed that III is an analysis of what a causal power is. According to III, the fact that a universal U1 gives to the object x at time t the power to cause the instantiation of a universal U2 is the fact that it is metaphysically possible that the state of affairs of instantiating x the universal U1 at time t causes the instantiation of U2 in some object, in some time. But according to the principle of Causal Possibility, the state of affairs s is metaphysically possible if and only if there is any object z at some time t’ that instantiates a universal U3 such that the state of affairs of instantiating z universal U3 at time t’ can cause the state of affairs s. So, it turns out that U1 confers on x at t the power to cause the instantiation of a U2 if and only if there is some actual object z at some actual time t’ which actually instantiates some universal U3 such that the state of affairs of instantiating z the universal U3 at time t’ can cause the state of affairs that: the state of affairs of x instantiating the universal U1 at time t causes the instantiation of U2 in some object, at some time. This formulation is convoluted because there is the talk of ‘causing ... to be caused . . .’ But, given the transitivity of the causal relationship, ‘causing to be caused’ is simply causing. The condition requested for the existence of a causal power is that the contemplated causal connection could be accessible with respect to the causal powers of actual objects according to the universals that these objects actually instantiate. Suppose the power to cause the instantiation of U2 conferred by the universal U1 on the object x at time t. If it is the case that actually x instantiates U1 at t, the space opened by the actual instantiation of U1 is sufficient for the causal power in question. If the state of affairs of x instantiating the universal U1 at t is not actual, then the causal power in question will be grounded on the fact that such a state of affairs can be caused by actual states of affairs. The causality of the state of affairs of x instantiating the universal U1 at time t is, by transitivity of the causal relationship, something that can be caused by an actual state of affairs. It is clear that in any of these hypotheses, the idea of a causal ‘power’ is being assumed from the outset as what determines the space of possibilities. Thus, what happens from the causal perspective of modality is that trivially causal powers satisfy analysis III, but not because the notion of causal power is ‘analysed’ by previous modal and causal notions. For this reason, no counterfactual qualification is required, such as that contained in the analysis V.

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Dispositions?

§ 65. It may have been noticed that in the previous discussions, no “dispositions” have been spoken of when it seems obvious that a ‘causal power’ must be understood as a ‘disposition’. And it is true that for a common speaker, the two expressions “causal power” and “disposition” seem to be simply synonymous. Saying that “x has the causal power of producing an F” seems to be the same as saying that “x has the disposition to produce an F” If by “disposition” one is going to understand the same thing that has been analysed here as a ‘causal power’, there is no problem in using henceforth “disposition”, “causal power” or “power” indistinctly. The problem is that there is a long philosophical discussion in the last century about the nature of a ‘disposition’ that can be a source of important misunderstandings. In spite of the multitude of different philosophical positions that have been defended about the nature of a ‘disposition’, how dispositions have been characterized obliges us to differentiate them from ‘causal powers’, as they have been characterized here. For much of the last century, there has been a discussion about whether dispositions could (or could not) be reduced to counterfactual facts. This discussion seems to have closed with the criticisms of Charles Martin (see Martin 1994) and others (see Mumford 1998, 36–63, Molnar 2003, 82–98, Bird 2007, 18–42).29 The subsequent discussion has been concentrated on the connection that dispositions have to natural laws. Positions of Humean inspiration—or more Humean inspiration—have argued that dispositions should be reduced to ‘categorical’ properties and natural laws—which must be contingent to such properties (see Prior et al. 1982; Armstrong 1997, 69–84). At the opposite extreme, some have argued that all properties are dispositional (see Bird 2007).30 Others have argued, on the other hand, that some properties are categorical and others are dispositional (see Molnar 2003). This question will be considered below about the conditions of identity of universals (see § 75–77), but for now, it is not necessary to enter into it. For what is of interest here, it is assumed that a disposition is a property that, in some way, makes the occurrence of a certain manifestation “necessary”, given a stimulus. This ‘necessity’ is grounded on the fact that a counterfactual conditional of the type could is true, or it is grounded on a natural law that connects by a primitive nomological relation two universals or is a regularity of types of events. It may be that the ‘necessity’ is

The fundamental criticism presented by Martin appeals to finks. A ‘fink’ is a worker in a union paid by the company to avoid strikes. Suppose there is electric current through a cable. In principle, a voltmeter will show that there is current through the cable if it approached it. It happens, however, that a device has been introduced which, whenever a voltmeter approaches, cuts off the current. It seems that there is a disposition in the cable, but the counterfactual conditional: if the voltmeter approached the cable could show that there is an electric current is false, because the device will prevent the current from manifesting as it normally does. Dispositions cannot be reduced, therefore, to counterfactual conditionals. 30 A variation of this position is that of those who have argued that all properties have a ‘categorical’ or purely qualitative aspect, and another ‘dispositional’ aspect (see Heil 2003, 75–125; 2012, 53–83). 29

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restricted to possible worlds ‘close’ to the world of evaluation, but it is something that must be valid for ‘all’ possible worlds, whether or not in a certain range. The provisions have been understood as satisfying the principles of Causal Power II and IV. But the difficulties that come with these analyses have already been explained. What is interesting about a causal power is that it makes possible the occurrence of an effect, not necessary.31 In this way, if by “disposition” one is going to understand what has been characterized here as a “causal power”, there is no problem. But if it is going to be understood as something reducible to (could) counterfactual conditionals or something that makes the occurrence of a manifestation ‘necessary’, then clearly causal powers are not ‘dispositions’.

9.1.5

Do Transcendent Universals Confer Causal Powers?

§ 66. It is convenient now to return to the central point of this discussion. According to the Eleatic principle, there is only that which produces a difference in the causal powers of something. If transcendent universals are acceptable entities in our ontology, then they should generate some difference in the causal network of the world. We are now in a position to understand more precisely in what this ‘difference’ would consist. There are two major ways in which a ‘causal power’ can be understood, as explained: (i) from a combinatorial perspective, according to counterfactual analysis V, or (ii) from a causal perspective of modality, as a fundamental entity, satisfying principle of Causal Power III. That is, an entity is supposed to make a difference, whether it might cause something—in the modal neighborhood of possible worlds close enough to the world of evaluation—or whether something can cause something—without restrictions to the vicinity of worlds ‘closest’ to the world of evaluation. The formulation of the Eleatic principle, which follows Armstrong’s formulation (see Armstrong 1997, 41), has causal connotations that must be carefully avoided. When it is said that every entity must ‘produce a difference in the causal powers of something’ this should not be understood as ‘causing’ something to come to have a causal power that it did not have before, or no longer have a causal power that it previously had. It is more accurate to maintain that the existence of the entity in question must ‘ground’ the existence of a causal power or, at least, it must be something on which the existence of the causal power is ‘dependent’. The formulation of the Eleatic principle also has temporal connotations that should be avoided. It is not that the existence of an entity should cause something that at one time did not

31

Perhaps, one of the reasons why this type of approach has been preferred is the prevalence of reductivist theories of causality during the last century. Since causality should be reduced to something else, one is inclined to think that structures in which, given the occurrence of the cause, the effect must occur, are appropriate for this reduction.

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have a causal power, to have it at another time; or that something that at one time had a causal power, then at another time will not have it. How the ‘difference in the causal powers of something’ occurs is simply because the existence of the entity in question in a possible world grounds the causal power of something in that world. It does not matter if to what has conferred the causal power possesses such power at all times when it exists in that world or does not possess it at all times when it exists in that world. Another unfortunate association that should be avoided, is that when it is indicated that the entity in question must establish a causal power or be that in which that causal power is dependent on in a possible world, it should not be inferred from this that there must be other possible worlds in which it does not confer such power.32 How ‘a difference in the causal powers of something is produced’ is done, is by grounding the fact that something has a causal power or being something that those causal powers are dependent on. The fact that this occurs in all possible worlds or only in some is not relevant. Given these preventions, a response to the objection can now be given. Transcendent universals fully satisfy the demands of the Eleatic principle. The existence of a transcendent universal ‘produces a difference in the causal powers of something’ trivially because it partially grounds all causal power that any object has and it is something on which every causal power is ontologically dependent. Of course, if one does not accept the existence of transcendent universals for other reasons, there is no contribution to be expected from them for the causal network of the world— causal contributions of what does not exist cannot be expected. But if one admits transcendent universals, and the objection is precisely considering the causal contribution that these universals would make, then one must admit that they fulfill the theoretical functions that have been attributed to them. And among these theoretical functions is—naturally—to confer any causal power. This is evident from the outset by the formulations of the analyses of Causal Power III and V—and it would be the same with any of the others—because something has a causal power because it instantiates some universal at some time. Something cannot have causal powers without instantiating universals. Then, if there are no universals, there are no causal powers at all. Indeed, from a combinatorial perspective of modality, something has a causal power because the relevant state of affairs might cause something in the ‘closest’ 32

In the case of the causal powers of contingent entities it seems obvious that there are possible worlds in which such entities possess those powers and possible worlds in which they do not, either because the causal powers in question are contingent for those entities, or because the entity itself is contingent—if the causal power is essential to it. The prevention indicated above has to do with the case of essential causal powers for a necessary entity. The obvious example is God. God has always been characterized as an omnipotent entity. The causal power of God is maximal so that everything depends causally on Him. But it is not an accidental feature of God that he might not have had if the circumstances had been different. God is essentially omnipotent. He is—to put it in some way—the most important ‘piece’ of the causal fabric of the world. What grounds that causal power, or what it depends on—if it depends on something—“produces a difference in the causal powers of something”, even though there are no possible worlds in which God does not have such maximal causal power.

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worlds to the world of evaluation. If one admits transcendent universals, one state of affairs causes another because there is a qualitative character in the object in question—if one adopts a regularist or counterfactual position of causality—and because there is also a qualitative character in the object that ‘receives’ the causal influence. And something has some qualitative character, from the perspective of an ontology of universals, precisely because there are universals. In the case of a combinatorial conception together with a non-reductive conception of causality, the connection is much more direct, since causality is a relationship between states of affairs, and a relationship is a universal. From a causal perspective of modality, on the other hand, an object has a causal power because it instantiates some universal or another. These causal powers are universals essentially tended towards what they would cause in different circumstances. The modal space is the space of different causal interactions to which the instantiations of these universals are essentially tended. It turns out, then, that under any reasonable conception of what a causal power is, transcendent universals ‘produce a difference in the causal powers of something’ by partially grounding all the causal powers of all objects and by being something of which all causal powers depend on. Probably, one reason why many have been inclined to think that transcendent universals would violate the requirements of the Eleatic principle is to contemplate a scenario in which the causal fabric of the world remains the same, but transcendent universals are ‘subtracted’. Thus, it seems that the Platonic universals do not make any causal contribution because if they did not exist, the causal network would continue as it is. But this is an illusion. Transcendent universals are necessary entities. There are no metaphysically possible worlds in which they do not exist. The ‘scenarios’ referred to are perhaps conceivable, but not metaphysically possible. And if there are no transcendent universals, there is nothing grounded on such universals or nothing dependent on such universals. The objects would not, therefore, have any qualitative character and no causal power. If one is postulating an ontology with transcendent universals, which fulfill the theoretical functions already presented with latitude in this work, then a scenario without universals is a scenario without causality and causal powers. Perhaps, an additional motivation for being inclined to think of transcendent universals as causally inert is that, in some sense, they are not ‘in’ particular states of affairs and ‘in’ particular objects. What is ‘in’ particular states of affairs and ‘in’ the particular objects are the particular instantiations of the universals. These instantiations have spatiotemporal localization, they are what enters the causal relations and they are what we perceive as ‘properties’ in a particular object. As will be explained below (§ 69), the ‘particular instances’ of universals are tropes, that is, particular properties. One could, then, have the impression that these tropes could exist by themselves, without any universal of which they are the particular instantiation. However, again, this is an illusion. In an ontology of universals, particular instantiations are essentially instantiations of a specific universal. If one is going to admit tropes in an ontology of this type, tropes must be conceived as entities ontologically dependent on the universals of which they are like a ‘shadow’ or a ‘reflection’. There

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are no self-subsisting shadows or reflections. Every shadow is a shadow of something and every reflection is a reflection of something. As can be seen, after this long examination, transcendent universals are not in violation of the Eleatic principle according to which everything that exists produces a difference in the causal powers of something. Understood this principle in a coherent way and under any credible conception of what is a causal power, it happens that transcendent universals partially ground all possible causal connections and, with it, all causal powers.

9.2

Lack of Economy

§ 67. There is a second line of objection against transcendent universals connected with questions of theoretical economy. It is well known that ceteris paribus, a theory should be preferred if it postulates fewer entities or fewer fundamental categories of entities than another. This is why we speak of ‘quantitative’ economy and ‘qualitative’ economy. One theory may be preferable to another because it requires a smaller number of entities of some kind. One theory, too, maybe preferable to another because it requires postulating fewer fundamental categories of entities. These requirements of ‘quantitative’ economy and ‘qualitative’ economy may be in tension with each other. A less quantitatively economical theory can be more qualitatively economical, and vice versa.33 What has been argued in this respect is that the theories that postulate transcendent universals incur in an important lack of quantitative and qualitative economy concerning their alternatives (see Armstrong 1989a, 132–133). It is assumed that in the theories of immanent universals, they are postulated as something that is ‘in’ their instantiations and the objects that possess them. Immanent universals are entities that do not imply something ‘outside’ or ‘above’ the spacetime system. This makes these universals result —at least, prima facie— adapted to a ‘naturalistic’ ontology (see Armstrong 1997, 5–6). Platonic universals, on the other hand, are ‘transcendent’ to the spacetime system. They seem ‘foreign’, ‘distant’, and ‘separated’ from the concrete world that we all inhabit. If one is going to admit that these universals contribute to the causal network —as has been argued in the previous section (see §§ 61–66)— it is because such universals have particular instantiations that ‘exemplify’ them. Such instantiations are located spatially and temporally and are relata of causal connections. In some sense, one can argue that

33

The most characteristic example of this type of tension is that which has been presented with the possibilist modal theory of David Lewis. On the one hand, Lewis multiplies concrete entities to the infinite, because there is an indenumerable multitude of different possible worlds that are entities of the same nature as our world, only disconnected spatiotemporally from us and each other. This is a gross lack of quantitative economy. But, on the other hand, the multitude of possible worlds of Lewis offers reductions of modal facts, of causal facts and properties (see Lewis 1986, 1–69). These reductions—if they were effective—would offer an important qualitative economy.

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the tropes of the transcendent universals are ‘in’ the particular states of affairs and are ‘in’ particular objects. Here it becomes clear why it has been alleged that theories of transcendent universals incur in a lack of economy. They are, in effect, theories that must postulate tropes together with universals, while theories of immanent universals seem to postulate only universals.34 This implies that there is a quantitative multiplication of entities because while the Aristotelian postulates a universal multiply instantiated, the Platonic postulates a universal and also tropes for each of its instances. There is also a qualitative multiplication of entities, because while the Aristotelian postulates one category of entities —universals— the Platonic postulates two —universals and tropes.35 It is going to be shown here that there is no such lack of economy. Aristotelian theories are as economical —or as ‘uneconomical’— as Platonic theories. In particular, theories of immanent universals must also postulate tropes, in the same way as theories of transcendent universals (see Alvarado 2012a, b, 2013b). Before entering into these considerations, however, it will be useful to make a general reflection on the problems of ontological economy. In principle, to satisfy the same explanatory requirements, the most economical theory should be preferred. If there are two theories and both explain equally well certain facts that demand an explanation, then the one that proposes a smaller number of entities or a lower number of categories of entities should be preferred. It is fundamental, then, that the most economical theory is explanatory apt if the considerations of economy and simplicity are going to have relevance. If one considers the dialectical situation in the debate in metaphysical of properties, what can be noticed —considering all that has already been explained throughout this work— is that none position in metaphysics of properties can replicate the theoretical advantages of transcendent universals. It is not, then, that we are facing two different theories that equally satisfy certain theoretical roles. If this were the case, the most economical of them should be preferred, either quantitatively or qualitatively. We are confronted with two theories, one of which is far superior to the other in its explanatory aptitude and its theoretical performance to the other. If you like, considerations of economy can be added here as another element for weighing the advantages or disadvantages of the theories when compared to each other. In no way, therefore, considerations of economy can be assumed by themselves as decisive for the debate between Aristotelians and

34

Naturally, the same lack of economy is noticed when comparing theories of transcendent universals with theories that aim to reduce universals to classes of tropes and nominalist theories. The friend of tropes posits one category of entities, while the Platonist posits two. For each trope posited by the friend of the tropes, the Platonic must postulate the same trope and also some universal of which it is an instantiation. In comparison to nominalism, the lack of economy is manifest. 35 In this category count, particular objects are being set aside, of course. Typically, an Aristotelian such as Armstrong, along with postulating universals, also postulates particular objects (see Armstrong 1978, 1989a, 1997). He could also, in the abstract, postulate bundles of tropes or universals to fulfill their functions. The same alternatives are open, in the abstract, for the Platonist. Below (see Part III, §§ 88–95) an ontology of nuclear bundles of tropes will be defended.

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Platonists, without weighing the totality of theoretical functions that fulfill one and another conception. In the remainder of this section, it will be shown, however, that there are no theoretical economy advantages of the Aristotelian over the Platonic. This has to do crucially with the systematic problems generated by the so-called “Bradley’s regress”.

9.2.1

Bradley’s Regress

§ 68. The so-called “Bradley’s regress” is a systematic problem that arises whenever, in some way, a plurality is intended to be something unified (see Briceño 2016; Maurin 2012; Bradley 1897, Chap. 3).36 It arises for a multitude of philosophical positions very different from each other.37 When it comes to universals, the problem takes the following form: a universal can be instantiated in one object or another, and an object can instantiate a universal or another.38 A universal doesn’t need to be instantiated in a specific particular object, although if it is an immanent universal, it must be instantiated in some or other particular object to exist.39 This has already been explained above (see § 18), but it will be useful to recall it now in its general lines. Suppose that in the possible world w1 the object a1 instantiates the universal U1, and the object a2 instantiates the universal U2, while in the possible world w2 the object a1 instantiates the universal U2 and the object a2 instantiates the universal U1. There are no differences between the possible worlds w1 and w2 as to what particular objects and what universals exist in them, but there are differences as to what states of affairs occur in those worlds. Why does an object and a universal enter to form something one, such as a fact, situation, or state of affairs? One might be inclined to argue that what distinguishes w1 from w2 is that in w1 the object a1 instantiates the universal U1 and does not instantiate the universal U2, as in w2. Also, in the possible world w1, the object a2 instantiates the universal U2 and not the universal U1, as in w2. Apparently, therefore, the facts about what is or is not under the instantiation relationship are what determine the differences between w1 and w2. More precisely, the fact that a1 is U1 is grounded on the fact that a1 instantiates U1. The same goes for

36

Gaskin notes that this type of problem has also been presented previously by Avicenna, Abelard, Scotus, Ockham, Buridan, Gregory of Rimini and Suarez (see Gaskin 2008, 314). 37 The general form of the problem is that, if there are is a plurality of entities, let x1, x2, ..., xn, which are somehow ‘one’, there must be something that works as the unifier of x1, x2, ..., xn, let this ‘unifier’ be z. But z can only fulfill its function if it is, in some way, unified with x1, x2, ..., xn. Appealing to another unifier would generate a vicious regress. 38 With the exception, of course, of properties essential for the object (§ 2). These will be left aside to simplify the discussion in what follows. 39 If these are universals that are essential for an object, the object in question may not exist. This has the exception, of course, of the essential properties of objects of necessary existence (§ 2). These cases will be set aside to simplify the discussion in what follows.

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other cases. ‘Instantiation’, however, is a relationship. If one defends an ontology of universals, this relationship should be treated like any other, that is, as a multigrade relational universal.40 But for a universal —whether relational or not— it is not necessary to be instantiated in such entities rather than others. In the possible world w1, the instantiation relation connects the universal U1 and the object a1, while in w2 it connects the universal U1 and the object a2. Therefore, the existence of the relation of instantiation in w1 is not sufficient to ground that a1 has U1 and that a2 has U2 and not the other way around. One, then, could at this point be inclined to suppose that what differentiates the possible worlds w1 and w2 are not simply the facts about the instantiation relation in objects and universals, but the facts about the instantiation of the instantiation relation in objects and universals. That is, in the possible world w1 the relation of instantiation is instantiated in the ordered pair and not in the ordered pair , such as in w2. The fact that a1 has U1 is grounded on the fact that a1 instantiates U1. And the fact that a1 instantiates U1 is grounded on the fact that the relation of instantiation is instantiated in . It is obvious that here again appeal is made to a relation of instantiation between instantiation and < a1, U1>.41 But a relation of instantiation must be understood as any other from a metaphysics of universals. It must be a relational universal that is not necessarily instantiated in certain entities rather than in others, and so on. It follows an infinite chain of facts of instantiation, none of which is fundamental. It is, therefore, a vicious infinite regress. There are analogous difficulties for all positions in metaphysics of properties. There has already been an occasion to consider some of these cases of Bradley’s regress for resemblance nominalism (§ 2), for theories of classes of tropes (§ 26) and theological nominalism (§ 36). For example, in predicate nominalism, it is argued that the ground for the fact that an object x is F is the fact that to x is predicated truly the predicate “is an F”. But the fact that to x is predicated truly “is an F” is a relational fact. It should be treated in the same way as any other relational fact by a nominalist of predicates. This is, the fact that to x is predicated truly “is an F” must be grounded on the fact that to the ordered pair is preached truly “to --- is predicated truly ---”. It is clear, however, that this is another relational fact that must be grounded on the predication with truth of some predicate, etc. The regress of Bradley, then, is not a difficulty that specifically affects theories of universals, although it has been presented it that way sometimes. It is well known how David Armstrong gave this type of objection against each position in metaphysics of properties except for his own (see Armstrong 1978, 18–21, 41–42, 53–56, 69–71). When it comes to explaining the nature of the connection between an object and the immanent universals that that object possesses, Armstrong argues that

Indeed, it is assumed that the relation of ‘instantiation’ must connect n-adic relational universals—for any n—and the n objects that fall under them. The case of monadic universals is discussed above to simplify the discussion, but what is said applies to arbitrary n-adic relationships. 41 See § 18 concerning a single relation of instantiation infinitely instantiated in itself versus infinite relations of instantiation of different logical types different from each other. 40

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Aristotelianism is a ‘non-relational realism’, for no relation between the particular and the universal object is being postulated (see Armstrong 1978, 108–111). The connection between universal and particular would be a case of the famous ‘formal’ distinction of Duns Scotus. It would be the connection between entities, in some sense inseparable from each other, but really different. Similar ideas have been proposed by Peter Strawson (“non-relational tie”; see 1959, 167–173) and by Gustav Bergmann (“nexus”; see Bergmann 1967, 3–21). In all cases, it refers to a ‘connection’ or ‘link’ that, in some way, is not a ‘relationship’. Appealing here to the Scotist ‘formal distinction’ is hardly a way to clarify the nature of this ‘link’. The difficulties that exist to specify what Scotus understood by such thing are well known (see Cross 1999, 149). Lewis pointed out that, if Armstrong is legitimated to maintain that ‘instantiation’ is an ontologically special fact, a ‘primitive’, which should not be treated like any other relational fact, then all other positions in metaphysics of properties should be allowed to do the same with their preferred ‘primitives’ (see Lewis 1983, 20–25). If ‘instantiation is a ‘non-relational tie’, why could not the nominalist maintain the same about ‘resemblance’, or ‘predication’, or the Platonic about a special bond of ‘participation’? It has become common to argue that the solution to Bradley’s regress is to postulate some ‘primitive’ fact. Defenders of universals will say that their ‘primitive’ is ‘instantiation’. Nominalists will maintain, for their part, that it is ‘resemblance’, ‘predicating truly’, ‘judging truly’, ‘belonging to a class’, or ‘being part of’. In this way, the fact that x is F is grounded on the fact that x instantiates the universal of being F, but here ‘instantiation’ is a primitive fact that does not admit any further explanation. There should be no search for something on which such a fact should be grounded. It has already been noted above that this position is unfortunate (see § 18). When speaking of a “primitive”, it seemed that this was a matter of choosing one predicate or another without this having important consequences. For example, standard extensional mereology can be formulated using as primitive the predicate “being a proper part of” or “being an improper part of”. The choice will have to do with pragmatic or aesthetic motivations but does not generate any difference for the strength of the theory that is formulated using any of these predicates. There is a feeling here that, in some way, all the discussion of metaphysics of properties had to do with the proper selection of a ‘vocabulary’ to express the ‘same facts’. But these analogies are most unfortunate. The qualitative character of each existing object is grounded on instantiation facts, according to an ontology of universals. Qualitative characters are not artifacts of our choice of vocabulary. Nor is the artifact of our vocabulary the causal fabric of the world. To hold that ‘instantiation’ is a ‘primitive fact’ is especially implausible in an ontology of universals. In fact, in an ontology of this kind, every relation is the exemplification of a universal. Instantiation is a relationship between universals and individuals. It is not sufficient justification to

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treat it differently —as a relation that is not a relation— the desire to avoid the regress of Bradley.42

9.2.2

Tropes of Instantiation

§ 69. There is a very simple way to block Bradley’s regress, however, without having to resort to more or less extravagant primitive facts. This alternative has already been presented by several philosophers (see Mertz 1996, 184–195; Meinertsen 2008; Wieland and Betti 2008; Maurin 2011; Alvarado 2012a, 2014b). Recall that the regress of Bradley arises because it is expected that the fact that an object instantiates a universal is grounded on another ontologically more basic fact. If instantiation is a universal relation like any other, the existence of the object, the universal and the eventual relation of instantiation is not enough to ensure that, in effect, the object instantiates the universal in question. Suppose now that what we call “the instantiation of the universal in the object” is an entity whose nature is precisely that of being the particular exemplification of that universal in that particular object. The ‘particular exemplification’ of a universal is a property, but particular, as particular as the object in which ‘it is’. Thus, the fact that any object x has the nature F—that is, the fact that x is F—is grounded on the fact that the universal being an F is instantiated in x. But this ‘instantiation’ is not a universal relationship. It is exactly the instantiation trope of the universal of being F in the object x. This is a trope that cannot exist but as the instantiation of the universal being F and cannot exist except as that instantiation in the object x. When considering this trope ‘F-@x’, its existence is sufficient to ground the state of affairs of x being F. This trope ‘F@-x’ is ontologically dependent on the universal being F and the object x, so, if the trope ‘F-@-x’ exists, then the respective universal and object must also exist. In this way, the trio of the universal being F, the object x and the trope ‘F-@-x’ ground the fact that x is F, and do not require, subsequently, something that is its grounding. The regress of Bradley is effectively blocked. Many theories of tropes have maintained that tropes—or most of them—are independent entities that can exist jointly or separately (see Williams 1953a, 1953b; Campbell 1981, 1990: Maurin 2002; Ehring 2011). A trope that essentially is the instantiation of a universal in a particular object is not consistent with these

42

Several other strategies have been proposed to evade or block Bradley’s regress that will not be discussed here. Grossmann, for example, has argued that relationships —such as instantiation— make the connection by themselves, without the need for anything else (see Grossmann 1983, 167–170). But if the relation in question is a universal, then it is not sufficient to produce the required ‘unification’. Gaskin and Orilia, on the other hand, have argued that an infinite sequence of facts grounded on each other would not be a problem. A finite sequence of ground would be problematic, but the leap to infinity would generate an acceptable grounding structure (see Gaskin 1995, 2008, 314–318; Orilia 2009). The problem is that in an infinite sequence, there would not be a first fundamental ungrounded element. This fact makes the regress equally vicious.

