Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Vagueness, ontology, and the social world
(USC Thesis Other)
Vagueness, ontology, and the social world
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
VAGUENESS, ONTOLOGY, AND THE SOCIAL WORLD
by
Edgardo Jaime David Castillo Gamboa
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(PHILOSOPHY)
December 2024
Copyright 2024 Edgardo Jaime David Castillo Gamboa
ii
A la memoria de Juana Collave
y Sotero Gamboa
iii
Acknowledgments
On February 22, 2017, I received the call informing me that I had been accepted into USC's
philosophy Ph.D. program. That moment was one of great excitement and marked the beginning
of a chapter of my life that concludes with this dissertation. Whatever merits this work or its author
may have, they would not exist without the contributions of many people: mentors, colleagues,
students, friends, and family. Allow me to acknowledge such contributions below and to apologize
to those who, by oversight or lapse of memory, I fail to mention.
I want to start by thanking my dissertation advisor, Gabriel Uzquiano. Gabriel has been an
invaluable presence since my first day at USC and is, undoubtedly, the person with whom I have
discussed philosophy the most over the past years. Virtually every idea I have had during this time
has been the subject of a conversation with him, including, of course, each of those presented in
this dissertation.
Gabriel uniquely combines uncompromising philosophical rigor with unwavering kindness
and patience. The lessons I have learned from him are countless. From him, I learned to truly
appreciate the relevance of formal tools for philosophical argumentation and to seek to understand
philosophical ideas in their full generality, and not just in their application to specific problems. I
also learned to be patient with my own ideas without lowering my expectations of them. Thanks
to him, I learned how to combine the careful understanding of a philosophical point with the
development of my own ideas. When my turn comes to serve as an advisor, I hope to do for my
students as much as Gabriel has done for me. That is quite a tall order.
I am infinitely grateful to have had Jeff Russell, John Hawthorne, Alexis Wellwood, and Barry
Schein on my dissertation committee, each of whom has made a profound contribution to my
development as a philosopher.
iv
The first time I engaged philosophically with Jeff was during a lecture he gave on the
metaphysics of modality. I remember being so impressed by the clarity and depth of his
presentation that I was immediately determined to work with him. From that moment on, we
worked together on my area exam and on this dissertation. In addition to providing detailed and
insightful comments on multiple drafts of this and other projects, his draft on vague existence
inspired the main idea of Chapter 2, and the technical results I present there are the outcome of
many iterations polished with his help.
John was my mentor during my second year, when I attempted (emphasis on “attempted”) to
write a paper on vagueness and metaethics. The impact John had on me during those early meetings
cannot be overstated. The meticulousness with which he read every line, the ease with which he
articulated complex ideas, and the way he raised objections to what I believed to be robust points
were remarkable and made me better appreciate how difficult and fascinating philosophy can be.
His continuous guidance throughout the years has not only been a source of intellectual growth
but has also set an extraordinarily high standard for my writing. I hope at least some portions of
this dissertation meet that standard.
Alexis’s presence has been pivotal during my years at USC. In addition to being her student in
multiple seminars, I had the privilege of collaborating with her on a variety of projects, including
co-authoring a paper and organizing a summer school. Throughout our time working together, she
has consistently been a model of hard work and dedication, as well as kindness and generosity. I
am especially grateful for her constant challenge to explain the project of metaphysics—a recurring
topic in our conversations that we never fully managed to finish but which has influenced me
deeply. Her questions during my dissertation defense gave me hope that I had at least made some
progress toward meeting her challenge.
v
I am grateful to Barry Schein for his willingness to serve as the external member of my
committee and for the valuable lessons I received from him in his semantics courses. Those who
have been his students are familiar with his characteristically incisive notes and the passion with
which he defends his convictions.
During my years at USC, I greatly benefited from the intellectual and professional guidance of
an amazing team of philosophers. I am particularly grateful to Mark Schroeder for his constant
advice throughout the various stages of the program, which has been invaluable in shaping my
academic journey. To Shieva Kleinschmidt, I am deeply thankful for her insightful lessons on
pedagogy, which have significantly influenced my approach to teaching. I am also grateful to her
for the course that sparked my conversion to metaphysics, setting the course for much of my
subsequent research. I also owe much to Scott Soames for his monumental efforts in building the
department into the vibrant intellectual community it is today, as well as for the great impact his
visits to Peru had on the philosophers of my generation. I am also grateful for memorable seminars
and conversations over the years with Andrew Bacon, Stephen Finlay, Jeremy Goodman, Robin
Jeshion, Jake Nebel, Jon Quong, Deniz Rudin, Jim van Cleve, and Ralph Wedgwood.
I am also deeply grateful to my fellow graduate students at USC, with whom I have shared
these long years both inside and outside the classroom. The camaraderie and intellectual exchange
we have experienced together have been central to my growth. To the members of my cohort—
David Clark, Noah Gordon, Jasmine Gunkel, Madiha Hamdi, Mahmoud Jalloh, Anthony Nguyen,
and Andrew Stewart—and to Simon Blessenohl, Irene Bosco, Antonio Cleani, Megha Devraj,
Sean Donahue, Maegan Fairchild, Paul Garofalo, Jen Foster, Zach Goodsell, Brian Haas, Frank
Hong, Philip Li, Eleonore Neufeld, Laura Nicoară, Dan Pallies, Quyen Pham, Wen Kin San, Yasha
Sapir, Vishnu Sridharan, Douglas Wadle, Levy Wang, and Shane Ward, thank you. A special thanks
vi
to the administrative staff of the department, without whom we would undoubtedly be lost: Brian
Eckert, Angie Guerrero, Natalie Schaad, John Nikolai, Michele Root, and Amanda Velasco.
Over these years, I have also had the opportunity to learn from philosophers at various
institutions through conferences, summer schools, and other events. Among a long list, I highlight
José Antonio Navarro, Oscar Piedrahita, Dashiell Shulman, John Schindler, and Evan Welchance.
I am grateful to the metaphysics community at Rutgers University, especially Karen Bennett,
Jordan Bridges, Rose Fonth, Carolina Sartorio, Alex Skiles, Jonathan Schaffer, Trevor Woodward,
and Dean Zimmerman, for the incredible experience of the 2023 Metaphysical Mayhem and the
2023-2024 metaphysics reading group. Special thanks go to Ted Sider. As this dissertation attests,
Ted’s work in metaphysics has been a tremendous influence on my own thinking. Getting to know
him and working with him during my time at Rutgers has been a true privilege. I was incredibly
pleased to discover that his philosophical genius was matched by his excellence as a teacher and
as a human being.
Intellectual activity, particularly the kind that is not closely tied to the most urgent concerns in
life, can lead to profound isolation and to questioning the purpose of one’s practice. I am fortunate
to have been spared from these predicaments thanks to my friends in the Peruvian philosophical
community, especially the members of Theorema and the faculty team of philosophy MA program
at the National University of San Marcos. Their dedication to philosophy and education has shown
me that intellectual work can be deeply connected to community, purpose, and a shared
commitment to making our portion of the world a better place. Even if the questions of metaphysics
end up being ultimately meaningless, I have no doubt that my work with them, and the
relationships I have cultivated, are not.
vii
I want to start by thanking Eduardo Villanueva. Eduardo’s return to Peru after his Ph.D. at USC
must be one of the most important events in Peruvian philosophy in recent decades. He created a
space, the Círculo de Investigación en