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positions. However, there is a long philosophical tradition that has conceived tropes as entities ontologically dependent on others. The postulation of tropes with the nature that has been proposed here is in accordance with this venerable tradition. On the one hand, philosophers such as Charles Martin and John Heil have defended tropes that depend ontologically on a substratum (see Martin 1980; Heil 2003, 137–150; 2012, 53–83). These tropes can only exist as determinations of the substratum to which they determine, although in these ontologies universals are not postulated. It is assumed that classes of similar tropes must fulfill their functions. On the other hand, philosophers such as Edmund Husserl, Peter Simons or Markku Keinänen (see Husserl 1913, Untersuchung III, §§ 20–22; Simons 1994; Keinänen 2005, 2011) have postulated tropes that are ontologically dependent on each other. The bundles that satisfy the functions of particular objects in these ontologies are made up of pluralities of tropes that are all of them—and they alone—ontologically dependent among themselves. Finally, philosophers such as John Cook Wilson (1926), Norman Kemp Smith (1927a, b, c) or Jonathan Lowe (2006, 20–33) have postulated tropes that are essentially the instantiation of a universal. Lowe’s position postulates, in addition to ‘modes’—that are essentially the instantiation of a universal— ‘substances’—that are, in turn, essentially the instantiation of a ‘kind’. What Lowe calls “kinds” are universals of substance. All these positions comprise the tropes as ontologically dependent entities, either of the substrata they are characterizing, or of other tropes with which they form a bundle, or of universals of which they are essentially an instantiation. What is being done here, therefore, by introducing a trope that functions as the ‘unifier’ of a state of affairs, is not foreign to the tradition of trope theories. In recent work, Sebastián Briceño has criticized this way of blocking the regress of Bradley (see Briceño 2016). He argues that invoking a trope that is essentially the instantiation of a universal in a specific particular generates a dilemma: (i) either the fact that this trope is connecting universal and object requires of a grounding—in which case, one is again in the presence of the regress—, (ii) or else it is the nature of the trope to make the connection, so nothing is required that grounds or explains it. In this second case, however, no explanation would be offered, since the trope of instantiation is itself a kind of complex unit. By postulating something that is, in itself, a complex unit, Bradley’s challenge is not being answered. The requirement that Briceño imposes for an explanation to be acceptable, however, is excessive. Tropes that are essentially the instantiation of a universal in a particular object are fundamental, although ontologically dependent.43 According to Briceño, if they are not grounded, then they are not an adequate explanation. But one cannot pretend that every entity is grounded on another. Some entity or another must be fundamental. Ontologies differ among themselves concerning which categories of entities are those considered fundamental. Some will postulate particular objects. Others will

43

They are, in effect, ‘emergent’ entities according to the terminology introduced above (see § 4).

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posit tropes. Others will posit universals and substrata.44 By postulating tropes and universals as fundamental entities, therefore, nothing theoretically illegitimate is being done. They are the categories that seem most reasonable to satisfy the different theoretical functions that have been discussed here at length. The mere fact that they are fundamental does not make them inappropriate.45

9.2.3

Primitive States of Affairs

§ 70. A different attitude towards Bradley’s regress—at least, apparently—has been to argue that facts, situations or states of affairs must be considered fundamental entities. The regress of Bradley arises because one is looking for something on which a state of affairs is grounded. Neither the universal, nor the object—or objects—that make up such a state of affairs are that ground. A universal relation of instantiation would not either. It seems legitimate in a situation of this kind to simply give up looking for a ground of states of affairs. Something analogous is what is done when one postulates universals instead of trying to reduce them to resemblance classes of objects, resemblance classes of tropes or whatever. Something analogous is also done when fundamental causal relations are postulated instead of trying to reduce them to regularities of events or forms of counterfactual dependence. Not everything can be explained. Explanations must conclude at some point or another. Grounding chains—as forms of ontological ‘explanation’—also. Of course, one can criticize a theory for postulating too many fundamental facts, or fundamental facts that are obscure or unfeasible. The postulation of fundamental facts should be weighed with the theoretical advantages that such postulation brings. This weighting can finally throw a positive or negative verdict. There is no problem in principle, however, with such a postulation. According to this strategy, instead of looking for what entity or entities are those in which a state of affairs is grounded, it can be maintained from the beginning that states of affairs are a fundamental category of entities,46 as are universals and Briceño, on the other hand, postulates a ‘negative primitivism’ that accepts complex entities such as a lump of sugar, but without demanding a metaphysical explanation of such unity (see Briceño 2016, 70–73). A ‘primitivism’ whether or not it is ‘negative’, is postulating precisely certain entities as fundamental, without the need for further grounding. And it’s about things like sugar cubes, i. e., instances of complex units. 45 What leads Briceño to consider tropes to be inadequate to resolve Bradley’s regress is that they would be a form of ‘multiple-unity’ and Bradley would be arguing that such a thing is incoherent in itself. A plurality cannot be identical to something one, because if it were so, it would cease to be a plurality (see Briceño 2016, 49–51). Whatever the merit of this claim against the intelligibility of mereological fusions, classes, or sets —paradigmatic forms in which a plurality is ‘unified’— this does not seem sufficient reason to declare the tropes that are here being postulated incoherent. The fact that a trope is ontologically dependent on a universal and an object does not make that trope a plurality. 46 These would be ‘emergent’ entities, as explained: not grounded, but ontologically dependent. 44

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particular objects. If Bradley’s regress showed that states of affairs could not be grounded, then we should simply conclude that they are ungrounded. A position of this kind is one that has been defended by Herbert Hochberg (see Hochberg 1978, 339–346) and, very notoriously, by David Armstrong in the last decade of the last century (see Armstrong 1997, 113–127). Armstrong presents an ontology in which the world is populated by states of affairs. The universals are ‘modes of being’ of objects. The objects are the ‘particularizers’ of states of affairs. Neither universals nor thin individuals can exist outside existing states of affairs. We conceive their separate existence by a kind of ‘abstraction’. When thinking about a universal property, we put into parentheses all the states of affairs that they are integrating. The same happens with thin individuals. Every ‘thin’ particular is included in a ‘thick’ particular, which is the fusion of all the states of affairs that overlap in the same ‘thin’ particular. The question here is what difference exists between a trope that is essentially the instantiation of a specific universal in a specific particular and a fundamental ‘state of affairs’, whose identity is fixed by the universal that integrates it and the particular object that integrates it—or the particular objects that make it up, if it is a relational universal. The trope in question is intrinsically the exemplification of the universal in an object and the Armstrongian state of affairs as well. A state of affairs depends ontologically on the universal and the particular object—or particular objects—that constitute it, and the trope of instantiation does as well. An Armstrongian state of affairs works as a truthmaker of true propositions, as causal relata and, in many cases, as the object of our perception. An instantiation trope fulfills exactly the same functions. Then, it seems evident that the difference between a fundamental or ‘primitive’ state of affairs and a trope that essentially is the instantiation of a universal in a particular object is only a difference of name. Philosophers like Armstrong, then, are also postulating tropes, but under another denomination. They intend to reject the introduction of tropes in their ontology of universals— because they are something “not economical”—but they postulate an additional category of fundamental entities, ‘states of affairs’. If there is a cost of economy in postulating tropes together with universals, the same cost of economy is that which is incurred when incorporating ‘states of affairs’. It turns out, then, that the situation of an Aristotelian position over universals is the same as that of a Platonic position. The difficulties generated by the regress of Bradley in both cases require the same maneuvers. Leaving names aside, the proposed solutions are the same solution. A universal, by itself, is not instantiated in one object rather than another. Ontologically, something additional is required: the ‘exemplification’, ‘presence’, ‘participation’ or ‘instantiation’ of the universal in the particular. It is these ‘instantiations’—whether they are called “tropes” or “states of affairs” is irrelevant—that we perceive as properties of objects, what interacts causally and what it works as truthmakers of most true propositions. The accusation of lack of economy directed against Platonic theories of universals is, then, simply false.

9.3 The Epistemological Problem

9.3

233

The Epistemological Problem

§ 71. The third type of objection against transcendent universals has to do with epistemological considerations. Platonic universals would be entities epistemologically disconnected from our cognitive capacities. We would have no way of knowing its existence unless one is willing to admit ‘special’ cognitive faculties appropriate to the ‘world of forms’. It is well known that Plato argued that the soul in a disembodied state before it has been ‘imprisoned’ in some body, has been able to contemplate ‘forms’ directly. Then one forgets such forms. Sense perception serves as a ‘memory’ of the forms that were directly known before (see Plato 1982, 81a-86c; Plato 2009, 69e-84b). For any contemporary philosopher, an explanation in these lines is simply a myth, which can be appreciated for its aesthetic virtues or certain moralizing resonances, but whose verisimilitude is almost nil. An ontology of transcendent universals that forces us to accept something like ghosts to explain how we know of them is an ontology for which our complete natural science is a reason for rejection. On the contrary, if these were the epistemological consequences of the postulation of Platonic universals, one could try to seek refuge in some form of skepticism. This would be equally unacceptable. We claim to know—or, at least, to have well-justified beliefs—about many aspects of the world around us. This knowledge must be knowledge of natural laws that, as explained above (see § 14), is knowledge of universals. If the universals are transcendent to our knowledge, then all our natural science is a gigantic illusion. It has been highlighted above how metaphysics must be in harmony with epistemology. A radical skepticism of the kind that has been described would be in violation of such a demand. Such skepticism would be, besides, self-refuting. In effect, the reason for adopting such a skeptical position would be the impossibility of knowing universals. But if this is the case, we do not have reasons in the first place to postulate universals that have to be transcendent to our knowledge. One way of formulating this epistemological critique can follow the lines of what was stated by Paul Benacerraf (1973) about the tensions between epistemology and metaphysics in the philosophy of mathematics. On the one hand, one is inclined to postulate an ontology of abstract mathematical objects. On the other, however, one should try to make that ontology coherent with a ‘reasonable’ epistemology. For Benacerraf—as also for many people today—a ‘reasonable’ epistemology is an epistemology according to which it is a necessary condition for a rational subject to know that p to have a causal connection with the entities described by the proposition that p (see Benacerraf 1973, 22–25).47 This epistemological restriction establishes that entities whose existence make them transcendent to what can be given to us with some form of causal transaction should not be postulated. Peacocke then generalized this tension between metaphysics and epistemology and called it the “integration challenge” for a given domain (see Peacocke 1999, 1–12). Whatever 47

Note that this causal restriction is a necessary condition for knowledge. It is not intended here to ‘analyse’ the concept of ‘knowledge’ in causal terms.

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the domain in question, if one maintains that there are facts of type D, then one must also be able to give an adequate explanation about how we might have knowledge of the facts of type D or, at least, beliefs sufficiently justified about facts of type D. That knowledge does not need to be exhaustive. It is not required that every fact of type D is known or knowable, but we must have some epistemic access to some type D facts. Otherwise, our ontology about such facts becomes doubtful. In this work, we have appealed to this challenge of integration several times about philosophical positions that make our knowledge of natural laws impossible or exceedingly difficult (see §§ 19, 33). What we see now is that the same kind of considerations applies, but against the ontologies of Platonic universals. It is important to keep in mind that—in general—there is no unique strategy to face an integration challenge concerning the ontology of certain facts of type D. In the abstract, one could solve the tension by modifying the ontology, but also by adjusting the epistemology. For example, for the mathematical case, perhaps the adjustment required is not to “cut out” the most recondite areas of set theory, but to explore epistemologies that do not depend on a causal constraint. There are also more radical options, either skeptical or not-factualist. One could declare facts of type D unknowable or one could declare that there are simply no facts of type D.48 Therefore, if one were to discover that there really is a challenge of integration in an ontology of Platonic universals, that is still not a reason, in itself, to declare them non-existent. It is a reason to look for a reasonable solution to the challenge. And the adjustment can be made both in metaphysics, as in epistemology, as in both. It is clear, however, that a theory of transcendent universals will not yet be well received without being accompanied by a reasonable epistemology that explains how we know such universals and how, moreover, natural science has as its main purpose the exploration of such universals. What is going to be done in this section is to show that there is nothing epistemologically extravagant in the postulation of universals. We access them in our ordinary dealings with the world and, of course, also when sophisticated empirical or formal research methodologies are applied. To show this, we will attend in some detail to how our perceptual beliefs become justified and the role that “a priori intuitions” or “a priori reflection” play in shaping such justified beliefs. It is not intended to present any great novelty. Fortunately, there is already an important stream of work in epistemology that has been thematizing a priori knowledge and on which one can rest (see, for example, BonJour 1998; Boghossian and Peacocke 2000; Peacocke 2004; Chudnoff 2013; Casullo and Thurow 2013). These works show an important degree of convergence with the epistemological work on these same issues in the phenomenological tradition (see Husserl 1913, Untersuchung II; 48

Peacocke describes seven different ways in which an integration challenge could be solved or alleviated (see Peacocke 1999, 7–12). Some of those forms seem to be already included in the options presented above. For example, Peacocke argues that one could maintain that the meaning of propositions that enunciate facts of type D can be treated not as having definite truth-conditions, but only certain assertion-conditions. But this implies adopting an anti-realist position regarding facts D, which is a drastic modification of the ontology of D-type facts.

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1948).49 It will be shown that the integration challenge is solved for the ontologies of Platonic universals through an adequate epistemology of our a priori intuitions. Of course, it is not possible to present here a complete epistemological theory of our a priori intuitions. It will be necessary to take many issues for granted, and many discussions will be ignored. Knowledge is to be understood simply as the ‘most general’ factive mental state (see Williamson 2000, 21–48). A mental state Φ concerning p is factive if and only if, if the rational subject S has Φ to p, then p. Knowledge is the most general factive mental state because: (i) it is a factive mental state—if S knows that p, then p—, and (ii) if S has a factive mental state Φ to p, then S knows that p. It is not going to be assumed any way of ‘analysing’ the notion of ‘knowledge’ and, therefore, of answering Gettier’s problem. Contemporary epistemology is traversed by discussions between ‘internalist’ and ‘externalist’ positions.50 Nothing of what is going to be explained here depends on a position concerning this controversy. It will be necessary, however, to adopt a foundationalist conception of justification and knowledge. According to a foundationalist conception, a rational subject S is justified in believing that p because she has a non-inferential justification that p (for example, she has perceived that p, or has the intuition a priori that p), or is justified in believing that p because p is inferred from propositions for which S also has justification. The chains of inferential justification must be finite and must end in some other non-inferential justification. What is going to be shown in this section is that there is a cognitive mechanism of non-inferential justification that offers us a priori justifications by intuition. It is the exercise of these cognitive abilities that unfolds in the knowledge of universals.

9.3.1

Perceptual Beliefs

§ 72. I see a red apple on table.51 The kind of perceptual evidence I have about the table and the apple is justification to come to believe that there is a red apple on the table. That evidence is made up of information about the chromatic distributions in my visual space. The information is temporarily integrated so that different ‘aspects’ of the apple and the table that are offered at different times are unified as ‘aspects’ of the same object.52 For what interests here, there is a huge amount of sensory

49

I am grateful for the help of my colleagues Eric Pommier and Mariano Crespo, who have guided me in the treatment of these issues in the phenomenological tradition. 50 In an ‘internalist’ conception, if the rational subject S knows that p, then S knows that she knows that p. The same goes for justification. An ‘externalist’ position, on the other hand, relaxes this requirement for reflexivity. It is not necessary to know that one knows to know. It is not necessary to know that you have a justification, to be justified in believing something. 51 The focus will be on visual perception to simplify the discussion, but what is said about it is applicable, mutatis mutandis, to other forms of perception. 52 For example, the visual image that I am receiving from the table is that of a rhombus, given the perspective in which I am to the table. What immediately appears to me about the shape of the table,

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information that one is receiving when one sees what one is seeing. Not only one is receiving visual information about the apple, the table, and the color of the apple. One also receives information about the relative size of the apple and the table, the direction and the angle of incidence of the light on the apple and the table, on multiple irregularities on the surface of the table and the apple, and the shape of the apple and the table. In a normal situation, in addition, there will be many other objects in the vicinity that will also enter the visual field. One would be inclined to say that there is infinite visual information in the visual field—if one supposes that the visual field is a continuum—but for all we know the visual information comes into discrete packets of photons in time and is processed by discrete packets generated by the excitation of a finite number of photoreceptor cells in the retinas, and then by a finite number of neuronal synapses. While still finite information, it is disproportionately large compared to that to which a subject can attend. Our attention will focus only on a minor part of that information. Most of it will be forgotten as irrelevant. What the subject concentrates on will depend not only on the qualitative nature of what is offered to us phenomenally but especially on what is important or relevant to him. When making this selection of visual information, there is already an ‘active’ or ‘spontaneous’ component from the subject, which is not merely ‘passive’ to what it sensibly receives. Then there is a ‘leap’ between the visual information received, attended and processed cognitively and the perceptual beliefs that the subject can come to form due to such visual information. A belief is a propositional attitude. Someone who believes something believes that something is the case, and what is described as the object of such belief is something that must be expressed by a proposition. The same perceptual information that has been given attention to can serve as a justification for a multitude of different beliefs. Seeing what one is seeing one can justifiably believe, for example, that there is a red apple on the table, but also that there is an apple, there is a table, there is something red on the table, there is something heavy on a rectangular object, there is a shade of red on that surface, there is something with a shape close to a circle, and so on. In the same way, the belief that there is a red apple on the table could be justified by an infinite multitude of different perceptual information. For example, if my perspective on the apple on the table were slightly different by being a millimeter more to the right than I am, I would still have perceptive evidence that there is a red apple on the table. What is interesting here to consider is that believing that there is a red apple on the table I am judging that what is offered to me sensibly falls under certain concepts. This implies several things. In the first place, it means that someone who comes to have the belief that p must be able to understand the content of such a proposition p. Secondly, that in forming a perceptual belief one is judging not only about the quality of what is

however, is that it has a rectangular, non-rhomboidal shape. Our cognitive processing already discounts the perspective. If one were asked about what form we ‘see’ the table has —and if there are no additional explanations that make one aware of the perspective at the time of answering— one will say that it is rectangular.

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offered perceptively, but also, and at the same time, judging about the content of the concepts involved. Nobody can come to believe that there is a red apple on the table without understanding, in some way, what it is to be a red apple on a table. There are conditions in which such a proposition is true, conditions in which it is false. One who judges that things are such that the proposition is true—or false, as the case may be—is exercising an understanding of such truth conditions. Understanding what it is to be a red apple on a table requires understanding, in general, what an apple is, what it is to be red, what a table is, and what the relationship of being over is about since there is a red apple over it. The proposition is made up of the combination of these concepts, or of what is understood by these concepts. If the reflection about the content of a concept is a case of a priori reflection,53 then it is clear that in the formation of each perceptive belief a priori intuition about the content of the concepts being judged is required. This does not mean that perceptual beliefs have a priori justification. If the justification of a proposition requires at least some sensory experience, then it counts as a posteriori. It is clear that the justification of our perceptual beliefs is a posteriori, but it requires a component that is a priori. There is a component of ‘spontaneity’, if you will, a certain ‘form’ put by us on something that works as an ‘hyletic’ component. It has been insisted several times already in this work that not every concept is correlated with an authentic property, just as not every term of our languages is correlated with an authentic property. There will be expressions that designate different authentic properties in different contexts of use, and there may be authentic properties for which we have no expression to designate it. It cannot be argued, therefore, that the semantic value of our predicates is always a property, or that the content of our concepts is always a property. These precautions, however, are not a reason to think that there is a complete disconnection between our concepts and authentic properties. Our concepts have been forged by their ability to judge what we think are salient features of reality, according to the best way we understand such reality. If we have chromatic concepts, or concepts of mass, or concepts of forms, or concepts of spin, it is because it seems—according to the best information we have about the world—that there are authentic properties that determine how things are and that correspond to what we are saying with such concepts.54 By making 53

For much of the last century, many philosophers, especially in the analytical philosophical tradition, have viewed a priori justifications with suspicion. The usual way of accommodating the a priori as a legitimate source of justification has been to suppose that it has to do with nothing more than the reflection on the content of our concepts, or about the meaning of the terms we use (see BonJour 1998, 28–61). A contemporary view, represented especially by Christopher Peacocke and Paul Boghossian (see Boghossian 2000, Peacocke 2004) continues to insist on the idea that it is the understanding of the content of our concepts what establishes the reliability of a priori reflection. Although it is very doubtful that a priori intuitions are limited in this way, it can be seen that the least that must be included in it precisely is what should be exercised in the formation of perceptual beliefs. 54 It is notorious, indeed, that the concept ‘grue’—something that is green and is examined before the year 3000 or is blue and is examined after the year 3000—is a concept forged to make a purely philosophical example. This is the standard example of a concept to which an authentic property

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perceptual judgments of the most elementary type, therefore, it becomes apparent that already in the way in which things are shown to us with their sensible qualities, there is something ‘in’ those things that admits being the content of a concept. When one has justifiably judged that there is a red apple on the table, the phenomenon that is presented to one is that of something to which a demonstrative indication could be made that has certain quality—this apple that is red. One could also make a demonstrative indication to this quality that the object possesses—this red color of this apple. That quality is located spatiotemporally and seems to have a particular character such as the object of which we say that possesses it. It is what has been called a “trope”. What is important here is that one could also explicitly consider not the particular property of the object, but the quality itself, of which the particular trope is an instance. It is part of the appearance of the red apple on the table that it is a red apple. The quality of being red, in itself, seems to be able to be exemplified, or given, or instantiated, in multiple objects. It is not, in effect, the particular trope ‘of’ an object, but that which is instantiated in such a trope. A reflection on what is shown in any ordinary perceptual phenomenon will reveal this same structure (see Husserl 1913, Untersuchung II, §§ 1–3, 21–23): an object to which an ostensive indication can be made, a trope of the object located where the object is located, and a quality instantiated in that trope and that, by itself, is repeatable (or seems to be). It is not surprising, therefore, that our natural languages have been structured grammatically in a way that seems to be the reflection of the structure of such perceptual phenomena.55 This may seem surprising at first glance. Indeed, if this is the structure of what is presented in any perceptual phenomenon —at least for rational subjects like us— then the postulation of universals should be obvious. But it is everything less than obvious. All the long argumentation that has developed in this work does not seem idle, and it is because there are many reasons why it seems that universals should be rejected. It happens, however, that the metaphysical positions that have rejected the universals, the different forms of nominalism or the different theories of classes of tropes, are theories that are trying to explain that these phenomena are mere appearances. From the outset, it seems to us that there are qualities in things and that, in and of themselves, they can be instantiated in many objects. This appearance is what has been called the “problem” of the one over many (see § 10). It is a cannot be correlated. In our cultures, acquire importance concepts that seem appropriate to understand the determining features of reality, that is, authentic properties, or what we believe are authentic properties. There are cases, of course, in which a concept that seemed to have an authentic property as content does not really have it. Even in these cases, however, it has been forged because it was assumed that there is the same nature in all specimens. 55 It is also not strange that—at least according to several interpretations—one of the ways in which Aristotle has designated an ousía or substance is as a tóde ti—something to which one can make an ostensive indication as ‘this’ (tóde) but with a quality that answers the question about ‘what’ (ti) it is (see Categories 5, 3b10–12). Some people interpret this expression differently, as a “something particular” (ti) that is of ‘this’ general type (tóde) (see Wedin 2000, 163, sub 10). In any case, under any of these readings, the idea is the same: a particular that has some or other ‘character’ or ‘quality’.

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‘problem’, indeed, because if one wants to reject the existence of universals, then one should present something that fulfills the functions of being the ‘unification of the multiple’ without being a universal.

9.3.2

A Priori Intuitions

§ 73. What is shown in ordinary perceptual experience is that something is instantiated that can be described as a multiply repeatable ‘quality’, which must be distinguished from its particular instance located spatiotemporally in the perceived object. These ‘qualities’ are what in this work has been called “universals”. How a rational subject comes to be in cognitive contact with the universal is through a process that could be characterized as a “taking out” or “abstracting” the universal from its particular instantiation. It could also be described as ‘bracketing’ the particularity of instantiation —as well as the other concomitant factors— to leave ‘in focus’ only the universal. By ‘putting into brackets’ one could also say that there is a ‘reduction’ of the phenomenon by leaving aside everything that is not the universal in itself. The cognitive abilities that are exercised when performing this ‘abstraction’ or ‘putting into parentheses’ are different from the cognitive capacities that are being exercised when we sensibly perceive a particular object with its tropes. There is no causal chain that is connecting our ‘senses’ with the universal, although there is a causal chain that is connecting our ‘senses’ with the object that instantiates such universal and with the particular instantiation of the universal in that object. The ‘abstraction’ operates, so to say, on what sensitive experience offers, without being reducible to it. In the absence of a more adequate expression, we can designate that cognitive process of ‘abstraction’ or of ‘putting into parentheses’ as an “intuition”. By using this expression, one is implicitly appealing to a visual metaphor. The Latin term intuere from which it originates means “to see”. We speak here of “intuition”, therefore because it is a ‘seeing’ the universal, but not with the senses, but with intelligence. It has been traditional to differentiate between the a posteriori and a priori sources of justification. A posteriori justifications are those offered by sensitive experience. What intuition provides here is not a justification by sensory experience and can, therefore, be designated as ‘a priori’. Considering the explanations that have been given here about how is our ordinary experience with the world, we might be inclined to think that we are only offered instantiated universals. These are those universals with whose tropes we can have some causal relation. Would not this be a reason to accept only instantiated universals? It is a fact, however, that not only are instantiated universals offered by a priori intuition —although these are a paradigmatic case— but also non-instantiated universals. We know a lot about triangles in a Euclidean geometry, for example, although we have never been

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experientially exposed to a Euclidean triangle.56 We know a lot about circumferences, and we have never been exposed experientially to one. It is perhaps because of this that a priori intuition has sometimes been described as ‘reminiscence’ (see Plato 2009, 72e-78b). When we remember something, there is some present experience that turns our attention to something past. That present experience will bear some resemblance to the past, or some notorious contrast to the past that cognitively ‘triggers’ the memory. When we experience something sufficiently similar to a triangle, the a priori intuition referred to a triangle is cognitively ‘triggered’. Thus, not only does intuition offer us instantiated universals, but also universals that are not.57 A priori intuitions work in many respects such as ordinary perceptual experience. In principle, if there is no defeater, one is justified in believing that p if one has the intuition that p (see Chudnoff 2013, 83–113). It cannot be assumed that what intuition offers is infallible. One could have an intuition of something false. Gottlob Frege, for example, had the intuition that the axiom V of the Grundgesetze der Arithmetik was consistent, but it was not. A ‘defeater’ of what is shown by intuition a priori is also something that is shown by intuition a priori —at least, normally. For the same reason, one could have the intuition that p and, however, not believe that p. Maybe the intuition that p should be weighed with other more robust intuitions that not-p (see Chudnoff 2013, 25–80). Although there is certain appearance that, for example, for any predicate there exists a class of all and only the objects that satisfy it, other intuitions show us that such a supposition leads to contradiction —in fact, let the predicate “x 2 = x”; it turns out that the class of all and only the objects that satisfy it does not satisfy it, and the class of all and only the objects that do not satisfy it satisfy it. One cannot argue, then, that a priori intuitions are not epistemologically reliable, because the reason why one is going to question what certain intuitions seem to show is what other intuitions seem more reliable to show. It has been traditional to distinguish between sense perception and intelligence. Sense perception is the cognitive faculty —or the set of cognitive faculties— by which one obtains perceptual information from our environment. Powers of this type should be attributed to animals, according to our best knowledge about their

56

More precisely, it is extremely improbable that we have been experientially exposed to any perfect triangle. Of course, we have had a multitude of sensitive experiences of things that are similar or close to a triangle. It is far more likely that none of these forms is the form of a perfect triangle, with interior angles that add exactly two right angles. 57 And some of these are universals that could be instantiated, although they are actually not. Also, some of them are universals that could not be instantiated. Let the universal to have exactly 10 gr of mass. Using operations on universals, the universal to have exactly 10 gr of mass and not have exactly 10 gr of mass can be generated. More precisely, be the universal to have exactly 10 gr of mass designated as ‘Ux’. Neg(Ux) is the negation of Ux, i. e., that universal that instantiate all and only those objects that do not instantiate U. Let Conj(Ux1, Neg(Ux2)) be the conjunctive universal of having something that is U and having something—not necessarily the same object—that is Neg (U ). Now let Refx1x2(Conj(Ux1, Neg(Ux2))) ¼ Conj(Ux1, Neg(Ux1)), that is, the universal that something instantiates if it is U and it is at the same time Neg(U ). Nothing could instantiate such a universal, due to how it has been constructed.

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behavior. Intelligence, on the other hand, is the faculty —or the set of faculties— responsible for our ‘superior’ cognitive activities. It is the understanding that is operating when mathematics is done or, in general, ‘theory’ is done. ‘Intelligence’ is the faculty of universals. It is the faculty —or the set of faculties— able to ‘see’ the universals in their particular instances and contemplate their content.

9.3.3

Universals Epistemologically Transcendent?

§ 74. The objection directed against transcendent universals is that their postulation would require introducing extravagant epistemic capacities. What has been explained in the previous sections is that already to generate our most basic perceptual beliefs is put into exercise a cognitive faculty —or a set of cognitive faculties— that allow us to have a priori intuition of the universals that are instantiated in a particular or simply some universal, whether or not it is instantiated. Of course, a nominalist or a friend of the tropes will not be willing to accept this description of the functioning of our cognitive abilities. From these perspectives, there are no universals, so we could not get to know them. The question here, however, is that, if there are universals, then there is a reasonable explanation about how our cognition works and about how the knowledge of universals is for us something habitual. We exercise such capacity in our daily life and when we do natural science. There is nothing extravagant or exorbitant in such abilities. The objection is that postulating universals leads to an unacceptable epistemology. What has been shown here is that there is no such thing. Postulating transcendent universals does not generate any dramatic imbalance in the challenge of ‘integration’ for metaphysics of properties. On the contrary, it is the rejection of universals that produces such an imbalance, because it is necessary to explain why we have the ‘illusion’ that there are qualities that, due to their intrinsic nature, can be multiply instantiated, when, according to their ontologies, there are no such things. Whoever rejects universals must be able to show how such phenomena should be considered ‘mere appearances’. The situation is that it is the postulation of universals the position that, from an epistemological point of view, simply requires taking the phenomena at face value. The opposite positions, however, require on our part skeptical attitudes or, at least, severe critical precautions against what our ordinary cognitive capacities seem to offer us. It is not surprising that then induction —the standard way in which information about natural laws is obtained— becomes an insoluble problem, or almost (see § 15). A closer examination of the issue shows, therefore, that the tables are inverted. It is not the defender of universals that is in explanatory debt concerning epistemological matters. The one who rejects universals is who is in debt. Something similar happens when one considers the contrast between Aristotelian and Platonic positions concerning universals. An Aristotelian should not admit the knowledge we claim to have of the essential characteristics of a triangle in a Euclidean geometry unless we are exposed to triangles actually instantiated. But we are not. Our mathematical knowledge and our knowledge of mathematical models —without which natural

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science would not work— becomes a mystery. What phenomena seem to show us must be explained as an ‘illusion’ or as ‘mere appearance’. It seems obvious to any mathematician that his inquiries are about a domain of structures that do not need to be instantiated to exist. The Aristotelian must try to reconstruct what the mathematician has shown, without incurring ontological commitments with Platonic universals. A Platonist, on the other hand, should do nothing more than admitting what the phenomena seem to show. Again, it happens at this point that the one who posits Platonic universals is simply accepting what is shown to us in the phenomenon. Whoever rejects this position must, instead, seek a more or less sophisticated philosophical re-interpretation to dismiss as deceptive what the phenomena seem to show us. Is not this, however, an epistemology that subverts the empirical methodology of the natural sciences? In effect, science is being presented here as a search for universals. But neither telescopes nor microscopes seem adequate instruments to discover universals. To maintain that what is done in laboratories or particle accelerators is to look for universals seems, at least, a very confusing description. Nothing that has been sustained here, however, goes against empirical methodology. For quite some time now, it has been highlighted by many philosophers and historians of science that the enterprise of empirical science is not a blind ‘gathering’ of information about what is observed. The experiments are ‘questions’ directed to nature to show the properties that really determine how things are. The advancement of science requires a joint exercise of a priori reflection and observation. The postulation of transcendent universals, therefore, far from being a subversion of natural science, helps to understand much better why their methodologies are —in general— epistemologically reliable, why the use of mathematical models is so fruitful and why inductive practices are a rich source of information about the world, among other things. This concludes the examination of the main objections against transcendent universals. It has been seen that all of those rest —in general— on errors or on neglecting important aspects of the issues involved. It is not effective that transcendent universals are ‘causally inert’. They are not in violation of the Eleatic principle. All causal powers that has some object is due to the instantiation of some universal or another, so it depends ontologically on such universal. It is not effective that the theories of Platonic universals are less economical than the theories of immanent universals. The theories of Platonic universals require positing tropes that are essentially the instantiation of the universal in a particular object —or particular objects if it is a relational universal. These tropes are located in space and time and are the relata of causal connections. It happens, however, that the theories of Aristotelian universals must also do so because of the requirements that come from the famous regress of Bradley. Finally, the postulation of transcendent universals does not require adopting an extravagant epistemology. The epistemic access that we have of universals, basically through their tropes, is ordinary. We have such access every time we consider a perceptual experience, as it happens for any rational subject like us.

References

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In this way, the most reasonable alternative is to postulate transcendent universals.

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Campbell, K. (1981). The metaphysic of abstract particulars. Midwest Studies in Philosophy, 6, 477–488. Campbell, K. (1990). Abstract Particulars. Oxford: Blackwell. Chudnoff, E. (2013). Intuition. Oxford: Oxford University Press. Cook Wilson, J. (1926). Statement and inference, with other philosophical papers. Oxford: Clarendon Press. Cross, R. (1999). Duns Scotus. New York: Oxford University Press. Divers, J. (2002). Possible worlds. London: Routledge. Ehring, D. (2011). Tropes: Properties, objects, and mental causation. Oxford: Oxford University Press. Gaskin, R. (1995). Bradley’s regress, the copula, and the Unity of the proposition. The Philosophical Quarterly, 45, 161–180. Gaskin, R. (2008). The unity of the proposition. Oxford: Oxford University Press. Grossmann, R. (1983). The Categorial structure of the world. Bloomington: Indiana University Press. Heil, J. (2003). From an ontological point of view. Oxford: Clarendon Press. Heil, J. (2012). The universe as we find it. Oxford: Clarendon Press. Hochberg, H. (1978). Thought, fact, and reference. The origins and ontology of logical atomism. Minneapolis: University of Minnesota Press. Husserl, Edmund (1913). Logische Untersuchungen. Herausgegeben von Ursula Panzer, The Hague: Martinus Nijhoff, 1984. Traducción al español de Manuel García Morente y José Gaos, Barcelona: Altaya, 1929. Husserl, Edmund (1948), Erfahrung und Urteil, Hamburg: Claassen. Traducido por Jas Reuter, Experiencia y juicio, Madrid: Editora Nacional, 1980. Jacobs, J. D. (2010). A powers theory of modality: Or, how I learned to stop worrying and reject possible worlds. Philosophical Studies, 151, 227–248. Keinänen, M. (2005). Trope theory and the problem of universals. Helsinki: Philosophical Studies from the University of Heksinki. Keinänen, M. (2011). Tropes: The basic constituents of powerful particulars? Dialectica, 65(3), 419–450. Kemp Smith, N. (1927a). The nature of universals – I. Mind, 36(2), 137–157. Kemp Smith, N. (1927b). The nature of universals – II. Mind, 36(3), 265–280. Kemp Smith, N. (1927c). The nature of universals – III. Mind, 36(4), 393–422. Kim, J. (1976). Events as property exemplifications. In M. Brand & D. Walton (Eds.), Action Theory (pp. 159–177). Dordrecht: Reidel. Reprinted in Supervenience and Mind. Selected Philosophical Essays, Cambridge: Cambridge University Press, 1993, 33-52. Lewis, D. K. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, 65, 113–126. Reprinted with postscripts in David Lewis, Philosophical Papers, Volume I. New York: Oxford University Press, 1983, pp. 26–46. Lewis, D. K. (1973a). Counterfactuals. Oxford: Blackwell. Lewis, D. K. (1973b). Causation. The Journal of Philosophy, 70, 556–567. Reprinted with postscripts in David Lewis, Philosophical Papers. Volume II. New York: Oxford University Press, 1986, pp. 159–213. Lewis, D. K. (1983). New work for a theory of universals. Australasian Journal of Philosophy, 61, 343–377. Reprinted in David Lewis, Papers in Metaphysics and Epistemology. Cambridge: Cambridge University Press, 1999, pp. 8–55. Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell. Lewis, D. K. (2004). Causation as influence. In John Collins, Ned Hall & L. A. Paul (eds.). Counterfactuals and Conditionals. Cambridge, Mass.: MIT Press, pp. 75–106. Expanded version of the article with the same name in The Journal of Philosophy, 97(2000), 182–197. Lowe, E. J. (2006). The four-category ontology. A metaphysical foundation for natural science. Oxford: Clarendon Press. Martin, C. B. (1980). Substance substantiated. Australasian Journal of Philosophy, 58, 3–10.

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Martin, C. B. (1994). Dispositions and conditionals. The Philosophical Quarterly, 44, 1–8. Maurin, A.-S. (2002). If tropes. Dordrecht: Kluwer. Maurin, A.-S. (2011). An argument for the existence of tropes. Erkenntnis, 74(1), 69–79. Maurin, A.-S. (2012). Bradley’s regress. Philosophy Compass, 7(11), 794–807. Meinertsen, B. (2008). A relation as the unifier of states of affairs. Dialectica, 62(1), 1–19. Mertz, D. W. (1996). Moderate realism and its logic. New Haven: Yale University Press. Molnar, G. (2003). Powers. A study in metaphysics. Oxford: Oxford University Press. Mumford, S. (1998). Dispositions. Oxford: Oxford University Press. Mumford, S. (2004). Laws in nature. London: Routledge. Oddie, G. (1982). Armstrong on the Eleatic principle and abstract entities. Philosophical Studies, 41, 285–295. Orilia, F. (2009). Bradley’s regress and ungrounded dependence chains. A reply to Cameron. Dialectica. 63(3), 333–341. Peacocke, C. (1999). Being known. Oxford: Clarendon Press. Peacocke, C. (2004). The realm of reason. Oxford: Clarendon Press. Plato, Meno (1982). Menón. In Diálogos II. Introducción, traducción y notas de F. J. Oliveri. Madrid: Gredos. Plato, Phaedo (2009). Fédon. Traducción, notas e introducción de Alejandro G. Vigo. Buenos Aires: Colihue. Plato, The Sophist (1982). Sofista. In Diálogos V. Introducción, traducción y notas de N. Luis Cordero. Madrid: Gredos. Prior, E. W., Pargetter, R., & Jackson, F. (1982). Three theses about dispositions. American Philosophical Quarterly, 19, 251–257. Psillos, S. (2002). Causation and explanation. Montreal: McGill-Queen’s University Press. Psillos, S. (2009). Regularity theories. In H. Beebee, C. Hitchcock, & P. Menzies (Eds.), The Oxford handbook of causation (pp. 131–157). Oxford: Oxford University Press. Ruben, D.-H. (2012). Explaining explanation. Boulder: Paradigm Publishers. Updated and Expanded Second Edition. Sider, T. (2002). The Ersatz Pluriverse. The Journal of Philosophy, 99, 279–315. Sider, T. (2006). Bare Particulars. Philosophical Perspectives, 20, 387–397. Simons, P. (1994). Particulars in particular clothing: Three trope theories of substance. Philosophy and Phenomenological Research, 54, 553–575. Strawson, P. F. (1959). Individuals. An essay in descriptive metaphysics. London: Methuen. Tooley, M. (1987). Causation. A realist approach. Oxford: Clarendon Press. Vetter, B. (2015). Potentiality. From dispositions to modality. Oxford: Oxford University Press. Wedin, M. V. (2000). Aristotle’s Theory of Substance. The categories and metaphysics Zeta. Oxford: Oxford University Press. Wieland, Jan Willem & Betti, Arianna (2008), “Relata-Specific Relations: A Response to Vallicella”, Dialectica, 62(4), 509–524. Williams, D. C. (1953a). On the elements of being: I. The Review of Metaphysics, 7, 3–18. Williams, D. C. (1953b). On the elements of being: II. The Review of Metaphysics, 7, 71–92. Williamson, T. (2000). Knowledge and its limits. Oxford: Oxford University Press. Zalta, E. N. (1988). Intensional logic and the metaphysics of intentionality. Cambridge, Mass: MIT Press.

Chapter 10

Identity Conditions for Transcendent Universals

Abstract It has been argued above that it is essential for a universal the nomological relations into which they enter. Even more, it has been argued that there is just a unique nomic network of necessary existence. Nevertheless, some have maintained that the conditions of identity for universals that result from these ideas are incoherent. It is explained here that this problem of incoherence can be dealt with, but assuming that the nomic structure is ontologically prior to the universals that enter in it. Universals are nodes in this unique necessary nomic structure. § 75. After this long journey considering several lines of argument in favor of transcendent universals and several traditional objections against them, in this section, some collateral questions will be examined before going directly into the topics of Part III on the nature of particular objects. It has already been seen above that it would not be reasonable to suppose that the causal powers conferred by a universal upon their instantiations were contingent upon these universals (see § 51). The natural laws that a universal will integrate—or that must be identified with such universal—are essential to it. To suppose that universals confer causal powers for anything would lead to the modal identification of any universal with any other if the causal powers are any guide to the identity of properties in different possible worlds. This still leaves open the alternative, however, that the conditions of identity of a universal are constituted by the causal powers that it confers and, also, by a primitive quidditas. What is going to be discussed in this section has to do fundamentally with the relevance of this moderate quidditist alternative. It will be convenient to remember some points about the nature of the causal powers that will be relevant to the discussion that follows. The causal power that is conferred by the universal U to the object x in the possible world w1 is not restricted by the causal connections in which x enters into that possible world w1—or the causal connections of which the state of affairs of x instantiating the universal U is a cause in w1. A causal power determines the metaphysical possibility of such causal

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_10

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connections.1 For U to confer a causal power to x in w1 what is required is that in some possible world w2 a causal connection occurs. If such a causal connection occurs in the same world w1 the condition would be satisfied, but, of course, it does not need to happen. This applies both to the combinatorial conception of causal powers and to the conception of causal powers according to a ‘causal’ theory of modality. For a universal U not to confer a causal power to an object x what is required, therefore, is that in no possible world x is involved in a causal connection of the relevant type—or, if it is a combinatorial conception, it is required that in none of the possible worlds closest to the world of evaluation, x is involved in a causal connection of the relevant type. When it is pointed out, then, that it is essential for the universal U to confer the causal power C what is being affirmed is that in every possible world w in which something instantiates U is the case that: it is metaphysically possible with respect to w the causal connection of the type specified by C. Note that, in this way, the essential character of the attributions of causal powers for a universal is almost trivially true in the ‘causal’ conception of modality, and almost trivially false in the combinatorial conception of modality. Indeed, suppose that a universal was conferring a causal power contingently in a ‘causal’ conception of modality. In some possible worlds, it confers such power, and in others, it does not. By hypothesis, suppose that the universal U1 confers in w1 the power to cause a U2, while in w2 it does not. As the attribution of a causal power is the attribution of a modal character, U1 confers in w1 the power to cause a U2 because there is a possible world w3 accessible to w1 in which the instantiation of U1 causes the instantiation of U2. In contrast, U1 does not confer in w2 the power to cause a U2 because there is no possible world accessible to w2 in which the instantiation of U1 causes the instantiation of U2. It has already been seen, however, that there is a possible world, let w3, in which the causal connection required exists between the instantiation of U1 and an instantiation of U2. This scenario would be consistent if w3 were accessible to w1, but not to w2. What happens in this scenario has general value. The only way that contingency was coherent for the causal powers of a universal—from the perspective of a ‘causal’ theory of modality—is if there were breaks in the accessibility relations, that is, if there were metaphysically possible worlds that would not be in some cases accessible to each other. However, to suppose such a thing is unreasonable if what is involved here is metaphysical modality. It has already been explained above why it should be assumed that accessibility relations should be taken as symmetric in the case of the metaphysical modality (see § 42). They should also be assumed to be reflexive and transitive. Accessibility relationships must be reflexive because otherwise, it would not be valid the principle ab necesse ad esse valet consequentia, that is, if something is necessary, then it is the case.2 The accessibility relationships must 1

More precisely, according to a combinatorial conception of modality, a causal power is grounded on modal facts about possible causal connections (see § 63). In a ‘causal’ conception of modality, on the other hand, the order of grounding is the inverse. Metaphysical possibilities about causal connections are grounded on causal powers (§ 64). 2 The modal principle is [□p ! p] and is an axiom characteristic of normal modal systems of type T. Suppose that [□p] in the world w. Then, according to the standard semantic clauses of modal

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be transitive because otherwise there would be coherent scenarios in which something necessary is contingently necessary.3 However, something contingently necessary is something simply contingent. If the accessibility relationships are reflexive, symmetric, and transitive, then all possible worlds are accessible to each other. The accessibility relationship is, in this case, an equivalence relation.4 It cannot be that w3 is accessible to w1 and not accessible to w2, or vice versa. In this way, it is really not possible for a universal to confer a causal power only contingently. When it comes to the combinatorial conception of causal powers, however, the situation reverses completely. It seems that no universal can confer a causal power essentially. Whatever the causal power may be, it might not have conferred it. Recall that, in this conception, a universal U1 confers to the object x in the world w the power to cause a U2 because, in at least one of the possible worlds closest to w, the instantiation of U1 causes an instantiation of U2. For a universal to confer a causal power, it should essentially happen that in all metaphysically possible worlds the class of worlds closest to any one of them is such that, in at least one of these nearby worlds, the causal connection required is produced. It is evident that supposing such a thing would be excessive. When considering all possible worlds and the ‘neighborhoods’ of worlds close to each of them, worlds very different from ours must be considered. For example, one might assume that the property of having a negative electromagnetic charge essentially confers the power to attract objects of positive electromagnetic charge. It happens, however, that—from a combinatorial perspective—one must admit possible worlds in which, for example, nothing has a positive electromagnetic charge. All the worlds closest to these will be worlds in which attraction will not occur. Then, in these worlds, the property of having a negative electromagnetic charge does not confer the causal power of attracting objects with positive electromagnetic charge. Recall that it has been preferred to formulate the notion of causal power, from the combinatorial perspective of the modality, using a counterfactual might to avoid other problems. If the restriction is not made to the possible worlds ‘closest’ to the world of evaluation, what results is that any universal

logic, in all the worlds accessible to w, p is the case. If the accessibility relationships were not reflexive, that is, if w were not accessible from w, then there would be no guarantee that p would be true in w. Part of the minimum content of metaphysical necessity is that what is necessary is effective. Another way to see why accessibility relationships should be reflexive is that otherwise, it would not be the case that ab esse ad posse valet consequentia, that is [p ! ◊p]. Indeed, if p is true in w, then—given reflexivity—it is possible that p because at least there is a possible world accessible to w in which p is the case, that is, w. 3 The transitivity of the accessibility relationships guarantees the validity of the modal principles [□p ! □□p] and [◊◊p ! ◊p]. If [□p ! □□p] were false, then it should be the case that [□p] and [◊Ø□p]. Given [□p] it trivially follows that [◊□p], so it would be contingent that [□p]. 4 More precisely, there could be several classes of possible worlds, each of which would satisfy the condition that all and only the possible worlds of the class would be accessible to each other. Assuming several different equivalence classes of possible worlds would be very unlikely if the modality in question is metaphysical modality. A modal logic in which accessibility relations satisfy these conditions of reflexivity, symmetry, and transitivity in accessibility relationships is called an “S5-type logic”.

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confers (almost) any causal power. If one were to assume that the conditions of identity of a universal are grounded on the causal powers that those universals confer upon their instantiations, then it should be held that all universals should be identified. This explanation will not be repeated here, but in a combinatorial perspective, no matter what universal it is, for (almost) any other universal there will be possible worlds in which an instantiation of the first causes an instantiation of the second. It is evident, therefore, that the combinatorial conception of causal powers does not seem coherent with the assumption that universals confer essentially certain causal powers to their instantiations. If such causal powers are understood from a combinatorial perspective, either it is evident that such an essential attribution is false, or the very notion of causal power is trivialized in such a way that it is useless. A theory, in fact, according to which I am (almost) omnipotent is a false theory, because—obviously—I am not. The unacceptable consequences to which radical quidditism leads have already been explained (see § 51). As explained above, one way to ‘repair’ the combinatorial conception of causal powers in a way that is consistent with the essential character of the causal powers for a universal would be to introduce ‘by brute force’ a restriction on the allowable combinatorics for the generation of the modal space of possibilities, admitting only the combinatorics consistent with the essential character of specific natural laws. However, from a combinatorial perspective, these restrictions are a ‘foreign body’, an obscure necessary connection between different existents, disgusting to any philosopher of Humean inclinations. The discussion will be conducted in what follows, therefore, concentrating attention on a ‘causal’ conception of modality and causal powers (see § 64), in which the idea that universals confer essentially certain causal powers is consistent.

10.1

Structuralism

§ 76. The question that must be considered here, then, is whether it is reasonable to suppose that, together with the causal powers conferred by universals on its instantiations, there are primitive quidditates as constituents of the conditions of identity of that universals. From the outset, no universal could confer causal powers different from those it actually confers, but the theoretical alternative that must be examined is whether it should be accepted that two or more different universals confer precisely the same causal powers. Consider the following principle: [Identity of Universals]

For all universals U1, U2: U1 ¼ U2 if and only if, for every causal power C: U1 confers C if and only if U2 confers C.5

More precisely: [□8X□8Y□ ((X ¼ Y ) $ 8C8x ((X confers C to x) $ (Y confers C to x))]. The variables ‘X’ and ‘Y’ have as range universals; the variable ‘C’ has as range causal powers; the variable ‘x’ has as range objects.

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If there are primitive quidditates for each universal, then this principle of identity would fail from right to left. Although the identity of U1 and U2 implies that all causal power conferred by U1 must also be conferred by U2 and vice versa, there could be two universals numerically different from each other, but with the same ‘causal profile’. This hypothesis is not as severe as that in which the same universal has completely different profiles in different possible worlds—hypothesis with severe problems of coherence as explained—but it is still very implausible. There are, at least, three primary reasons why moderate quidditism must be rejected, just as radical quidditism has been. A first reason for rejecting moderate quidditism is epistemological and closely follows lines already explained above (see §§ 51–52, 71–74). Universals that only differed by their respective quidditates would be completely indiscernible. Postulating quidditates, in addition to the usual causal powers is analogous to postulating an ‘evil genius’ that generates the same experiential appearances that we have but whose difference from veridical experiences is undetectable. How we experientially perceive a property is by the display of causal powers conferred by such property in its instantiations and how they interact with our senses. For example, if a surface appears to us as red, this is due to the causal power of the surface to which we attribute that color to reflect in our retinas light at specific wavelengths. We access universals because we interact with their instantiations. It has been usual in discussions about functionalism in the philosophy of the mind to present as counter-example to the ‘causal’ specification of mental states the case of the ‘inverted spectrum’ to argue that there is something more than the causal component to specify a state mental. Perhaps here one could be inclined to think of such a case to argue in general terms that there is something more than the causal powers for the identity of a property. It has been argued that a subject to whom the shades of red of the chromatic spectrum would appear to him phenomenally as the shades of blue, and the shades of blue appear phenomenologically to him as the shades of red, would behave as any of us would, would call “red” to the same things we call “red” and “blue” to the same things we call “blue”. It is assumed, then, that the web of causal connections in which a mental state is immersed is not sufficient to ground its phenomenal character (see Byrne 2015, § 3.1). However, leaving aside the differences in behavior that a subject may present, a possible world in which the red surfaces look blue is a scenario in which there are different causal powers for such surfaces. It is not a situation in which causal powers and causal interactions remain identical, but there is a variation in phenomenal appearance. It is a scenario with other causal powers and other causal interactions. To postulate a quidditas over the causal powers conferred by a universal is to postulate something that exceeds the phenomenology of universals, how we ordinarily accede to them and, then, is to suppose a ‘surplus’ utterly alien to our cognitive abilities.6 Therefore, it is an ontological component for whose application there is no justification. It is

6 The situation is very different when it comes to particular objects or tropes. The particular character of a particular is to be this, something to which—in principle—an ostensive indication can be made.

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reasonable, therefore, not to postulate it. Holding this is not embracing some form of verificationism about quidditates. It is not being held that there can only be what—in some way—one can come to know. What is working here is the requirement of ‘integration’ between ontology and epistemology that has been discussed above (see § 71). If postulating quidditates is to postulate the complete inadequacy of our cognitive abilities to access universals, then we have reasons to reject such a postulate in principle, since obviously, we have some access to universals. Otherwise, there would be no natural science. A second reason for rejecting moderate quidditism is that it requires making intelligible the distinction between the quidditas and the ‘causal role’ of a property (see § 51). Indeed, in the perspective of moderate quidditism, each universal confers essentially certain causal powers and has a primitive quidditas. It is assumed that different quidditates may be satisfying the same ‘causal role’. Therefore, that ‘causal role’ should be intelligible regardless of the quidditates that are joined to it.7 But it does not seem possible. Suppose that the causal role that the universal U1 must satisfy is specified by its position in the following nomic network: [N(U1, U2)], [N (U2, U3)] and [N(U3, U1)]. Suppose there is another universal U4 that satisfies the same nomic network, but that is numerically different from U1 by having a different quidditas. What does it mean to say here that “it satisfies the same nomic network”? It means that U4 is connected with U2 and with U3 in the same way as U1 does. The ‘nomic network’, however, is specified by the universals U2 and U3. A nomic network that had the same abstract ‘structure’, but in which, instead of being integrated by U2, was integrated by another universal U5, would be a different nomic network altogether. Also, in order for it to be a different universal, it is enough for it to have a different quidditas. Then, if really the identity of a universal has to do, even in part, with a quidditas, the identity of the nomic networks in which the universals can enter is fixed by the quidditates of the universals that integrate them. Therefore, the idea of a ‘causal role’ independent of the quidditates is not intelligible, because there are no ‘causal roles’ without relying on quidditates from the outset. The third problem for moderate quidditism is that it brings with it a problem of ‘arbitrariness’ in the modal space of metaphysical possibilities. It is assumed that there is a totality of metaphysical possibilities and possible worlds. This totality has, of course, an infinite cardinality, but it is something to which some cardinality can be

A universal, on the other hand, appears phenomenologically as a ‘quality’ that by itself is multiply instantiable. 7 Following an analogy already presented, the ‘causal role’ works for a universal such as the ‘description’ of a particular object, while the quidditas would function as a ‘proper name’ of an object. For radical quidditism, it is contingent for a universal—whose identity is given by its quidditas—what causal role it is satisfying. Here, on the other hand, the causal role is necessary for a universal, but different universals—whose identity is given by their quidditates—could satisfy the same causal role, although no universal could have a causal role different from the one it possesses.

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assigned.8 From a combinatorial perspective of metaphysical modality, this modal space is fixed by the totality of universal properties that may or may not be instantiated in different objects. From a causal perspective of modality, this modal space is fixed by the totality of ‘causes’ of which the instances of universals were a cause. In either case, the extension of the modal space of possible worlds depends on which universals exist. If there are numerical differences between universals simply because of their quidditates, then it cannot be supposed that the totality of universals is fixed on which the totality of possible worlds must depend. For each causal role, there can be an indeterminate multitude of different universals satisfying it. Assigning a cardinality to this crowd would be completely arbitrary. For a ‘causal role’ C, for example, why should there be two different universals satisfying C instead of three? Alternatively, why not twenty instead of a denumerable infinity? Why not an infinity with the cardinality of the continuum? The assignment of any limit to the plurality of universals that satisfy the same causal role lacks any justification. And it is not that a finite number of different universals cannot be satisfying the same causal role. Nor is it a matter of an infinite number. Either there is an entirely arbitrary limit—a brute fact—or there is no limit. Either option would be very unreasonable. The first for its obscurity and the second because then there would not be a definite totality of possible worlds. At most, it could be a plurality ‘indefinitely extensible’, so that for any totality that is considered, there would be possible worlds not considered in such a totality. The problem that this would have is that such indeterminacy would cause the indeterminacy of modal facts in general. In standard modal semantics, a proposition p is necessary if and only if it is true in all possible worlds. The quantifier ‘all’ must have as range the totality of possible worlds. If there is no such totality, then the modal attribution of necessity will be meaningless. One perhaps could argue that the attributions of metaphysical possibility would remain intelligible, because for the possibility of p, it is sufficient that p is true in at least one possible world, but this is a small consolation.9 Assuming that there are quidditates for universals in addition to their causal roles has the risk that it can blow up all the modal facts or a good part of them. There are many reasons, therefore, for rejecting moderate quidditism, just as there are for rejecting radical quidditism. The conditions of identity that must be postulated for universals are those that are specified in the principle of Identity of

8 It is a tough question to answer what is such cardinality. It depends, among other things, on the general conception of metaphysical modality. If one supposes, for example,—from a combinatorial perspective—that spacetime is a continuous structure and that each point of spacetime can be filled or empty, then the cardinality of the modal space of possible worlds should be the cardinality of the whole power set of the continuum. Assumptions just a little more complicated—and realistic— make these accounts vary dramatically. 9 And it is, also, doubtful. The modal operators of necessity and possibility are dual among themselves so that these equivalences must be valid: [□p $ Ø◊Øp] and [◊p $ Ø□Øp]. It seems reasonable to suppose that satisfying these equivalences is part of the minimum content of the notion of ‘metaphysical possibility’. But if this is so, then the problems for the intelligibility of metaphysical necessity will also infect the metaphysical possibility.

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Universals. A conception of this type has been called “causal structuralism” (see Hawthorne 2001), since the identity of a universal is dependent on all the other universals with which it forms a nomic network. It has already been explained that there is only one nomic network (see § 52), so the identity of a universal is ontologically dependent on all other universals. Analogous positions have been proposed in the philosophy of mathematics and the philosophy of physics. It has been proposed that mathematics is not about a domain of ‘abstract objects’, but only about structures that can (or cannot) be satisfied by a multitude of different types of objects. The natural numbers, for example, would be anything that satisfies Peano’s axioms. The number 2 is anything that has the same ‘role’ in such a structure (see Shapiro 1997, 71–177). In the case of the philosophy of physics, it has been argued that, at least at the most basic level, physical entities are not ‘objects’ endowed with monadic properties and relationally connected in different ways. They are ‘structures’ in which relationships have a fundamental character (see Ladyman and Ross 2007, 130–189; French 2014, 101–323). This ‘physical structuralism’ will be discussed with more detention in Part III of this work (see §§ 93–94). For what matters here, it suffices to consider that a form of ‘structuralism’ for universal properties is not something utterly unusual due to these other antecedents.

10.2

The Problem of Regress of the Conditions of Identity

§ 77. A structuralist conception of the conditions of identity of universals has faced important criticism. Perhaps the one who has raised this criticism in the most perspicuous way is Jonathan Lowe (see Lowe 2010). Suppose, in a completely general way, that the identity of A—it does not interest here to which category it belongs—depends ontologically on B. That is, A is the entity that it is because previously B is the entity that it is. Now, suppose that the identity of B is ontologically dependent on C, but the identity of C is ontologically dependent back on A. In a scenario like this, A is not a ‘basic’ entity at least as far as ontological dependence is concerned. Its being refers to other entities. These other entities—B and C in the example—depend on A. It seems that in such a scenario, none of the entities possess determinate identity conditions. The chain of remissions must end at some point, and ontologically independent entities should constitute this point. However, here, none of the entities is ontologically independent. It cannot be that they are dependent on themselves, because—as explained above (see §§ 4, 86)—, ontological dependence is irreflexive. A situation of this kind is what seems to occur with the conditions of identity of universals, as they have been proposed in the principle of Identity of Universals. The identity of a universal U1 is fixed because it confers the power to cause a U2. Thus, U1 is ontologically dependent on U2. However, U2 is the universal that it is because of its ontological dependence on other universals, and so on. Either there will be an infinite sequence of ontological dependence between universals, none of which is independent, or else there will be a circle of dependence between

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universals. The first is a vicious regress, and the second is a vicious circle inadmissible for the intelligibility of identity conditions for universals. It has been pointed out, however, that no problem should be seen in the hypothesis that the identity of universals is determined by their ‘position’ in the structure formed by the totality of causal roles of all universals (see in especial, Bird 2007, 132–146).10 Suppose that there are universals U1, U2, and U3 connected by a series of nomological relations that can be abbreviated as the triadic relation N*. One could argue that the conditions of identity of U1 are fixed by being the only universal X in [N*(X, Y, Z)] with two universal Y, Z. For this procedure to work, it is required that the structure imposed by N* be such that the positions of the universals that are being connected cannot be ‘confused’. It must be a structure non-invariant by permutations of its ‘constituents’. Note that the identity of U1 is not dependent on the identity of U2 and U3, since—in turn—neither U2 nor U3 have identity conditions independent of the structure they are integrating. It is the complete structure that establishes the identity conditions of the universals that integrate it. One way to understand this with a little more precision is through graph theory, which is precisely a mathematical theory that studies abstract structures (see for a general presentation, Balakrishnan and Ranganathan 2012, 1–35). A graph is defined as an ordered pair , in which V is the set of vertices, nodes or points and E is the set of sides, lines or arcs. I will talk about “nodes” and “lines” in the following. The nodes will be designated by the variables ‘n1’, ‘n2’, . . ., ‘nn’ and the lines will be designated by the variables ‘l1’, ‘l2’, . . ., ‘ln’. Intuitively, lines are dyadic relationships that connect nodes and are mathematically represented as sets of pairs of nodes, or as sets of ordered pairs of nodes. A line l ¼ {n1, n2} represents a symmetric relationship between nodes n1 and n2. A line l ¼ represents an asymmetric relationship11 between nodes n1 and n2. The converse of l will be designated as l*. If l ¼ , l* ¼ . Sometimes asymmetric or ‘directed’ lines have been called “arcs”. A ‘directed graph’ is a graph consisting of at least one directed line. For what interests here, some graphs are invariant by automorphisms. An ‘automorphism’ on a graph G is a function that assigns to each node of G a node of G.12 A graph that is invariant by automorphisms is a graph in which the position of the nodes is not sufficient to ground its identity. A graph that does not satisfy this condition is called an “asymmetric” graph. There should not be any problem about

10

Bird notes that this objection of regress or circularity has been proposed in several ways, not all of which have the same systematic interest. Some have argued that assuming structural identity conditions would generate an incoherence (see Bird, 2007, 132–133). Others have argued that structural identity conditions would make universals unknowable (see Bird, 2007, 133–135). These other forms of objection will not interest for the reasons that Bird himself exposes in the cited passages. 11 Recall that the ordered pair is defined as {{x}, {x, y}} à la Kuratowski. 12 An automorphism on the graph G that maps each node of G on itself is a trivial automorphism. What is interesting here are non-trivial automorphisms, of course, in which at least to one node of G is assigned a different node.

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the identity of the nodes grounded on their position in a structure if such a structure is an asymmetric graph. If a ‘nomic network’ is to be represented through a graph, some preventions must be made, but they do not modify the main issue. The possible causal relations that connect universals are asymmetric. The appropriate graphs to represent the structure of a nomic network must be, then, directed graphs. A universal confers multiple causal powers on its instantiations. If one is going to represent the universal by a node in a graph, it will be necessary to connect it by multiple lines directed to other nodes. A graph in which the nodes are integrating more than two lines is called a “multigraph”. A nomic network, then, must be represented by a directed multigraph. There are symmetrically directed multigraphs—that is, invariant by non-trivial automorphisms—but there are also asymmetrically directed multigraphs. What is required is that the only nomic network can be represented by an asymmetric directed multigraph to ensure that the position of each universal in the network is sufficient to ground its identity conditions. John Hawthorne has argued that the position of a universal in a nomic network is not sufficient to ground its identity conditions, since the nomic network can be ‘symmetric’ in the indicated sense—that is, it could be represented by a directed multigraph invariant by non-trivial automorphisms (see Hawthorne 2001). For example, suppose the following nomic network: N(U1, U2), N(U1, U3), N(U2, U4), N(U3, U4), N(U4, U1). The position of U2 and U3 is entirely interchangeable in this structure. U2 is the universal that occupies the following position: N(U1, X) and N (X, U4). However, U3 should be characterized precisely in the same way. Then, if their position in a nomic network fixes the conditions of identity of a universal, we should conclude that U2 and U3 are the same universal, which is absurd. However, there are several problems with this scenario. Although one may be inclined to think about the conditions of identity of a universal in a way analogous to how the problems associated with the principle of the identity of the indiscernibles for objects have been treated, they are very different questions. According to the principle of identity of indiscernibles, two objects that have exactly the same properties must be the same object. A counter-example to this principle is a scenario in which two objects have exactly the same purely qualitative properties (see Adams 1979; Alvarado 2006, 2007). Such a scenario seems entirely possible because an object can have a specific profile of purely qualitative properties, and what can have an object, two or n objects can also have. The causal powers conferred by a universal, on the other hand, are initially modal attributions and are of such a nature that, if a universal confers them, then it confers them necessarily. The nomic network is unique, as has been explained, and it could not be different from how it is. It is confusing to say, therefore, that “there could be indiscernible universals concerning their position in the nomic network”. Either it is a necessary fact that there are universals that, because of their position in the nomic network, are indiscernible, or else there are no such universals, and it is metaphysically impossible that there are. No assumptions can be made about how is the nomic network of all universals really. What universals exist is a question that must be decided by empirical research—at least, for the most part. It is doubtful that we now have a complete picture of all

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universals and all their nomic connections. The question, then, of whether or not the nomic network is an asymmetric structure, cannot be decided a priori, but it is reasonable to expect it to be, precisely because the conditions of identity of the universals seem to be determined by the causal powers of universals, according to everything that has been argued above. There is an additional problem, however, with the assumption of identity conditions for universals given by the causal powers that these universals confer. In a conception of this type, all universals would be ontologically dependent on all the rest. Given the transitivity of ontological dependence, this implies that all universals would be dependent on themselves, which conflicts with the irreflexivity of dependence. It would also happen that all dependence relations between universals would violate the asymmetry that such a relation has. This type of problem has been presented in all structuralist positions, and several have proposed that the way to solve it is through a relaxation of the basic characteristics of ontological dependence (see Ladyman and Ross 2007,148–159; Linnebo 2008; French 2014, 178–183; §§ 93–94), for example, admitting a form of symmetric or non-asymmetric dependence, or admitting a non-irreflexive form of dependence, or relaxing the transitivity of dependence relationships.13 What several have been inclined to maintain is that in a structuralist conception—in which the identity of an entity is given by its relations with the other entities of the structure—there would be a mutual dependence between the constituent entities of the structure and the structure itself. On the one hand, the identity of the constituent entities depends on the structure, but on the other hand, the structure also depends on its constituents. An important motivation for adopting a position of this type is the traditional idea according to which relationships are ontologically dependent on their relata. Relationships have been treated as ‘accidents’ dependent on ‘substances’ (see, for example, Aristotle, Categories, chap. 8). There has been even a traditional pressure to ‘reduce’ relations to their ‘foundations’—which should be monadic properties of the relata (see for a general overview, Mertz 1996, 3–172).14 However, to suppose that ontological dependence is something different from a strict order—i. e., an irreflexive, asymmetric, and transitive relationship—is beyond discussion, for the reasons already indicated above (see §§ 4, 87). Ontological 13

Suppose, by hypothesis, that a nomic network N* is integrated by the universals U1, U2, and U3. It is true, for example, that U1 depends on U2 and U2 depends on U1. Here it happens that: (i) the asymmetry of dependence is violated, and (ii) by transitivity of dependence, it will turn out that U1 will be dependent on itself. A repair of (i) requires relaxing asymmetry. A repair of (ii) requires either relaxing irreflexivity or relaxing transitivity. Recall that a relation R is said “asymmetric” if it is necessary that, if Rab, then it is not the case that Rba. It is symmetric when it is necessary that if Rab then Rba. It is non-asymmetric when it is not necessary that, if Rab, then it is not the case that Rba. Something analogous is true for reflexive, irreflexive and non-reflexive, transitive, intransitive and non-transitive relationships. 14 A sample of these trends is that in graph theory, as we have seen, lines, arcs, or sides are defined as sets of nodes. A set is ontologically dependent on its elements so that defining lines in this way presupposes—in some way—the ontological dependence of lines on the nodes that these lines connect.

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dependence is a relationship that is capturing a form of ontological ‘priority’ and must, therefore, be irreflexive. It is obviously transitive, and asymmetry follows from irreflexivity and transitivity. A reflexive relationship of ‘weak dependence’ occurs between two entities x and y if and only if either x depends on y or x ¼ y. This ‘weak dependence’, however, is anti-symmetric. So, if x weakly depends on y and y weakly depends on x, then x ¼ y. If one were to assume that universals are all weakly dependent on one another, then we should admit that all universals are a single universal, which would be absurd. The solution to the problem of reciprocal dependencies between universals is not to introduce some form of non-standard ontological dependence—which would be to introduce an ontological dependence that is not ontological dependence—but by assigning a clear order of determination between universals and the nomic network. The universals are dependent on the nomic network. The nomic network is ontologically prior. As already indicated several times, the nomic network is a single entity of necessary existence. It is not ‘constituted’ by universals that then connect nomologically, although this is the perspective one is inclined to adopt, given the way universals are epistemologically offered to us by natural laws. Such a way of conceiving the nomic network again would force to introduce an inverse dependence. The nomic network does not result from a ‘construction’ of simpler ‘constituents’ or ‘elements’. What are universals, in this perspective? They are nodes of the great unitary structure that is the nomic network. These nodes are grounded on the structure and depend ontologically on the structure. There are several important precisions to make regarding this conception. In the first place, universals are not here eliminated or reduced to the nomic network. It is not being held that there are no universals and that what we mistakenly designate when we want to refer to universals does not really exist. Nor is it being said that universals should be identified with the nomic network and that, therefore, when we want to refer to universals, we are really referring to the only nomic network.15 What is being said is that the nomic network grounds universals and that universals depend ontologically on the nomic network. If there is really grounding, then there are universals, there is also the nomic network, and these are entities numerically different from each other. What can make this perspective less acceptable is, in the first place, the inclination we have historically had to think of a structure as a sort of ontological ‘construction’ that has to be ‘built up’ from more basic constituents. Secondly, this perspective also hinders the inclination we have had to think that relationships must be ontologically derivative to the ‘things’ that are related. One could describe these ‘inclinations’ as ‘intuitions’. The evidential value of an ‘intuition’, however, must be weighted by how it can be integrated coherently into an explanatorily fertile theory and with other intuitions. What happens here is that precisely other intuitions and the explanatory

The ‘reduction’ of universals to the nomic network, for that matter, is difficult to understand, since there is a plurality of universals and only one nomic network. It only makes sense to complete an identity statement flanked by an expression designating a plurality, by another expression that also designates a plurality.

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References

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value of the general conception that has been defended in this work impose to leave aside what they seem to show. Instead of having a structure constituted by dyadic relations that, in turn, are dependent on the nodes, what happens here is that there is an ontologically basic structure that grounds its nodes. Such nodes are what we call “universals”. The order of ontological priority is the reverse of what one would be inclined to admit from a traditional perspective. The nomic network is not an n-adic relation, because there is nothing in which it must be ‘instantiated’. It is a ‘subsistent’ entity, abstract, necessary, and timeless. The idea of a unitary nomic network is an important novelty for the usual ontologies and deserves a detailed development that cannot be done here. It is what imposes taking the proposed identity conditions for transcendent universals seriously. This concludes Part II of this work. Universals are transcendent or Platonic, whose existence does not depend on having instantiations. It remains to consider how particular objects should be understood if there are Platonic universals.

References Adams, R. M. (1979). Primitive Thisness and primitive identity. The Journal of Philosophy, 76(1), 5–26. Alvarado, J. T. (2006). Esencias individuales e identidad primitiva (J. Ahumada, M. Pantaleone, & V. Rodríguez, Eds.), Epistemología e Historia de la Ciencia 12, (pp. 31–36). Alvarado, J. T. (2007). Esencias individuales e identidad primitiva. Analytica, 11(2), 155–195. Aristotle, Categories (2009). Categorías. Sobre la interpretación. Introducción, traducción y notas de Jorge Mittelmann. Buenos Aires: Losada. Balakrishnan, R., & Ranganathan, K. (2012). A textbook of graph theory. New York: Springer. Bird, A. (2007). Nature’s metaphysics. Laws and Properties. Oxford: Clarendon Press. Byrne, A. (2015). Inverted Qualia. In Ed Zalta (Ed.). Stanford Encyclopedia of Philosophy. https:// plato.stanford.edu/entries/qualia-inverted/. Accessed on 18.01.17. French, S. (2014). The structure of the world. Metaphysics and representation. Oxford: Oxford University Press. Hawthorne, J. (2001). Causal Structuralism. Philosophical Perspectives, 15, 361–378. Reprinted in Metaphysical Essays, Oxford: Clarendon Press, 2006, 211–227. Ladyman, J., & Ross, D. (2007). Every thing must go. Metaphysics naturalized. Oxford: Oxford University Press. Linnebo, O. (2008). Structuralism and the notion of dependence. The Philosophical Quarterly, 58, 59–79. Lowe, E. J. (2010). On the individuation of powers. In A. Marmodoro (Ed.), The metaphysics of powers – Their grounding and their manifestations (pp. 8–26). New York: Routledge. Mertz, D. W. (1996). Moderate realism and its logic. New Haven: Yale University Press. Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. Oxford: Oxford University Press.

Part III

Particulars

Chapter 11

Substrata and Bundles

Abstract After the defense of Platonic universals it is necessary to consider how are particulars to be conceived, given that their characters are grounded in what universals those particulars instantiate. There have been traditionally two great alternatives for understanding the nature of particular objects. Under one alternative the ‘particularity’ of a particular is dependent on a substratum that is the subject of instantiation of different universals and its principle of unity. Under other alternative, particulars are ‘bundles’ or ‘collections’ of properties, either universals or tropes. This chapter examines the problems that these traditional conceptions face. On the one hand, it is argued that the difficulties leveled against substrata are not compelling. On the other hand, it is argued that the difficulties affecting the theories of particular objects as bundles of properties seem to demand structures of ontological dependence, like those postulated by Husserl in his idea of ‘pregnant wholes’. § 78. It has been justified in the previous chapters that there are universals and that universals have an existence independent of having or not instantiations. This still leaves open the question about how it is that these universals affect the concrete states of affairs that intervene in the causal network and that are, also, what is ordinarily offered to us in our knowledge of the world. The question of how particular objects—or what we usually consider as such—should be understood is also open. In effect, the postulation of universals brings with it problems inverse to those which a nominalist position brings. For the nominalist, the existence of particular objects is obvious. It is a fundamental fact of the world that there are particular objects. There are open problems regarding the mereological structure of such objects—i. e., how objects are constructed from other objects that must be their parts—or concerning their conditions of persistence over time, but there is no problem about how particular objects should be structured with the properties that those objects possess, simply because there are no properties. For a nominalist, however, the problem arises of explaining how numerically different objects can possess the same nature—the problem of the one over many (see § 10)—and how is it that the same object can have different natures—the problem of the many over one (see § 11). If the nominalist maintains that there are no universals, then something © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_11

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must fulfill its functions. It must be an entity that does not include anything other than particular objects and classes of particular objects. When things are considered from the perspective of the friend of the universals, the tables are inverted. It is evident that there are universal properties. Many objects can have the same nature because they can instantiate the same universal. It is evident that the same object can have different natures because it can instantiate several universals different from each other. What is not apparent now is what status should be granted to particular objects. A particular object must be made up of a structure, including universal properties and something of a particular character, but there are several theoretical alternatives very different from each other to understand this structure. Whoever proposes universals must explain what this structure is. This is a philosophical problem that the Platonist cannot evade because the particular objects that are ordinarily offered to us possess properties. They have, for example, a size, a mass, and a shape. The surfaces of an object are presented with colors and offer resistance if we exert any force on them. Universals are, on the other hand, abstract entities whose nature does not require to be instantiated in any spatiotemporal region rather than in another or, even, does not require being instantiated in any spatiotemporal region at all. What connection exists, then, between abstract universals and the particular objects that are offered to us in our everyday experience? What are those particular objects? A multitude of alternatives have been proposed to explain the nature of a particular object, compatible with the postulation of universals: (i) it has been proposed that there are only universal properties and that particular objects are bundles of universals; (ii) it has been proposed that there are universal properties and substrata—also called “bare” or “thin” particulars—that instantiate those properties; (iii) it has been proposed that there are universal properties and particular tropes that make up bundles; and (iv) it has been proposed that there are universal properties, particular tropes and, in addition, substrata that possess those tropes and form with them particular objects.1 Given what has already been argued in Parts I and II above, alternatives (i) and (ii) can be discarded. In the first place, alternative (ii) can be rejected because the sole postulation of universals and substrata is incapable of explaining the existence of the particular instantiations of such universals due to the regress of Bradley, as explained above (see §§ 67–70). The fact that a universal is instantiated in a particular object or several particular objects is not grounded on its universal and substratum constituents. States of affairs are something added to them. This is why it has been argued

1

Besides these theoretical alternatives there are conceptions that are not considered here: (v) the conceptions of bundles of tropes as particular objects, but without these tropes being essentially the instantiation of a universal, and (vi) the conceptions of substrata and tropes that these individuals possess, without the tropes being essentially the instantiation of a universal. In any of these alternatives, there are no universals, so they are incompatible with the theory that is being defended here. What has been typical for defenders of tropes is to prefer one of these two positions. If one considers the four positions (i)–(iv) indicated above together with these two additional (v)–(vi), there are six alternatives. To these six alternatives should be added conceptions of particular objects without tropes or universals, which is the type of position defended by the nominalists.

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that states of affairs are ‘primitive’ entities, ontologically emergent with respect to their constituents—i. e., ontologically dependent on universal and substratum, but not grounded on them. Such ‘primitive’ states of affairs are particular instances of the universal in the object or objects in question, that is, particular properties or tropes. The alternative (i) can also be discarded from now on. The conception of objects as bundles of universals has been among the least popular. In recent decades, however, it has received renewed attention and theoretically exciting work has been done to make this alternative more acceptable (see Cover and O’Leary-Hawthorne 1998; Hawthorne and Sider 2003). Perhaps some of the more traditional objections have been hasty. Even granting this, however, theories of bundles of universal have not substantially improved in likelihood. A traditional criticism of these theories is that they would require the acceptance of the principle of identity of the indiscernibles, that is, that two objects that possess exactly the same properties are the same object. By contrast, two objects numerically different from each other must differ concerning what properties they possess.2 In effect, if an object is nothing more than a plurality of universals ‘co-present’ with each other, then the identity of an object seems to be determined by what are the universals that make up the bundle. If what appears to be two bundles includes precisely the same universals, then they must be the same bundle. Two truly different bundles must differ by at least one universal that integrates one of them and does not integrate the other. From the work of Max Black (Black 1952; also, especially, Adams 1979; Alvarado 2006, 2007, 2016), however, it has seemed more reasonable to assume that it is metaphysically possible that there are two or more objects that have the same purely qualitative properties, against the principle. There does not seem to be any substantive reason to reject the metaphysical possibility of, for example, two spheres perfectly homogeneous with one another in all monadic properties and relationships they may possess. To answer this objection, it has been argued that the same bundle of universals could be multi-located in different regions of spacetime. The scenarios proposed by Black of different indiscernible objects can be treated as scenarios of the same plurality of universals instantiated in different spatiotemporal regions. Each universal can be instantiated in different objects. What is true for a universal, also applies to several universals. The whole reality of a particular object could be understood, then, as the ‘localization’ of a plurality of universals. However, a brief reflection on what might be a ‘localization’ of a universal allows us to understand why the theories of bundles of universals should be left aside compared to alternatives (iii) or (iv) indicated above. The ‘localization’ of a bundle of universals in a region of spacetime is the fact that these universals are all 2

There are several qualifications to be made to the principle of identity of indiscernibles: [8x8y ((Fx $ Fy) ! (x ¼ y))]. It should be understood as restricted to purely qualitative properties, as explained above. Otherwise, the principle is not only true but logically valid in first-order predicate logic with identity. In effect, let two objects a and b. If the objects are numerically different from each other, then a will have the property of being a [λz (z ¼ a)], while b will not possess it, because a 6¼ b.

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instantiated or exemplified in such a region—however the ‘regions’ of spacetime are understood. What is required, then, to make the theories of bundles of universals viable are ‘instantiations’, ‘exemplifications’ or ‘realizations’ of a universal or of several universals. However, a ‘particular instantiation’ of a universal is a trope. Similar considerations could be made about the primitive concept of ‘co-presence’ used by Russell as what unifies a bundle of universals (see Russell 1948, chapter 8). The fact that universals are ‘co-present’ with each other is the fact that they are instantiated by ‘the same’, or instantiated in the ‘same’ spacetime region. However, a particular instantiation of a universal is a trope. It turns out, then, that the alternatives open to conceiving the particular objects are the alternatives (iii) and (iv) if one must accept universals and their instantiations, as has already been explained in the previous chapters. Particular objects must be understood, either as bundles of tropes that are essentially the instantiation of a universal or else they are a structure of tropes and a substratum where, again, tropes are essentially the instantiation of a universal but now in a specific substratum (see Alvarado 2014). The contrast occurs between ontologies with substrata and ontologies of trope bundles. This is what will be discussed next.

11.1

Substrata or Bundles?

§ 79. As explained, then, there are two great theoretical options open to the defender of universals when it comes to an understanding of the nature of particular objects: these should be understood as bundles of tropes that are ‘co-present’ with each other or have another form of connection, or should be understood as the structure that results from a particular substratum and the tropes that the substratum possesses. The first alternative is similar to the type of position that has been defended by those who have traditionally proposed trope ontologies (see, among others, Williams 1953a, b; Campbell 1981, 1990; Denkel 1996; Maurin 2002; Nef 2006; Ehring 2011). The second alternative, on the other hand, is similar to the type of position that has been defended by philosophers such as Charles B. Martin or John Heil (see Martin 1980; Heil 2003, 2012). What is peculiar here is that the tropes that form bundles or are determining a substratum are essentially the instantiation of a universal. There are antecedents for positions like these with tropes and universals, but they are not too many. Edmund Husserl (1913, Untersuchungen II and III), John Cook Wilson (1926) and Norman Kemp Smith (1927a, b, c) have postulated theories of tropes that are essentially the instantiation of a universal. More recently, Jonathan Lowe has postulated an ontology of ‘four categories’ that includes substances, modes, and universals (see Lowe 2006).3 There are not many who have postulated

3 Lowe further argues that two types of universals must be differentiated: those whose instantiation are substances and those whose instantiation are ‘modes’ or tropes (see Lowe 2006, 87–100).

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theories similar to the one that is going to be presented here because usually, the defenders of tropes have believed they could dispense with universals. For what will be discussed in this section, then, it will be useful to make a comparative examination of the ontologies that comprise objects as property bundles and the ontologies that postulate a substratum as a constituent of particular objects, in addition to their properties. In what follows, the ‘properties of an object’ are its tropes. Traditionally it has been assumed that ontologies of substrata and ontologies of bundles of properties are incompatible with each other. Who posits a substratum would be postulating something additional to the properties of the object. It would be appropriate for an ontology of property bundles, on the other hand, to maintain that objects are ‘nothing more’ than the properties that ‘constitute’ them—however the nature of such ‘constitution’ is finally understood. Substratum theories postulate a category of entity in addition to the category or categories postulated by bundle theories. This contrast between theories of bundles and theories of substrata would be connected with an epistemological contrast between positions more ‘empiricist’ and more ‘rationalist’. A particular object is offered to us empirically by its ‘sensory’ properties. To perceive something as an apple is to perceive specific colors, textures, flavors, and aromas. It seems more economical and more in line with an empiricist perspective to hold that the apple is nothing more than those qualities. On the other hand, when a substratum is postulated—what possesses such qualities—something that exceeds what the experience offers us seems to be postulated. It would be something that should be postulated for purely theoretical reasons, as the explanation of sensory appearance, or as the best explanation of sensory appearances. Before going directly into the discussion of the comparative merits of ontologies of bundles and ontologies of substrata, it will be convenient to define some concepts. In general, the relationship between an object and the properties that the object possesses has been called “instantiation”. But the ‘instantiations’ of universals are tropes, as explained above (see § 69), so it is necessary to reserve a different expression to designate the relationship between an object and the tropes that the object possesses. Jonathan Lowe, for example, has distinguished between the relations of ‘instantiation’, ‘characterization’ and ‘exemplification’ (Lowe 2006, 34–51). As has been explained, Lowe postulates an ontology of ‘four categories’ with substances (substrata), modes (tropes), attributes (universals whose instantiation is a mode) and kinds (universals whose instantiation is a substance). Substances and modes are particular. Attributes and kinds are universal. The relationship that exists between an attribute and a mode, as well as the relationship between a kind and a particular substance is called “instantiation”. The relationship between a substance and its modes, on the other hand, is called “characterization”.4 The relation between a substance and the attributes whose modes characterize that substance is called “exemplification”. Lowe’s distinction between ‘instantiation’ and ‘characterization’

It is also ‘characterization’ for Lowe the relationship that exists between a kind and an attribute. This is a relation between universals and not a relationship between a particular substance and its particular modes. 4

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is useful but will not be enough. Both the defender of an ontology of trope bundles and a defender of an ontology of substrata agree that objects ‘own’, ‘possess’ or ‘have’ properties. Objects appear to us with properties. An object has been called, therefore, a “thick individual” as opposed to a “thin individual” which is the substratum that is ‘characterized’ by its tropes (see Armstrong 1978, 113–116). Both the friends of bundles and the friends of substrata agree to admit ‘thick particulars’. It will be convenient, therefore, to reserve a term to designate the relation of an object and its properties and to differentiate this relation from the specific relation of a substratum with its tropes. Thus, for the sake of accuracy, the relationship between an object and the universals that determine it will be called “instantiation”. These instances are tropes of the object. The relationship between an object and its tropes will be called “possession”.5 This relationship is neutral between ontologies of trope bundles and ontologies of substrata. The relationship between a substratum and its tropes will be called “characterization”. As will be explained below (see § 82), there are several ways in which how a plurality of tropes makes up a bundle has been conceived. Suppose that the position according to which a trope bundle is its mereological fusion is adopted. One could, then, formulate the contrast between ontologies of substrata and ontologies of bundles in the following way: Basic explanation of an ontology of substrata: for an object x and nature F, there is the substratum of x, let s(x), and a trope correlated with the nature F, let T(F), such that: the fact that x is F is grounded on the fact that T(F) characterizes s(x). Basic explanation of an ontology of bundles: for an object x and nature F, there is a trope correlated with the nature F, let T(F) such that: the fact that x is F is grounded on the fact that T(F) is part of x.

In both cases, it must be assumed, of course, that the respective tropes are essentially the instantiations of some universal. It can be seen here that in both cases the explanandum is the same, that is, the fact that an object has some nature. Since the trope that is mentioned is the instantiation of some universal, the grounding relationships referred to here are specifications of the basic grounding relationship postulated by an ontology of universals. It remains that the fact that an object has a nature is grounded on the instantiation of a universal. It should also be noted that the explanation scheme of an ontology of bundles will vary according to how they are conceived. If, for example, a primitive relation of ‘co-presence’ is proposed, then the explanans of the explanation scheme will be that T(F) is co-present with such and such other tropes instead of being part of a mereological fusion. The concept of ‘substratum’ can be analysed in terms of the characterization relationship. Something is a substratum if and only if it can be characterized by What is called here the “possession” of a property by an object has been called “manifestation” (manifestación) by Juan Luis Gubbins (see Gubbins 2016). ‘Manifestation’ seems more appropriate to indicate the relationship of an object with its sensory properties by which it can be perceived. As the properties may not be experientially perceptible, at least not in a ‘direct’ way, I have preferred to speak here of ‘possession’.

5

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something and cannot characterize anything. In a correlative way, the concept of ‘trope’ can be analysed through the relation of ‘characterization’, if one admits substrata in the ontology. In effect, something is a trope if and only if it can characterize something and can be characterized by something.6 However, one could also take the concepts of ‘substratum’ and ‘trope’ as primitives and analyse ‘characterization’ from them. R is a relation of ‘characterization’ if and only if it satisfies the following formula: for every object x and all nature F, there is a substratum of x, let s(x), and a trope correlated with F, let T(F) such that: if x is F, then the fact that x is F is grounded on the fact that R(s(x), T(F)). In other words, the characterization relation is precisely the relation between a substratum and a trope that grounds the natures that possess the respective object. It has been explained above that there is a distinction between ‘modifier’ and ‘module’ tropes (see § 27, Garcia 2014, 2015a, b). Modifier tropes ground the character of an object, but they do not possess it. Module tropes ground the character of an object because they possess it. Both for those who are inclined to posit modifier tropes and for those inclined to postulate module tropes, tropes must satisfy the explanatory functions indicated above. The fact that a substratum is characterized by a trope or the fact that a trope is part of an object, ground that the object has the nature in question. If one wants to formulate the distinction in a neutral way between theories of substrata and theories of bundles7 it should be said that, in either case, modifier tropes ground that an object x has nature F, but the correlative trope T(F) is not F. By contrast, module tropes ground the same fact that x is F because T(F) is F. In the case of modifier tropes its function is to ground that the objects that possess them have a nature. In the case of module tropes, their function is to have a nature and to make the object that possesses them also, in a vicarious way.

6

More precisely, using C as an abbreviation for the asymmetric relationship to be characterized by: x is a substratum ¼ df (♢∃y Cxy ^ Ø♢∃y Cyx) x is a trope ¼ df (♢∃y Cxy ^ ♢∃y Cyx) It must be assumed in these definitions that the quantifiers have as range entities of any category. The definition of substratum presupposes that every substratum must be characterized by some trope or another in some possible world or another. This could be considered controversial by some (see § 81). Perhaps an analysis should be preferred in which the concept of ‘substratum’ is something of a category to which possibly belong entities that can be characterized by something and cannot characterize anything. It would not be required that all substrata are possibly characterized, but only that some substratum may be. Something similar could be said about the definition of trope. It is assumed that every trope must be characterized by some trope or another in some possible world or another, which could also be considered controversial. This definition could be modified analogously and analyse a ‘trope’ as something of a category to which entities that can characterize something and possibly can be characterized by something possibly belong. That is, it is not required that all tropes are possibly characterized but only that it is possible that some trope is. 7 Although, as explained above (see § 27), modifier tropes seem more appropriate for substratum ontologies, while module tropes seem more appropriate for ontologies of bundles.

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Difficulties of Substratum Ontologies

§ 80. The problems of substratum ontologies will be considered first. These are ontologies that have been seen with frequent mistrust in recent centuries due to various empiricist objections. These difficulties must be weighed, taking into consideration also the obvious advantages offered by an ontology of substrata. Our intuition is that an object must be a unified entity and not a mere aggregation of parts. A substratum precisely fulfills the function of unifying the particular object in a way that bundle theories can only replicate with difficulty8 because it is what the plurality of tropes is characterizing. The possession of tropes by an object is grounded on the characterization that the tropes make of the only substratum. A substratum seems to be what persists at different times in which the same object can have different properties, without ceasing to be the ‘same’. A substratum, too, seems to be what is invariant in different possible worlds in which the object exists, although it possesses different contingent properties in those worlds. Of all the difficulties that have ever been adduced against the ontologies of substrata, here I am going to focus attention on four main objections: (i) the epistemological objection according to which one could not even get to know a substratum, nor get to form the concept of a ‘substratum’; (ii) the incoherence objection according to which substrata would be entities that, considered by themselves, do not own any property, but should at least have the property of having no property, which is incoherent; (iii) the objection of over-determination according to which the introduction of substrata would require duplicating the properties, since both the substratum and the respective object should ‘own’ them equally; and (iv) the modal objection according to which a substratum should be an entity that could not be characterized by anything. As this seems unacceptable, the postulation of substrata would also be unacceptable. What is going to be argued here is that none of these criticisms is a compelling reason to reject the coherence of the idea of a ‘substratum’. This is not yet a defense of its existence. It is merely going to be shown that there are no reasons to discard them a priori. The theory that is going to be postulated here will be a conception that, at the same time, will understand objects as bundles of tropes and as integrated by a substratum—or something that works as such (see § 89). The first objection against substrata has its justification in certain empiricist premises. Many philosophers of this philosophical tradition have argued that we can only form an intelligible concept of what is offered to us in sensory experience. The contents of our thinking are sensory impressions or collections of sensory impressions. At some point, the idea that all the content of our thought must come from sensory experience has been formulated as the requirement that the ‘meaning’ of a synthetic statement must be the set of sensory experiences that would verify

8 In traditional theories of universals and particulars, substrata additionally fulfilled the important function of being what confers particularity to the object or ‘thick particular’ and the states of affairs that the substratum integrates (see Armstrong 1978, 102–132). This theoretical function is not relevant here since both the substratum and the tropes that characterize it are particular.

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it. An intelligible concept in this perspective should be something with specific empirical application conditions. It should be content that we could apply to the experience that is presented to us, in order to discriminate what falls under such content from what does not. The problem would arise for the concept of ‘substratum’ because there would be no sensory experience or set of sensory experiences by which one would be justified in admitting that there is a substratum or that there is no substratum. The existence or nonexistence of a substratum seems to be transcendent to what experience can offer us. It would seem, then, that there is no intelligible content about whom one can judge whether or not there are substrata. It would be a pseudo-concept. There are several reasons to reject this objection, however. The empiricist premises are implausible. If we could not conceive what is not offered to us by sensory experience, we would not have been able to form the concept of ‘electron’ or the concept of ‘superstring’, for example. All scientific theoretical entities are justified by their explanatory functions for what is shown in sensory experience and not by what is shown in sensory experience. Analogously, a substratum should be postulated to fulfill theoretical functions of unification of particular objects and not because it is something given by perception. Above, it has been indicated how it could be analysed using the concept of ‘characterization’ along with other modal notions and logical constants. Should the notion of ‘characterization’ for these same reasons also be declared unintelligible? Such a thing does not seem reasonable. Some have also argued that there is a sense in which substrata are offered to us experientially (see Martin 1980, 7–10). From the perspective defended by these philosophers, substrata are not something ‘extra’ to the properties of an object. Neither the substratum nor the properties are ‘parts’ of a whole. Applying mereological notions to a particular object would be a categorical error. The substratum is something ‘about’ the object. What we call an “object” is the “possessor of properties” along with the “possessed properties”. It would be inscribed in the same concept of ‘property’, therefore, that it must have a possessor. When offered empirically, such properties are already offered to us as possessed by something, which must be a substratum. As can be appreciated according to this perspective, not only substrata are offered to us experientially when experiencing sensory properties, but the ontologies of substrata are justified by the analysis of the concepts of ‘property’ and ‘substratum’. Those who reject such ontologies would be unaware of the content of such concepts or would be misunderstanding them. In the section that follows (see § 81), there will be an examination of different types of problems that affect the ontologies of bundles. Some of these problems are quite deep, but however profound they may seem, there is no difficulty of incoherence as these philosophers have suggested.9 Besides, an important part of what is going to be

9 Indeed, it is difficult to argue against a philosopher who refuses to admit that he understands the concept C—and any other concept by which C could be defined. However, conversely, the claim that a concept is unintelligible when many other philosophers seem to understand and judge correctly about its content is implausible—it is of little credibility, at least, if specific derivations

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explained is that there is a multitude of different metaphysical positions that could be proposed as forms of ontologies of trope bundles. There are different types of difficulties for different ways of understanding trope bundles. It does not seem reasonable, therefore, to maintain that the mere idea of a ‘property’ without the correlative idea of a ‘substratum’ of such property is not intelligible. The second type of objection is that the idea of a ‘substratum’ would be contradictory. It is assumed that a substratum—considered by itself, in abstraction from the other ‘components’ of an object—has no property. However, then, it is true to assert of a substratum that it does not possess any property. Then it would seem that it owns the property of not owning any property, after all. However, if it does not own any property, then it has at least one property, which clearly is incoherent. It seems clear, however, that, although it may be true to say of a substratum the predicate “does not possess any property”, there is no correlative property that is what makes a substratum something that does not possess any property. Not every predicate is correlated to a property. There is no authentic property of not owning any property (see Sider 2006, 392–393). The reason generally adduced to maintain that there is no real property of not having any property is that it would be a negative property, which should be rejected as a rule. It happens, however, that there is another problem here, and that is that there is no universal to have a property to which the negation operation Neg can intelligibly apply to generate the universal of not having any property. It would be a universal that would instantiate everything that instantiates a universal. However, if x instantiates the universal U, it is not required additionally the universal of having a universal X such that: x instantiates X. The universal of having a universal X such that: x instantiates X would also be, by hypothesis, a universal. Be this second universal U0 . Then again, the universal having a universal X such that: x instantiates X should be instantiated. One could argue that this universal is identical to U0 , which would not generate an ascending hierarchy of different universals. Still, that same universal U0 should be instantiated multiple times in the same object, which is equally extravagant. Every object that owns a property must possess an additional property, namely, the property of owning some property. Also, since he owns this property of owning some property, then it has the property of owning some property again. Recall that these properties are tropes. It would be, therefore, that for each trope possessed by an object, there would be an infinity of other derivative tropes. It does not seem reasonable, therefore, to postulate the existence of such property. So, since there is no property of owning a property, there is no property of not owning a property. There is a difficulty close to the previous one which has to do with the character of a substratum by which it is, in effect, an entity of such type. A substratum must have an intrinsic nature. It is true to say of a substratum that “it is a substratum” and it is not true to say of a substratum that “it is a universal” given that intrinsic nature. The objection is that a substratum possesses the intrinsic nature that it has because it

of some patent inconsistency do not accompany this argument. Thus, in this case, the claim of philosophers like Martin that a ‘trope bundle’ is not intelligible seems implausible.

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instantiates the universal of being a substratum. So, a substratum is what it is because it instantiates at least one universal. Then, the idea of a substratum would be incoherent, because something that, considered in itself, does not possess any property, should have at least one property. In the case of a substratum, however, no property to be a substratum that confers to substrata the character of such is required. The intrinsic nature of a substratum is not a matter of properties that it is instantiating or of tropes that it possesses. Similarly, a universal has the intrinsic character of being universal, not because it is instantiating a universal of being universal. The character of a substratum is something that, if it is to be grounded on something, should be grounded on an improper part of itself. It is something that will not change if a substratum is ‘accompanied’ or not by other entities in a possible world, or if there are more or fewer entities those that ‘accompany’ it (see § 5). Therefore, it can be said that it is an “intrinsic” character or nature. Another type of objection has been raised more recently (see Bailey 2012; a detailed discussion in Gubbins 2016). An object has properties. A substratum also ‘possesses’ properties. If we consider, then, an object x that is F, it turns out that two numerically different entities are F: x and the substratum of x, s(x). Suppose that in a room there is a perfect cube and nothing else. In principle, it seems evident that there is only one cubic object. Be a ‘definite property’ a trope that, if it is possessed by something, it is possessed only by one object. So, if F is a definite property, and it happens that x is F and y is F, then x ¼ y. It is assumed that there is a substratum that ‘possesses’ properties, and this fact explains why the object formed by such a substratum possesses those properties. Let the property in question be a definite property. But, by hypothesis, an object is numerically different from its substratum. It happens, however, that if both an object and its substratum ‘own’ the same definite property, then the object must be identified with its substratum, against the hypothesis. Foreseeing this objection above, care has been taken to distinguish the concepts of ‘possessing a property’ and ‘being characterized by a property’. An object ‘possesses’ properties, but the substratum of such object is ‘characterized’ by them. If one keeps the precaution of differentiating the relationship between an object and its properties and the relationship of a substratum and the properties that characterize it, Bailey’s argument can be blocked without difficulty. An object x has properties because such properties characterize s(x). The object must be understood as the substratum together with the properties that characterize it. There is, then, no duplication of the tropes of the object. The fundamental connection is the ‘characterization’ of the substratum by its tropes. The possession of tropes by an object is a fact grounded on such fundamental connection between substrata and properties. The properties ‘possessed’ by an object are ‘constituted’ by the fact that those same properties—recall that they are tropes—characterize the substratum.10

10

This point of view very naturally imposes itself if an object is the mereological fusion of properties and a substratum. There are preventions, however, concerning a mereological perspective that will be explained below (see §§ 84, 87).

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Bailey’s argument is instructive, however, because it highlights the neglect that has existed in describing the theoretical functions that a substratum must fulfill and whether or not it should be identified with what we ordinarily understand as a ‘particular object’. Many times, it is freely spoken of ‘instantiation’ or ‘owning’ properties to deal with the connections between objects and universals, the connections between objects and tropes, the connections between substrata and universals or the connections between substrata and tropes. It has already been seen that these connections must be differentiated carefully. Even making these distinctions with the proper caution, however, it turns out that we have two candidates to be what persists in time and what should (or not) exist identical in different possible worlds. If one considers the problems usually treated as questions about the ‘persistence of objects in time’ (see Haslanger and Kurtz 2006) or as questions about the ‘identity between different possible worlds’ (see Mackie 2006) there is talk of the same ‘object’, for example, that exists ‘whole’ at different times or in different possible worlds. However, is it about the persistence over time of objects or substrata of objects? Is it about the identity of objects in different possible worlds or the identity of substrata in different possible worlds? Our ordinary intuitions about particular objects seem to indicate that it is contingent for an object to possess some properties. Analogously, it seems that there are properties that an object possesses only at some times and not at others. The problem here is that, if one is thinking of an ‘object’ as, for example, the mereological fusion of a substratum and the properties that characterize it, it would not be metaphysically possible for an object to vary properties in time or to have accidental properties. A mereological fusion has its parts essentially. It would not be the mereological fusion that it is without the parts it actually possesses. If the objects are mereological fusions of a substratum and the properties that characterize it, then that object could not have different properties, nor could it exist at different times possessing different properties. However, adequate flexibility in how a ‘particular object’ is understood must preserve contingent properties. Below this will be one of the aspects to be addressed for the theories of trope bundles (see § 81) and for the theory of particular objects that will be defended here (see §§ 87–94). The last type of objection that is going to be considered has to do with the metaphysical possibility that no property characterizes a substratum. If substrata are a category of entities different from properties, then one should assume that substrata and properties are different existents that can exist jointly or separately. A substratum could be characterized or not by some property. Moreover, just as it could be characterized or not by any property, it seems that it could not be characterized at all. If this assumption seems absurd—in effect, what would be a substratum characterized by nothing?—then there are reasons to reject the existence of substrata. One could be inclined to accept this argument from a combinatorial modal perspective (see §§ 43, 63) in which the space of metaphysical possibilities is grounded on the mutual independence of fundamental entities. If substratum and properties are entities numerically different from each other, they should be independent of each other, which opens up the possibility of a substratum without essential properties or a substratum without properties. It has already been seen,

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however, that this is not the only general perspective about metaphysical modality (see § 64) since it can also be conceived as grounded on primitive causal powers of actual entities. Even accepting the combinatorial conception of modality, however, it cannot be seen why one could not accept something similar to what Armstrong calls “the principle of rejection of bare particulars” (see Armstrong 1978, 113), i. e., that every object must necessarily have some or other property.11 This leaves open the question of whether or not there are essential properties for a substratum—because it must be characterized by some or other property in each possible world, without requiring it to be the same property in all the worlds in which the substratum exists— but it excludes the possibility of an uncharacterized substratum. The assumption of a substratum not characterized, or without being characterized essentially, should not be seen, however, with such suspicion. Not only is it not a weakness of substratum theories that they leave open the metaphysical possibility of such assumptions, but it is an advantage of such theories that they do so. Think, for example, about the thesis of divine simplicity. It has been argued that God lacks any ontological composition (see St. Thomas Aquinas, (1952), I, q. 3). God could not be, under this assumption, a trope bundle or a substratum characterized by tropes. It is consistent with this thesis to maintain, however, that it is a substratum.12 It results, then, that all objections—or the main objections—against substrata can be neglected. This is not yet a reason to positively justify the existence of substrata. For this justification, a comparative weighting is also required regarding the ontologies of property bundles, which is what will be done next.

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§ 81. The great alternative to ontologies of substrata has been the ontologies of bundles of properties. There has already been occasion to mention the conception of particular objects as bundles of universals co-present with each other and the severe difficulties that this conception must face (see § 78). The situation of the ontologies of trope bundles, on the other hand, has been very different. In most cases, those who have been inclined to posit tropes have also been inclined to reduce particular objects

Armstrong differentiates between a ‘strong’ and a ‘weak’ version of the principle. According to the strong version, every object must possess at least one monadic property in every possible world. According to the weak version, every object must have at least one property in each possible world, be it monadic or relational (see Armstrong 1978, 113). 12 Theodore Sider has also indicated that spacetime points or numbers in a structuralist ontology of mathematics would be good examples of uncharacterized substrata (see Sider 2006, 393). It would be, in any case, substrata not characterized by any monadic property, because they should be characterized by relations with other points or numbers, as the case may be. They would be an exception to Armstrong’s principle of ‘strong’ rejection of bare particulars, but not to the ‘weak’ principle. 11

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to tropes satisfying certain conditions.13 The ontologies of substrata allow to easily satisfy the explanatory requirement to account for the unity of a particular object because it is precisely the only substratum of an object what is characterized by all the tropes possessed by the object. This explanatory requirement, however, is an important problem for the different ontologies of trope bundles, because—in principle—no trope of a bundle should be fulfilling the same function of unification. Along with the explanatory requirement of unification, an intuitively adequate theory of particular objects should also be able to explain how there are essential and accidental properties for an object. It should be a theory in which there are certain properties that an object can possess or not possess while remaining the same object, as well as properties that the object must possess to be the same object. Another explanatory requirement—related quite closely to the previous one, but which should not be confused with it—is that a theory of particular objects must leave room for different properties to be possessed by the same object at different times. These two explanatory demands can be solved without great difficulties by an ontology of substrata. As has been explained, although it has sometimes been suggested that a substratum should not have essential properties—by the application of combinatorial principles of metaphysical modality—nothing prevents a substratum from having essential characterizations. Different properties characterize the same substratum in different possible worlds. For persistence over time, the same substratum is characterized by different properties at different times. These explanatory requirements, then, could be summarized as follows: [Requirement I] [Requirement II] [Requirement III]

A particular object is something unified and not a plurality. A particular object possesses accidental properties in different possible worlds. A particular object possesses transient properties at different times.

A mere plurality of entities is not unified, nor ‘constitutes’ something one, nor ‘belongs’ to something one, nor are ‘parts’ of something one. Frequently, this requirement of unification expressed in Requirement I has also been manifested in the requirement that a particular object must be occupying a topologically connected spacetime region.14 As for Requirement II, it has been formulated in terms of accidental properties, since these are what allow the contrast between something that has to remain modally invariant between different possible worlds and what, on the other hand, varies without affecting this nucleus. Similarly, Requirement III has

13

Exceptions are, as indicated above, Charles B. Martin (see Martin 1980) and John Heil (see Heil 2004, 2012). It is not entirely clear, on the other hand, that Jonathan Lowe is also an exception (see Lowe 2006) since his ‘substances’ are essentially the instantiation of a universal of substance, which seems to be a peculiar type of trope. The position that will be defended here, on the other hand, will be a theory of bundles of tropes, but with a substratum (see § 89). 14 As it is usual, a region is represented by a set of points. A region is ‘connected’ if and only if it cannot be represented by the union of two or more open disjunct non-empty sub-regions.

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been formulated in terms of ‘transient’ properties. A ‘transient’ property is a property that an object possesses at one time and that it does not possess at another time. As in so many other areas of philosophical discussion, these requirements are justified by certain intuitions about what a particular object is. These intuitions are not untouchable. It could be rational to reject them if there were important theoretical advantages that could be gained otherwise. There have not been lacking, on the other hand, those who have rejected such intuitions or have sought some way to respect them verbally without admitting them in substance. Thus, for example, it has been proposed that accidental attributions to an object should be understood as attributions to a ‘counterpart’ of that object in another possible world (see Lewis 1968). Strictly, in this conception all the properties of an object are essential, but the mechanism of the counterparts allows us to ‘say’ the same that the person who posits accidental properties is trying to say by re-interpretation of the truth conditions of the sentences in which the modal attributions are made to an object. It has been proposed, too, that what we call “an object persisting in time” is really a mereological fusion of temporal parts (see Lewis 1986, 202–204; Heller 1990; Sider 2001; Hawley 2001). Each of these temporal parts is an object of instantaneous existence that, therefore, does not possess transient properties. However, anyone who maintains that the possession of different temporal parts grounds persistence in time at different times can accept, in a derivative sense, that objects have transient properties since they have different instantaneous temporal parts that have different properties. Here the intuitions that are justifying the Requirements I–III are not going to be questioned. In principle, without further considerations, it is better for a theory to be able to satisfy such requirements than not. Moreover, the theories of substrata can satisfy them, as has been explained. Several of the more traditional defenses of trope ontologies have maintained that a particular object must be identified with a mereological fusion of tropes that co-occupy the same region of spacetime (see Williams 1953a, 9–12; Campbell 1981, 481–483; 1990). This conception seems to be very simple. All that is required to generate objects based on tropes are the axioms of standard extensional mereology—which are sufficient to guarantee the existence of unrestricted mereological sums (see § 6)—and facts about the occupation of spacetime regions. Of course, there will be mereological fusions for any plurality of tropes, but only a minority of such fusions will constitute an authentic object since all mereological fusions of tropes that are not co-occupying the same spatiotemporal region will be ‘filtered’. A conception in these lines, moreover, seems to satisfy a principle of ‘plenitude’. Any spatiotemporal region that is considered will select a single mereological fusion of all the tropes located exactly there.15 In this way, it seems that each spatiotemporal region—no matter how strange it may be, or if it is disconnected—is occupied exactly by an object. However, despite its simplicity—

15

For the purposes of this characterization, an object x is located exactly in the region r if and only if x is located in r, there are no proper parts of x located in some region r’, disjoint from r, and there are no sub-regions of r in which a part of x is not located.

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or apparent simplicity—there are, at least, four significant difficulties that a theory in these lines must face. In the first place, it would be a theory that would imply a priori the metaphysical impossibility of non-physical objects. God would be metaphysically impossible because having no spacetime location could not be any trope. God is supposed to be a particular type of object. If, in general, particular objects are bundles of tropes, God should be a bundle of tropes too. As bundles are based on co-location, no object without spacetime location could exist.16 Of course, reasons have been presented to reject the existence of God—such as the problem of evil—but hardly the absence of location of God could count as one of those reasons, especially if there are other alternative ways of understanding the unity of a bundle that does not have these drastic consequences. Secondly, a theory of this kind would be very inappropriate to satisfy Requirement II, since all the properties of an object would be essential to it. In effect, the identity of a mereological sum is given by what its parts are. If the objects are mereological fusions of tropes, then, they could not be constituted by different tropes from the tropes that compose them. An object could not possess any property different from those it possesses, even if the difference contemplated was infinitesimal. If Napoleon Bonaparte measures 1.70 mts, then he could not measure 1.70000001 mts. He could not have a hair less than he has, not one hair more than he has.17 It is also a consequence of a theory of trope bundles of this type that no object could be located in a spacetime region different from the one it occupies. It could not last temporally an infinitesimal instant more, nor an infinitesimal instant less than it lasts. A third difficulty is that in a conception of this type, there could not be different objects located in the same spatiotemporal region. Each spatiotemporal region selects a single object that is exactly the one that occupies it. It happens, however, that there are objects that can be co-located, although they are numerically different. A neutrino can cross the region occupied by other objects without interacting with them. Neutrinos should be metaphysically impossible if the objects were mereological fusions of tropes co-located with each other, but it would be curious that the Standard Model of Particle Physics should be abandoned for a minor metaphysical inconvenience. The fourth problem with this position is that the ‘co-location’ relationship in the same spatiotemporal region is of fundamental importance for the formation of particular objects. However, the ‘co-location’ is a relationship, and should, therefore, be a trope like any other. It is not difficult to see that this would generate a vicious 16

One might be inclined, perhaps, to assign to God as location the totality of spacetime. The problem with this assumption is that, then, God should be confused with the entire spacetime system—an idea perhaps acceptable to a friend of Spinoza or Hegel, but not to me, at least. 17 There are no problems—at least, not in principle—with Requirement III. An object is a fusion of all and only the tropes co-located exactly in the same spatiotemporal region. If this is a temporally extended region, it can have different tropes at different times, thus saving the possession of transient properties. This issue will be considered more closely below (see § 84).

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regress. In effect, the relational co-location trope should be co-localized in the same region with the remaining tropes that make up a bundle to form an object, but now such ‘co-location’ of the co-location trope and the other tropes must also be a trope, etc. Of course, a defender of an ontology of bundles in these lines would be inclined to argue that co-location is not a relational trope but an ontologically primitive, fundamental fact, not grounded on anything. But this idea is especially incredible for an ontology of trope bundles. What can spacetime be in an ontology of this kind? It would not be reasonable to maintain that it is a ‘substance’ existing by itself, that is, a kind of great substratum or a set of substrata—the set of all the points of spacetime.18 Indeed, co-localization in a ‘same’ region of spacetime demands the existence of such region. It has been usual to conceive space and time either as ‘substances’ that, in some way, exist independently of the objects or processes that are located in them or as structures grounded on such objects and processes. If, in general, there are no substrata but only bundles of tropes, then neither should it be maintained that space and time are constituted by substrata—as could be the points of the spatiotemporal regions—or be identified with a substratum. Space and time must be structures grounded on tropes. One could argue, perhaps, that spacetime is, by itself, a special trope bundle, but if so, those tropes should be co-located with each other in the same spatiotemporal region to form a bundle—because this is what a trope bundle consists of. But that same bundle is supposed to be spacetime. Spacetime should be ontologically prior to spacetime, which is absurd. One could, perhaps, argue then that spacetime is a structure grounded on trope bundles and their mutual relationships. However, this is again an absurdity, because such bundles exist because the tropes that constitute them are co-located with each other spatiotemporally. Then, bundles of tropes depend on the existence of spacetime. However, spacetime is supposed to be grounded on those same trope bundles. Then spacetime and the bundles of tropes would be and would not be ontologically prior among them, which again is absurd.19 As can be seen, then, although the assumption that trope bundles are mereological fusions of all and only tropes co-located with each other in the same spatiotemporal region seems elegant and straightforward, many difficulties make this alternative unwise. A variation of this type of conception is one that has been rehearsed by philosophers such as Anna-Sofia Maurin (see Maurin 2002, 117–180) or Douglas

18

In the case of Campbell’s theory (see 1990, 135–155), it is argued that there is a single large trope bundle that is the mereological fusion of tropes corresponding to fields for each of the fundamental physical forces. It is assumed that spacetime is simply the extension of these giant fields. How can this theory consist of these tropes of fields being co-located in the totality of space-time? There does not seem to be any spacetime but the primitive distension of such vast tropes. It cannot be, therefore, that such tropes make up a bundle because they are co-located in the same region of spacetime, because the regions of spacetime—or the entire spacetime, for the same reasons—are entities grounded on the vast tropes of the bundle which is the only object of the world. 19 It would be a situation analogous to what happens with immanent universals and their instantiations. Universals are supposedly grounded on their instantiations, but instantiations are ontologically dependent on the universals of which they are, in effect, instantiations (see § 58).

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Ehring (see Ehring 2011, 98–135). In traditional trope ontologies, such as those proposed by Williams or Campbell, tropes are independent entities that can exist jointly or separately. A trope does not need to be part of the bundle of which it is, in fact, a part. Moreover, although tropes actually occur in bundles, nothing would prevent a trope of mass, for example, from existing separately without being ‘accompanied’ by other tropes or ‘co-located’ with other tropes. The proposals of Maurin and Ehring preserve this conception of tropes that are modally independent of each other, but instead of postulating facts of co-location for the unification of the bundle, they postulate a relationship of co-presence. The ‘co-presence’, in and of itself, does not bring with it demands for a specific spatiotemporal location, which allows more flexibility than traditional theories. Nothing prevents an object from having a spacetime location different from the one it actually possesses. Nothing prevents, either, that different objects share the same location.20 It is notorious, however, that a ‘co-presence’ relation is, in effect, a relation and, in a general trope ontology, relationships must also be conceived as tropes. A trope of ‘co-presence’ in these terms would generate a vicious regress. Indeed, a plurality of tropes forms a bundle because these tropes are co-present with each other. Then, there is a trope of co-presence integrating the bundle. Then, the trope of co-presence and the remaining tropes make up such a bundle because they are co-present with each other. However, this ‘co-presence’ is also a relationship, etc. There are, in principle, two ways to solve this problem. One could argue that ‘co-presence’ is a primitive ontological fact or one could postulate a special nature for the trope of co-presence so that— somehow—the vicious regress is avoided. The latter is what Maurin and Ehring have done, though in different ways. Maurin maintains that the relational trope of co-presence is essentially connected to the remaining tropes of a bundle (see Maurin 2002, 164–166). Since it is part of the essence of the trope of co-presence to be, in effect, the co-presence of certain tropes, then no additional fact is required to ground the existence of a bundle. The remaining tropes of a bundle are independent of each other so that they can exist connected to one bundle or another—or even exist ‘free’, without being connected to any bundle. The trope of co-presence, on the other hand, depends ontologically on

20

It will be noted that the relationship of a bundle of tropes with a spatiotemporal region—however a ‘region’ is understood—is a relationship and, for that reason, it should be treated just like any other trope. There is a general problem here for the ontologies of bundles that will be discussed below with detention (see §§ 93–94). If all the relationships of a bundle with others are part of a bundle of tropes, then a bundle must include all the others with which it has some relation. This would, of course, include all other objects that are at some spatial or temporal distance. It seems, therefore, that—at the very least—there could be no more than a single bundle of tropes in a spacetime. There could be no more than a single bundle of tropes for all tropes that have some causal connection with each other. The theories of trope bundles would seem, if this is the case, to be recommending a form of monism. Theories of trope bundles have been developed with monadic properties in mind, and relationships have not been considered with sufficient attention. A more careful examination of relationships becomes especially urgent if in some areas, a form of structuralism seems advisable, as will be explained below (see §§ 93–94). For the time being, monadic trope bundles will be considered to simplify the discussion.

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each of the other tropes. It could not exist without them and, therefore, the complete bundle that is an object could not exist without any of the tropes that actually make it up. It can be appreciated, therefore, that a theory in the lines proposed by Maurin, although would leave room for different spacetime locations, would not allow for the possession of accidental properties. Requirement II would not be satisfied. It would also imply introducing a profound difference between ‘ordinary’ tropes and tropes of co-presence. It is assumed that tropes are entities independent of each other. Such has been the assumption of the philosophical tradition in which the work of Maurin is inscribed. There is a special type of trope, however, that completely breaks with the ‘ordinary’ nature of a trope and are precisely the tropes required to avoid the vicious regress pointed above. The introduction of this differentiation is uneconomical, at the least. The relational trope of co-presence is suspiciously similar to a substratum that fulfills the functions of unification of the particular object and of being what is characterized by ‘normal’ tropes.21 There is a significant difference, however, between the structure proposed for a particular object by Maurin and the structure traditionally assumed in substratum ontologies. Maurin’s co-presence tropes reverse the usual order of ontological priority. In the philosophical tradition from Aristotle onwards (see Aristotle (2009), 5, 2a34–2b6) it has been assumed that the properties depend ontologically on the substratum they characterize, while the substratum is independent. In the scheme presented here, on the other hand, what would fulfill the functions of a substratum—the relational trope of co-presence— ontologically depends on the properties that ‘characterize’ it. The motivation to make this inversion is to preserve for ‘ordinary’ tropes their mutual ontological independence, but this is done at the price of extreme modal fragility for the objects constructed by such tropes. Paradoxically, mutually independent tropes can only construct an object assuming that it has to possess essentially all the properties that it possesses. Conversely, in traditional substratum theories, the modal independence granted to the substratum allows to preserve the intuitions pointed out in Requirements II and III. As will be shown below, one way to preserve the spirit of bundle theories for the understanding of particular objects is to renounce the idea of mutual independence between tropes. Before considering this family of positions, mention should be made of the solution offered by Ehring (see 2011, 128–135). It is a more radical proposal than that of Maurin to preserve ‘normal’ tropes independent of each other but also affected by much more severe difficulties. In the same way as Maurin, Ehring posits a relational trope of co-presence. It is a trope modally independent of the other

21

This, in itself, would not be a problem if, in this way, it is possible to show a profound unity between theories of bundles and theories of substrata. Something in these lines is what is going to be sustained below (see § 89). The problem here has to do with the additional theoretical costs imposed by Maurin’s solution. The closeness of Maurin’s relational tropes with substrata is due to several reasons. A substratum, by itself, has no qualitative character. A trope, on the other hand, is essentially a particular quality. However, the relational tropes of co-presence do not seem to have any qualitative character —with the preventions indicated in § 27—in the same way that happens with substrata.

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‘normal’ tropes, so the costs of economy implied by Maurin’s position are avoided. This trope of co-presence, however, is self-connecting to the remaining tropes of the bundle. It is assumed that a trope of co-presence is what explains ontologically why ‘common’ tropes of a bundle are unified among themselves to construct an object. Let tropes T1, T2, ..., Tn that make up the same object x. The fact that T1, T2, ..., Tn form x is grounded on the fact that T1, T2, ..., Tn are co-present with each other. ‘Copresence’ is, in turn, a relational trope, be it C. The fact that T1, T2, ..., Tn and C make up a bundle is, in turn, grounded on the fact that T1, T2, ..., Tn and C are co-present with each other. This ‘co-presence’ is a relational trope, but is the same relational trope C. In this way, T1, T2, ..., Tn make up a bundle because of C, and T1, T2, ..., Tn, C form a bundle also because of C. It is difficult to understand precisely what Ehring is proposing here. In the case of a universal, there are no great difficulties in supposing that the same universal is instantiated in different objects. For a relational universal R that is instantiated in x1, x2, ..., xn, nothing prevents it also being instantiated in other objects y1, y2, ..., yn. In the case of tropes, however, clearly the relational trope TR characterizing x1, x2, ..., xn must be numerically different from the relational trope TR’ characterizing y1, y2, ..., yn, if x1, x2, ..., xn and y1, y2, ..., yn are different from each other. The ‘co-presence’ of T1, T2, ..., Tn should be numerically different, then, from the ‘co-presence’ of T1, T2, ..., Tn, C, since it is connecting different relata. Ehring responds to this objection by pointing out that ‘co-presence’ must be understood as a multigrade property. In this way, it would seem that the same relational trope C relates T1, T2, ..., Tn, C at the same time (see Ehring 2011, 133). The problem here is that the difficulty that has generated the vicious regress subsists. It is assumed that one advantage of Ehring’s position is that the relational trope of ‘co-presence’ does not have a different nature from that of any other trope. It is modally independent of the other ‘common’ tropes that make up a given bundle. Although a trope of co-presence C is, in fact, co-present with tropes T1, T2, ..., Tn, it might not be. The fact that this co-presence is the same co-presence of C with T1, T2, ..., Tn is irrelevant. C might not be co-present with T1, T2, ..., Tn but with other different tropes. The fact that C is forming the bundle that it forms seems to claim for an ontological explanation. Moreover, there are only two strategies that seem to solve this demand, but these strategies go by already well-known lines. One could maintain that this trope is essentially connecting to T1, T2, ..., Tn and C to form a bundle, as Maurin has done, or one could give up an ontological explanation of this fact and maintain that the co-presence of T1, T2, ..., Tn and C is a ‘primitive’ and fundamental fact. In this second case, one cannot see why a trope of co-presence C should be postulated since it would suffice to argue that the co-presence of T1, T2, ..., Tn is a primitive and fundamental fact, not a relational trope of co-presence. As can be seen, therefore, the alternatives proposed by Maurin and Ehring for a relational trope of co-presence generate important difficulties. It seems more reasonable simply to postulate a ‘primitive’ and fundamental fact of co-presence that does not admit being treated as a trope. But on several occasions, it has already been indicated that the postulation of ‘primitives’ is usually suspect (see, for example, § 18). Many times, the introduction of primitives is the pretension to achieve the satisfaction of an ontological function without incurring in annoying ontological

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commitments. A ‘primitive fact’ of co-presence is a co-presence that fulfills functions of unification of an object just as a substratum does. It seems to be simply an additional category of entity in addition to tropes. Postulating such a ‘primitive’ is not more economical ontologically than postulating a substratum. There is a parallel philosophical tradition, however, in which tropes are not independent entities, but quite the opposite. Edmund Husserl is a very conspicuous representative of this tradition, but there are many other antecedents. Tropes ontologically dependent have been the general rule in the history of philosophy because a particular property has been conceived as an essential determination ‘of something’, that is, as a type of entity that cannot exist separate from a substratum or other properties. Husserl was perfectly aware of the problems of regress that have been noted above. The solution it offers is to think of bundles of tropes as ‘pregnant wholes’ (see Husserl 1913, Untersuchungen III, §§ 19–22), i. e., as ‘wholes’ formed by mutually dependent tropes. In other words, an ‘object’ is a bundle of all and only the tropes that are to each other under the relation of ontological dependence.22 The whole bundle can be qualified, in some sense, as ‘independent’. Since ‘independence’ has traditionally been seen as one of the characteristic features of a ‘substance’ (see Hoffman and Rosenkrantz 1994, 89–143), Husserlian ‘pregnant wholes’ seem acceptable candidates for ‘substances’. If the bundle of tropes that is an object is the mereological fusion of such tropes, it is clear that it will be dependent on the tropes that are its parts, in the same way as for any other mereological fusion. The bundle, however, is independent of any entity other than its parts and, therefore, can be said in some sense “independent”.23 It does not depend on anything disjointed from itself. Note that in the Husserlian proposal the requirement to postulate something as a ‘unifier’ of the bundle is eliminated since it is now part of the essence of each of the tropes of a bundle to be part of it. No trope could exist without any of the others with which it forms an object. All the tropes of a bundle depend mutually on each other. It can be seen that a conception of tropes according to which mutual ontological dependencies interconnect these avoids all the maneuvers to unify an object that have been attempted by the traditional defenders of trope ontologies. Trope bundles thus understood manage to satisfy, therefore, Requirement I. There are in this conception, however, several important difficulties. In the first place, objects remain extremely modally fragile, since all the properties that an 22

Husserlian terminology is different from the usual one today, of course. A trope is designated as a “moment”, and ontological dependence is called “foundation”. See, for a general presentation, Simons 1982, 1994, 557–560. 23 Simons says that: “a substance is a particular whole under the closure of dependence, and does not need to be invoked as a new basic category” (Simons 2000, 148). The ‘closure’ of the dependence relation is the class of all and only the entities that are connected by the relation of ontological dependence. A substance would be the mereological sum of all and only the entities connected by ontological dependence because all of them depend on others that are part of the sum, or they are entities on which some other part of the sum depends. A ‘whole’ of entities all connected by the ‘closure’ of a dyadic relationship is called by Simons an “integral whole” (see Simons 1987, 324–360). A pregnant whole is an integral whole whose ‘characteristic relationship’ is ontological dependence.

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object in fact has will be essential for it. Husserlian ‘pregnant wholes’ are not suitable to satisfy Requirement II. No object could have properties different from those it actually possesses, even if they differ only infinitesimally. Husserl seems to have been motivated to postulate mutual dependencies between tropes by the considerations of the connections that can be found between, for example, a color and the surface on which it is, or a musical tone and its volume (see Husserl 1913, Untersuchung III, §§ 4–11). No color can exist except on a surface, and a surface must have some color. No sound of a specific tone could exist without also having a volume and a timbre. No sound of a given volume could exist without also having a tone, etc. In these cases, however, it is not clear that there is rigid dependence between color and surface, or between tone, volume, and timbre. The indicated connections could be generic dependencies.24 It does not seem that Husserl has considered the nature of these connections with sufficient care, at least not in this text (see Husserl 1913, Untersuchung III; in this sense, Simons 1994, 559–560). In any case, the consideration of these connections is not enough to motivate rigid dependencies between the tropes of a ‘pregnant whole’. Second, the assumption that there is a plurality of tropes all of them dependent on each other requires postulating an ontological dependence relationship that, at least, is non-asymmetric, and then, either non-irreflexive or non-transitive. Recall that a relation R is said to be “non-irreflexive” if and only if it is not necessary for all x: not Rxx. It is “non-asymmetric” if and only if it is not necessary that, if Rxy, then not Ryx. It is “non-transitive” if and only if it is not necessary that if Rxy and Ryz, then Rxz. For a class of all and only the dependent tropes to be intelligible, it must be assumed that (i) either all the tropes are dependent on themselves—if the transitivity of dependence is preserved—, (ii) or no trope is dependent on itself—if transitivity is rejected. In any case, asymmetry must be rejected so that all tropes depend on each other. However, as has already been explained several times, ontological dependence is a strict order, that is, an irreflexive, asymmetric, and transitive relationship. The intelligibility of a ‘pregnant whole’ in the terms proposed by Husserl requires a quite radical reform of the notions of ontological priority. The intelligibility of

24

Recall that rigid ontological dependence is the standard ontological dependence as it has been described so far. It is said that “x depends rigidly on y” when, by virtue of the essence of x, x could not exist without the existence of y—that specific entity. It is said that “x depends generically on an F” when, by virtue of the essence of x, x cannot exist without the existence of some F (see § 4). One might, perhaps, ask at this point if a pregnant whole could not be characterized as something like a ‘weak pregnant whole’ whose characteristic relationship was generic dependence and not rigid dependence. That is, could not a cluster of tropes be conceived as a mereological fusion of all and only the tropes that are all generically dependent on each other? It happens, however, that the generic dependence of x on y—of type F—does not make necessary for the existence of x to be accompanied by the existence of y. The problems that have been described above have to do with the difficulties of explaining how a plurality of tropes actually makes up a single object. Generic dependence does not explain why these particular tropes make up a bundle, but only why there should be a trope or another of certain types forming a bundle. The fact that particular tropes make up a bundle would require an additional explanation that would have to go through the lines already known and described above.

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mutual ontological dependencies cannot be presumed. This will be discussed in more detail below (see § 86). Is it possible to refine an ontology of trope bundles in these lines to solve these difficulties? Several philosophers have thought that, although a theory of Husserlian ‘pregnant wholes’ has several problems, it is appropriate to think of tropes as dependent entities, that is, in the way they have been conceived by much of the philosophical tradition. This is what is going to be discussed in the next chapter.

References Adams, R. M. (1979). Primitive thisness and primitive identity. The Journal of Philosophy, 76(1), 5–26. Alvarado, J. T. (2006). Esencias individuales e identidad primitiva (J. Ahumada, M. Pantaleone, & V. Rodríguez, Eds.), Epistemología e Historia de la Ciencia 12, (pp. 31–36). Alvarado, J. T. (2007). Esencias individuales e identidad primitiva. Analytica, 11(2), 155–195. Alvarado, J. T. (2014). Sustratos versus cúmulos. Ontologías alternativas para objetos particulares. Límite, 9(29), 7–34. Alvarado, J. T. (2016). Esencias y espacio modal. Revista de filosofía, 15(1), 11–36. Aquinas, Thomas. Summa theologiae (1952). Cura et studio petri Caramello cum texto ex recensione leonina. Roma: Marietti. Aristotle, Categories (2009). Categorías. Sobre la interpretación. Introducción, traducción y notas de Jorge Mittelmann. Buenos Aires: Losada. Armstrong, D. M. (1978). Universals and scientific realism, Volume I, nominalism and realism. Cambridge: Cambridge University Press. Bailey, A. M. (2012). No bare particulars. Philosophical Studies, 158, 31–41. Black, M. (1952). The identity of indiscernibles. Mind, 61, 153–164. Campbell, K. (1981). The metaphysic of abstract particulars. Midwest Studies in Philosophy, 6, 477–488. Campbell, K. (1990). Abstract particulars. Oxford: Blackwell. Cook Wilson, J. (1926). Statement and inference, with other philosophical papers. Oxford: Clarendon Press. Cover, J., & O’Leary-Hawthorne, J. (1998). A world of universals. Philosophical Studies, 91, 205–219. Denkel, A. (1996). Object and property. Cambridge: Cambridge University Press. Ehring, D. (2011). Tropes: Properties, objects, and mental causation. Oxford: Oxford University Press. Garcia, R. K. (2014). Bundle’s theory black box: Gap challenges for the bundle theory of substance. Philosophia, 42, 115–126. Garcia, R. K. (2015a). Two ways to particularize a property. Journal of the American Philosophical Association, 1(4), 635–652. Garcia, R. K. (2015b). Is trope theory a divided house? In G. Galluzzo & M. J. Loux (Eds.), The problem of universals in contemporary philosophy (pp. 133–155). Cambridge: Cambridge University Press. Gubbins, J. L. (2016). Particulares verdaderamente desnudos. Una reinterpretación a las teorías de sustratos. Tesis de Licenciatura en Filosofía, Pontificia Universidad Católica de Chile. Haslanger, S., & Marie Kurtz, R. (Eds.). (2006). Persistence. Contemporary readings. Cambridge, MA: MIT Press. Hawley, K. (2001). How things persist. Oxford: Clarendon Press.

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Hawthorne, J., & Sider, T. (2003). Locations. Philosophical Topics (pp. 53–76). Reprinted in Metaphysical Essays (pp. 31–52), Oxford: Clarendon Press, 2006. Heil, J. (2003). From an ontological point of view. Oxford: Clarendon Press. Heil, J. (2012). The universe as we find it. Oxford: Clarendon Press. Heller, M. (1990). The ontology of physical objects. Four-dimensional hunks of matter. Cambridge: Cambridge University Press. Hoffman, J., & Rosenkrantz, G. (1994). Substance among other categories. Cambridge: Cambridge University Press. Husserl, E. (1913). Logische Untersuchungen, Herausgegeben von Ursula Panzer, The Hague: Martinus Nijhoff, 1984. Traducción al español de Manuel García Morente y José Gaos, Barcelona: Altaya, 1929. Kemp Smith, N. (1927a). The nature of universals – I. Mind, 36(2), 137–157. Kemp Smith, N. (1927b). The nature of universals – II. Mind, 36(3), 265–280. Kemp Smith, N. (1927c). The nature of universals – III. Mind, 36(4), 393–422. Lewis, D. K. (1968). Counterpart theory and quantified modal logic. Journal of Philosophy, 65, 113–125. Reprinted with postscripts in David Lewis, Philosophical Papers. Volume I. New York: Oxford University Press, 1983, 26–46. Lewis, D. K. (1986). On the plurality of worlds. Oxford: Blackwell. Lowe, E. J. (2006). The four-category ontology. A metaphysical foundation for natural science. Oxford: Clarendon Press. Mackie, P. (2006). How things might have been. Individuals, kinds, and essential properties. Oxford: Clarendon Press. Martin, C. B. (1980). Substance substantiated. Australasian Journal of Philosophy, 58, 3–10. Maurin, A.-S. (2002). If tropes. Dordrecht: Kluwer. Nef, F. (2006). Les propriétés des choses. Expérience et logique. Paris: Librairie Philosophique J. Vrin. Russell, B (1948). Human Knowledge. Its scope and limits, London: George Allen & Unwin (1977). Traducido por Néstor Míguez como El conocimiento humano. Su alcance y sus límites, Madrid: Tauros. Sider, T. (2001). Four-dimensionalism. An ontology of persistence and time. Oxford: Clarendon Press. Sider, T. (2006). Bare particulars. Philosophical Perspectives, 20, 387–397. Simons, P. (1982). The formalisation of Husserl’s theory of wholes and parts. In B. Smith (Ed.), Parts and moments. Studies in logic and formal ontology (pp. 113–159). München: Philosophia Verlag. Simons, P. (1987). Parts. A study in ontology. Oxford: Clarendon Press. Simons, P. (1994). Particulars in particular clothing: Three trope theories of substance. Philosophy and Phenomenological Research, 54, 553–575. Simons, P. (2000). Identity through time and trope bundles. Topoi, 19, 147–155. Williams, D. C. (1953a). On the elements of being: I. The Review of Metaphysics, 7, 3–18. Williams, D. C. (1953b). On the elements of being: II. The Review of Metaphysics, 7, 71–92.

Chapter 12

The Nuclear Theory of Trope Bundles

Abstract The previous chapter has discussed the main alternatives for the conception of particular objects. The problems that affect theories of trope-bundles have led to the idea of ‘pregnant wholes’ originally proposed by Husserl. This chapter considers in more detail the ‘nuclear’ theory of pregnant wholes defended by Peter Simons in a series of works. The nuclear theory preserves the advantages of the ‘pregnant wholes’ and correct some of its problems. Nevertheless, the nuclear theory faces important additional complications. On the one hand, it has to face the problem of ‘substantial change’. On the other hand, nuclear bundles of tropes are built by connections of ontological dependence between tropes, but these connections violate basic requirements of irreflexivity, asymmetry and transitivity. § 82. The so-called “nuclear” theory of trope bundles was initially proposed by Peter Simons in a series of works (see Simons 1994, 1998, 2000), but then it has been continued in different ways by several other philosophers (see, for example, Keinänen 2005, 2011; Keinänen and Hakkarainen 2010; Alvarado 2015, 2016). The theory that will be proposed here (see §§ 87–94) is a form of nuclear theory. Two central ideas guide this conception: (i) tropes are ontologically dependent entities, in the way foreseen in Husserlian “pregnant wholes” theory and much of the philosophical tradition, but (ii) there are essential and accidental properties for an object. There are properties that ‘fix’ the identity of the object and others that presuppose such an already ‘fixed’ identity. It is a conception, therefore, in which the unification provided by the relations of ontological dependence must also be adequate to establish the contrast between tropes that the bundle could not fail to possess without ceasing to be the bundle that it is, and other tropes that do not fulfill this function. What is postulated is that there are two different ontological strata in a bundle of tropes: the nucleus and the periphery (see Simons 1994, 567–569). It is the postulation of this ‘nucleus’ that gives its name to this conception. The nucleus of an object is a mereological fusion of tropes that satisfy the condition of being all of them ontologically dependent on each other, as happens with a Husserlian ‘pregnant whole’. The tropes that are parts of the nucleus are essential properties of the object © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 J. T. Alvarado, A Metaphysics of Platonic Universals and their Instantiations, Synthese Library 428, https://doi.org/10.1007/978-3-030-53393-9_12

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because if one of those tropes did not exist, no other trope of the nuclear bundle would exist. They also constitute the individual essence of the object in question (see Simons 1994, 567). In effect, these are essential properties, but, besides, they are particular properties. No other object could possess these particular properties without being exactly this same object. This does not, of course, preclude the metaphysical possibility of other objects possessing tropes that are essentially the instantiation of the same universals. That is, nothing prevents the existence of qualitative duplicates of a given object. The nuclear bundle, however, is made up of tropes that cannot be shared by other objects. In addition to the nucleus, there is also a periphery that is made up of tropes that depend ontologically on the tropes of the nucleus. The point is that the tropes of the nucleus do not depend ontologically on the tropes of the periphery. The peripheral tropes cannot exist if it is not ‘determining’ the nucleus that they, in fact, ‘determine’. The tropes of the nucleus, on the other hand, could exist accompanied by other peripheral tropes different from those that, in fact, accompany them. The peripheral tropes are, therefore, accidental to the object. In nuclear theory, objects can continue to be conceived as independent ‘substances’. These are mereological fusions of all and only the tropes connected by ontological dependence. Of course, not all tropes are dependent on each other. This only applies to the tropes of the nucleus. It is valid for all the tropes of the bundle either that they depend on tropes of the bundle, or that there are tropes in the bundle that depend on them. The complete bundle is formed by the ‘closure’ of ontological dependence and, therefore, does not depend on anything that is not part of it.1 Simons also notes that the nuclear theory offers great flexibility to understand the structure of different types of particular objects (see Simons 1994, 568–569). Objects with nuclei of different numbers of tropes can be given. Some with more nuclear tropes and others with less. In the limit case, there could be objects with a single nuclear trope. This hypothesis will be exploited below (see § 87). There could be objects with nuclear tropes and without peripheral tropes.2 It should generally be assumed that, although nuclear tropes do not rigidly depend on peripheral tropes, they may be generically dependent on peripheral tropes of a specific type. A physical object should, for example, have some mass and some shape, although not

An ‘integral whole’ in the terminology of Simons (see 1987, 324–335) is a mereological fusion of all and only the entities that are under a reflexive, symmetric and transitive relationship R. For an asymmetric relation R, the converse of R can be defined, let that converse be R*. Thus, even if R is not symmetric, the disjunction of R and R* is, let that disjunction be [Disj (Rxy, R*xy)]. Although R is not transitive, the ancestral of R is. And even if R is not reflexive, [Disj (Rxy, x ¼ y)] is. In this way, any dyadic relation R allows defining the characteristic relationship of an integral whole, that is, the ancestral of the disjunction of [Disj (Rxy, x ¼ y)] and its converse. This is true generally, hence it is valid for the relation of ontological dependence. The operation Disj can be defined as: [Disj (U1x, U2x) ¼ df Neg (Conj (Neg(U1x), Neg(U2x)))]. 2 Simons also indicates that there could be objects with periphery and no nucleus (see Simons 1994, 568), but this hypothesis is difficult to understand. Indeed, if peripheral tropes do not depend rigidly on a nucleus, on what do they depend? Either they depend on each other, in which case they would not differ from an object with a nucleus and without periphery, or they do not depend on each other at all, in which case they could not form a single object. 1

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necessarily this determinate mass and this determinate shape. There would be no principled objections to also admitting the possibility of ‘extra’ peripheral tropes for which the nuclear tropes do not have generic dependence. Another alternative is that peripheral tropes come in clumps, with a sub-core and a sub-periphery. Finally, in the periphery of an object there can be relational tropes that connect that object with others. Relationships would be overlaps of the peripheries of two or more different objects. The flexibility of nuclear theory would be especially appropriate for a ‘scientific’ ontology attentive to what our best empirical research teaches us about the structures of natural objects. Simons presents as a sample of this flexibility how the nuclear theory could accommodate what the Standard Model of Particle Physics seems to show (see Simons 1994, 569–574). In the Standard Model, the particles are ‘defined’ by specific physical quantities that are invariant and ‘identical’ for each type of particle. These magnitudes are the rest mass, the charge and the ‘quantum’ of spin. Every electron, for example, has exactly a mass of 0.511 MeV/c2, a charge of 1 and a spin quantum of ½; all quarks of the type ‘above’ have exactly a mass of 2.4 MeV/c2, a charge of 2/3 and a spin of ½; all quarks of type ‘down’ have exactly a mass of 4.8 MeV/c2, a charge of 1/3 and a spin of ½. The ‘identity’ of a particle seems to be determined by such properties. It seems reasonable, then, to assume that these tropes make up the nucleus of a particle. There are other properties, however, that a particle can have in a modally contingent and temporally transitory manner, such as its relative position, its kinetic energy, its momentum or the direction of its spin (not its ‘quantum’ of spin). These tropes seem to form, therefore, the periphery of a particle. The same nucleus may or may not be accompanied at different times by different tropes of momentum, position, kinetic energy, and direction of spin. In the Standard Model, a fundamental difference is made between ‘fermions’ and ‘bosons’. ‘Fermions’ are the particles that obey Pauli’s exclusion principle, according to which two or more fermions cannot be in the same state. Quarks and leptons are fermions—the best known of the leptons is the electron.3 Bosons, on the other hand, do not obey Pauli’s principle, so there may be a plurality of different bosons in the same state. The best-known boson is the photon.4 In this way, photons can be superimposed in the same state to generate a laser. On the other hand, a helium atom only admits two electrons in its innermost orbital, since these electrons have different states by the directions of their spins. It seems, prima facie that the nuclear theory would have enough flexibility to accommodate these empirical findings. Fermions seem to generate ‘exclusions’ in the properties that make them up, that is, they are particles that seem to determine specific properties that are determinable ‘collectively’, so to speak, and not ‘individually’. A macroscopic object must have a certain mass, but

3 Muons, tau particles, and neutrinos are also leptons. Neutrinos have a neutral charge, while the remaining leptons have a negative charge of 1. 4 Are also bosons: the gluon, the boson Z, the boson W, and the Higgs boson. Bosons have a quantum of spin of 1—except for the Higgs boson that has no spin—while fermions have all a quantum of spin of ½.

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the mass that possesses an object does not make it necessary that other different objects must have or not other specific masses. An electron in an orbital, however, by having a spin direction makes it necessary for another electron in the same orbital to have the opposite spin direction. This makes it reasonable to think that electrons in the same atom make up a unitary ontological structure.

12.1

Saturation Structures

§ 83. As has been indicated, the nuclear theory and the theory of ‘pregnant wholes’ have advantages not only to explain the unification of a particular object but also offer advantages to explain the connections between properties of the same object. In effect, if each trope is independent of any other and may or may not be accompanied by other tropes, then why are tropes always forming bundles? (see Denkel 1996, 31–32). Why do charge tropes always accompany mass tropes? Why, in the case of macroscopic objects, the mass is always accompanied by a certain shape and size? If tropes are really independent, then ‘free’ electromagnetic charge or mass tropes would be possible, without forming any bundle. Moreover, it should also be possible for many mass or charge tropes to form a bundle. Just as any trope can exist without forming a bundle with others, any trope can exist, forming a bundle with any other.5 One could argue that these are contingent facts (see Schaffer 2003), but then what we invariably find in our world would be an incredible coincidence. One could also argue that these connections are only ‘nomologically necessary’, but not metaphysically necessary (see Denkel 1997). It has already been explained above, however, that there are no metaphysically possible worlds with ‘other’ natural laws (see §§ 51–52, 76). Natural laws form a single nomic network of necessary existence. The justification we have for these connections is empirical, but this does not impede dealing, in fact, with necessary connections. As indicated above, tropes seem to be connected not only by ontological dependence but also by what might be termed as ontological “repulsions” or “exclusions”. It does not seem metaphysically possible for an electron, for example, to have a trope of mass, a trope of quantum of spin, but not to have a trope of charge. Nor would an electron have two mass tropes of 0.511 MeV/c2 instead of one. A macroscopic physical object cannot possess a form, certain spatial lengths, and not have mass. It

5

These cases have been referred to in the literature as cases of trope stacking and trope pyramiding (see Ehring 2011, 86–91; Schaffer 2001). The first type of case (stacking) is an “accumulation” of tropes and the second (pyramiding) is a “multiplication” of tropes. In cases of accumulation, the plurality is not empirically manifest. In cases of multiplication, it is. For example, let two objects x1 and x2 with surfaces of the same color. Object x1 has a color trope. The object x2, on the other hand, has two tropes of the same color. The duplication of color tropes on the same surface does not make any difference as to the color of the surface. This is a case of trope stacking. Suppose now that x1 has a mass trope of 10 gr, while x2 has two mass tropes of 10 gr each. Then, x2 will have a total mass of 20 gr, which will be empirically evident. This is a case of trope pyramiding.

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Saturation Structures

291

has been explained above how the relations between determinate and determinable properties make necessary certain connections and exclusions between properties (see § 12). A particular object of a specific type seems to require determinate properties under certain determinables and, in addition, seems to require only one property under each determinable, which implies that any other property under the same determinable should be excluded. Not only there is necessitation and exclusion between determinate properties under the same determinable, but also between determinables. The mass and spin of an electron make necessary the co-presence of a negative charge of 1. The spin of a photon excludes any mass or charge trope. The mass and spin of a neutrino exclude any charge trope. This complexion of mutual dependencies and exclusions can be described as a form of “saturation”, as Arda Denkel has aptly stated (1996, 188–194; 1997). An object of a certain type must ‘saturate’ all the determinate properties that correspond to it under specific types of determinable properties. It is complicated to explain these saturation structures if one assumes—as many of the advocates of trope ontologies have done—that these are entirely independent entities. The work of Markku Keinänen should be highlighted in order to specify in more detail how particular objects should be formed by ‘saturating’ determinable properties that correspond to them in the same spatiotemporal region (see Keinänen 2005, 369–383; 2011).6 The theory proposed by Keinänen deals with the nature of particular simple objects, that is, objects that do not have proper parts, nor are they constituted by other particular objects. A simple object must be of a specific type of sortal—let that sortal be K—because it has a nucleus formed by tropes that must be determinate properties of certain determinable properties D1, D2, ..., Dn. For each of these determinables, there must be one and only one determinate trope.7 These tropes are mutually dependent rigidly, according to what has already been explained about the nuclear theory. If there were only one nuclear trope, on the other hand, it would not be rigidly dependent on other tropes. Keinänen is careful to postulate, together with the rigid mutual dependence between nuclear tropes, generic dependence of these tropes on properties under certain determinables (see Keinänen 2011, thesis SNT2). This may seem an unnecessary addition, but it has the function of ensuring that each nucleus has a defined number of properties under certain determinables. It is not only inscribed in the essence of each of the tropes that make up the nucleus of an object that they can only exist forming a bundle with the other nuclear tropes, but also it is inscribed in their essence that they can only exist if they form a nucleus with some property or another under specific determinables. In terms of spatiotemporal localization, nuclear tropes are co-present in exactly the same region, but this co-presence is grounded on the spatiotemporal location of the substance they 6

The nuclear theory supplemented with these requirements of spacetime saturation is called by Keinänen as the “strong nuclear theory” (see Keinänen 2011, 422). 7 An important part of Keinänen’s work is devoted to arguing that the properties that make up a nuclear bundle must be dispositional (see Keinänen 2011, 423–428). It is not necessary to deal with this question here since it has already been explained that the identity of all property is fixed by the causal powers it confers, so all properties are already ‘dispositional’ (see § 65, 76).

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make up. Nuclear tropes are essentially part of the object of which they are a part. The spatiotemporal relationships between this object and other objects ground the spatiotemporal location of the tropes that it possesses and, for this reason, the fact that they are located exactly in the same spatiotemporal region. In the case of the periphery, on the other hand, there are no rigid mutual dependence relations between the tropes. The peripheral tropes depend rigidly on the tropes of the nucleus, while the tropes of the nucleus do not depend rigidly on the peripheral tropes. In some cases, nuclear tropes depend generically on peripheral tropes under certain determinable, but in other cases, they do not. These peripheral tropes are located exactly in a sub-region of the spatiotemporal region occupied exactly by the nucleus. That is, they occupy all or part of such spatiotemporal region. The ‘strengthened’ nuclear theory of Keinänen, then, postulates a specific priority profile for particular objects. Nuclear tropes with their rigid mutual dependence relations ground the type of entity that corresponds to the object in question or, as it has been called, its ‘sortal property’. This nucleus is prior to peripheral tropes that depend rigidly on the nucleus. The object formed by the nucleus is located spatiotemporally, whether this location is grounded by the relation that the nucleus has with a spatiotemporal region of independent existence, or whether it is grounded by the set of distance spatiotemporal relations with other objects. The spatiotemporal location of the nuclear tropes and peripheral tropes is grounded on the spatiotemporal location of the object that possesses them. Since the peripheral tropes are accidental properties of the object, as well as properties that could be transient for the object, they do not need to occupy exactly the same spatiotemporal region that the nucleus occupies. It is enough that they occupy a sub-region of the latter. In short, there is an ontological priority of nuclear tropes to the object they form, but then the spatiotemporal relations of the formed object have ontological priority to the spatiotemporal relations in which the nuclear tropes will enter. In several other cases, there has already been an opportunity to consider priority profiles that, in some way or another, seem to bring about some circularity (see §§ 47, 53–59). These cases have been shown, in general, as incoherent, so it will be convenient to carefully examine the priority profile proposed by Keinänen for simple objects and the tropes that make them up. This will be done below (see § 87), once other more general problems regarding nuclear bundles have been cleared. A more concrete example will now be considered to visualize how the saturation structures operate. Let an electron e. It has already been indicated that every electron has a mass of 0.511 MeV/c2, a charge of 1 and a spin of ½ essentially. The electron e must be conceived, then, as a mereological fusion of a trope of mass TM, a trope of charge TC and a trope of spin TS that are essentially the instantiation of the universals of having a mass of 0,511 MeV/c2, having a charge of  1 and having a spin of ½, respectively.8 Each of these tropes has: (i) a generic dependence to some or other

8

Of course, this is assuming that tropes are essentially the instantiation of universals, as has been proposed here. A philosopher like Keinänen, on the other hand, will argue that a mass trope of 0.511 MeV/c2 is such because it belongs to the relevant similarity class of tropes.

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Saturation Structures

293

trope of mass, charge or spin as the case may be, and (ii) a rigid dependence to TM, TC or TS, as the case may be. The electron e is a mereological fusion of TM, TC, and TS, which is grounded on such tropes and depends ontologically on such tropes. Derivatively, e must necessarily have some mass, some charge, and some quantum of spin, because to exist requires, for example, TM, and TM depends generically on some charge and some quantum of spin. A similar situation arises for TC and TS. The electron e, then, requires to exist to be ‘saturating’ the determinable properties of mass, charge, and spin. The nucleus of e demands such tropes and ‘excludes’ all other tropes. Even more noticeable is this saturation structure for peripheral tropes. An electron must possess some or other position, some or other momentum, some or other kinetic energy and some or other spin direction. The nucleus depends generically on determinate tropes under these determinables. The electron e, then, must also be ‘saturated’ concerning these other accidental properties. According to Keinänen’s ‘strengthened’ theory, the spacetime position or location of the electron e is a relation of e to a spatiotemporal region—or other objects, depending on the conception of spacetime adopted. The nuclear tropes TM, TC, and TS, are derivatively located because they are part of a mereological fusion that has a location. Can something similar be held about the other peripheral properties? The spin direction obviously has to do with the spin. The momentum obviously has to do with the mass. The spin direction requires a quantum of spin but also a spatial orientation. The spin could not have direction, then, if the object that owns it did not have a location. Momentum is a vector magnitude that results from the product of mass by velocity, but neither could there be velocity without spatial orientation.9 It turns out, then, that without the location of the object, there could be no speed and, thus, there could be no momentum and no kinetic energy. If Keinänen’s position is correct, then, not only the position is a property of the object that can only be derivative from the nuclear tropes. The same happens with the direction of the spin, momentum, and kinetic energy. These are properties of the object that are derivatively attributed to nuclear tropes—mass and spin, respectively. The cases that have been presented up to this moment have to do with fundamental physical particles, but what happens with macroscopic objects? What is the nucleus of a cat or a bacterium, for example? What is the nucleus of a planet or a galaxy? A unique type of answer for these questions cannot be expected from nuclear theory. Many of the macroscopic objects—perhaps most of them—should be understood merely as a mereological fusion of ‘simple’ objects that will possess their respective nuclei and peripheries. Perhaps some of these macroscopic objects cannot even be identified with a unique mereological fusion, but with a succession of

9

As is known, some have argued that velocity is a property that can be attributed to an object only if this object has traveled some given spatial distance at a given time—since it would be the quotient of distance and time. Others, on the other hand, have argued that speed is a property that can be attributed to an object regardless of any distance it has traveled. Speed in this conception is a ‘tendency’ to move locally that can be possessed by an object, regardless of whether or not it has traveled some distance. In any of these cases, ‘speed’ requires a spatial orientation because it is a vectorial magnitude (see Bigelow and Pargetter 1990, 62–82).

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different fusions at different times. A table or chair, as we ordinarily understand these artifacts, are objects that include different parts at different times. The constant interactions of a table with its environment cause a continuous loss of parts—a molecule of the surface, for example, when scraping it with the hand, or an electron. These losses of parts may be invisible to our perceptual capacities, but sufficient so that what we identify as ‘the table’ is a different mereological fusion of simple physical objects. A table is also conceived as a single object that persists being identical at different times. None of the mereological fusions that roughly coincide with what we identify as ‘the table’ at one time corresponds to what we assume is the ‘same’ table at different times. In different times it will have to be identified with different mereological fusions. These different fusions will coincide in most of their parts and in most of their properties, which makes it reasonable to postulate that some are ‘continuants’ of the others. The sequence of ‘continuants’, that is, the sequence of mereological fusions that roughly coincide in most of their parts and their macroscopic properties can be called, in some sense, as the “same” table (see Chisholm 1976, 97–104). The properties of a table or a chair will be grounded on the physical properties of its parts and their mutual relations. The mass of a table will be grounded on the masses of each of its parts since it will have a magnitude equivalent to the sum of all those masses. The shape of a table will be grounded on the spatial extension of its parts and the relative positional relations between them, and so on. These properties have been called “Gestalt properties”. Something similar should be held, perhaps, of a planet or a galaxy. Strictly, then, neither tables, nor chairs, nor planets, nor galaxies are nuclear bundles of tropes, but temporal sequences of mereological fusions of nuclear trope bundles. Faced with this same type of case, many philosophers have preferred to maintain that objects are mereological fusions of temporal parts and that all persistence in time must be constituted by the fact that different temporal parts exist exactly at different times (see Lewis 1986, 202–204; Heller 1990; Sider 2001; Hawley 2001, among many others). These philosophers start from the assumption that no object can exist more than a punctual instant of time. Every object existing in another moment of time must be numerically different. The idea of objects persisting in time being identical at different times is, for these philosophers, merely an illusion. What can be offered is a more or less close ‘substitute’ that is a mereological sum of objects whose existence is instantaneous.10 I do not share these assumptions about persistence over time. I do not see any difficulty in principle in accepting, at least in some cases, the existence of entities that persist identical in different times, or that exist ‘all entire’—as ‘simple extended entities’—in each time in which they exist. This is not the place to explain in detail the reasons for this theoretical option.11 In what follows

10

Some have proposed instead of a mereological fusion of temporal parts a sequence of temporal ‘counterparts’ that must also be of temporal instantaneous existence (see Sider 2001; Hawley 2001). 11 In any case, the perdurantist conception conflicts with the ordinary experience that each one of us has from his first-person perspective about himself, and is also in conflict with what seems to show us our best natural science. ‘Matter’ seems to be constituted not by sequences of temporally

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Saturation Structures

295

in this paper it is going to be assumed that there are ‘enduring’ objects, identical in different times, but what is proposed here could also be understood within the framework of a ‘perdurantist’ ontology. This is not a ‘neutral’ presentation concerning the persistence of particular objects over time, because one of the advantages of the nuclear theories that will be discussed has to do precisely with the ‘enduring’ of objects. This does not prevent, however, that this factor can be discounted by someone with more ‘perdurantist’ inclinations. Faced with cases such as those indicated about tables, chairs, planets, and galaxies other philosophers have preferred to postulate a nihilistic position about such putative entities (see van Inwagen 1990; Merricks 2001; Rosen and Dorr 2002). For these philosophers there are no tables, chairs, planets, or galaxies, but everything we say about tables, chairs, planets, and galaxies can be reinterpreted as statements about pluralities of simple objects configured ‘chair-wise’ or ‘planet-wise’, etc.12 I do not think it is necessary, however, to reject so radically the entities that result from standard extensional mereology. Mereological fusions are not fundamental entities in any case. It should not be such a severe problem to admit them as ontologically derivative entities. What has been raised above is that there are tables, chairs, planets, and galaxies, but they are not the ‘enduring’ objects that we usually consider them to be, but sequences of ‘continuants’, each of which is a mereological fusion of simple objects. These are objects grounded on simple objects and their mutual relationships.13 It also happens that the possibility of objects authentically ‘constituted’ by simple objects must be left open. Perhaps in the case of these complex objects, it is necessary to postulate a specific nuclear bundle different from the nuclear bundles of the simple objects that ‘constitute’ it. One of the advantages of the nuclear theory is to leave space for the treatment of different types of objects, according to what our best evidence shows us about its nature. If for a table or a planet it does not seem reasonable to postulate a specific nuclear bundle, it does not seem reasonable to conceive a bacterium or a person merely as a sequence of mereological fusions of simple objects that are ‘continuants’ with each other. Living beings seem to be authentic objects with nuclear bundles. People seem to be authentic objects with nuclear bundles. It is not yet entirely clear whether it is reasonable to admit atoms as authentic objects ‘constituted’ of elementary physical particles. Nor is it entirely clear whether molecules should be admitted as authentic objects ‘made up’ of atoms. These are open questions that must be answered, in no small extent, with careful

punctual distributions of properties, but by elementary particles temporally enduring. The perdurantist conception conflicts with experience and common sense massively. 12 These formulations may seem strange. The idea of nihilists is that qualifications like ‘chair-wise’ or ‘table-wise’ are adverbial qualifications indicating the arrangement of a plurality of mereologically simple objects (see van Inwagen 1990, 98–114). 13 If one restricts quantifiers to ontologically fundamental entities, then there are no tables, no chairs, no planets, no galaxies. Under this restriction, there are only universals and tropes. From this perspective, there are also no simple objects for a defender of the traditional nuclear theory, since simple objects are grounded on their nuclear tropes. There are only universals and nuclear tropes. If the quantifiers are unrestricted, on the other hand, there are tables, chairs, planets, and galaxies.

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attention to the empirical evidence we have about bacteria, atoms, molecules, and humans.14 There is, of course, much theoretical work to be done to develop appropriate ‘regional’ ontologies in these cases. For many, the notion of ‘material constitution’ is still too obscure (see Sider 2001, 141–161). Considering the question in an abstract way, space must be left for: (i) cases in which an object must be grounded on a specific configuration of simple objects with respect to which, however, it is not ontologically dependent, and (ii) for cases in which an object is ontologically dependent on the configuration of simple objects on which, however, it is not grounded. As seen above (§ 4), (i) is a case of ‘realization’, while (ii) is a case of ‘emergence’. Many times, when one speaks of “material constitution”, cases of “realization” and of “emergence” are treated indifferently as if they were the same, simply because in both cases there is a complexion of objects contributing to the existence of another and there is certain ontological priority of a level that is more ‘basic’ than another. Contemporary biological science has tended to assume that living beings are ‘realized’ in organic molecules configured in a certain way, but they do not depend ontologically on the organic molecules on which they are in fact ‘made’. This is how the relation of a bacterium with the molecules that ‘constitute’ it and the relation of a whale with the molecules that constitute it have been understood. A bacterium in a given time is constituted by a plurality of molecules configured in a certain way but could be constituted by other molecules.15 If this same bacterium had been exposed to other nutrients in a different environment, it would have made its own other molecules different numerically from the molecules from it has been nourished, even if they were of the same “type”. The form of ‘constitution’ that has been contemplated here is a form of ‘realization’.16

12.2

Persistence in Time

§ 84. The nuclear theory offers significant theoretical advantages also concerning the persistence of objects over time (see Keinänen and Hakkarainen 2010). Our ordinary conception is that an object is an entity that ‘endures’ by existing ‘entirely’ at different times or by being identical at different times. These intuitions are those that are included in Requirement III according to which the same object must have transient properties or, at least, it must be possible that the object possesses transient properties, that is, properties that it possesses at some times in which it exists and 14

It has been, moreover, the intuition of several nihilistic philosophers about material composition that, besides simple objects, there are living beings (see van Inwagen 1990, 142–168) or people (see Merricks 2001, 85–160). 15 With the exception, perhaps, of the molecules that constitute it the first moment of its existence, if one admits the necessity of origin. See § 88. 16 Cases of ‘emergence’ are much easier to identify. Every state of affairs is ‘emergent’ with respect to its constituents, as has already been highlighted on several occasions.

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Persistence in Time

297

does not possess at other times in which it exists. According to nuclear theory, an object exists at a time if and only if the tropes that make up its nucleus exist. Peripheral tropes may temporally coincide with the entire duration of the nucleus, but may not. In this case, peripheral tropes will be transient properties of the object. According to this, precise identity conditions for particular objects can be formulated, at least in the case of simple objects: [Temporal Identity] x-@-t1 ¼ y-@-t2 if and only if for every trope T: T is part of the nucleus of x-@-t1 if and only if T is part of the nucleus of [email protected] That is, if the nuclear tropes of an object persist between two times, the object persists between those times. If the object persists between two times, then the nuclear tropes of the object persist. The Husserlian theory of ‘pregnant wholes’ seems to replicate identity conditions similar to those specified in the principle of Temporal Identity, but without the qualification to nuclear tropes. The principle would be that an object persists between times t1 and t2 if and only if all the tropes of the object exist in those two times, without exception. Something similar should hold for bundles of tropes independent of each other, whatever the way these bundles are conceived. As in any of these theories, the tropes that constitute a bundle are essential to the bundle, it seems that these same tropes should also be temporally invariant for the persistence of the object. These conceptions would not satisfy Requirement III, which would constitute an important theoretical advantage of the nuclear theory with respect to all of them. There are ways, however, in which both the Husserlian theory of ‘pregnant wholes’ as well as the remaining theories of bundles of independent tropes could, in some way, be accommodated to Requirement III. First, of course, they could adopt a perdurantist position about the persistence of objects in time according to which an object persists between times t1 and t2 if and only if temporal parts are existing in those times. The trope bundles discussed above would be temporally instantaneous bundles existing in a single punctual instant of time. What one understands as a temporally distended object is a fusion of existing parts for each time of this distention and nothing else. This would require to adopt a contested position about persistence over time—a position that I believe to be false, as explained above. On the other hand, the nuclear theory would theoretically be neutral on this point, since it can work both from an endurantist perspective and from a perdurantist perspective, which is an additional theoretical advantage. Another way in which both the theory of ‘pregnant wholes’ and the theories of bundles of independent tropes could—eventually—accommodate Requirement III is postulating some form of ‘diachronic’ connection between the tropes of a bundle. In general, it has tended to be assumed that the co-presence of the tropes that make up a bundle, or the mutual ontological dependence relations of a pregnant whole, should

With more precision: [□8x□8y□ ((x-@-t1 ¼ y-@-t2) $ 8T ((T < n(x-@-t1)) $ (T < n(y-@-t2)))]. The variables ‘x’ and ‘y’ have as range particular objects. The variable ‘T’ has tropes as range. ‘n(x)’ is the nucleus of x. Recall that ‘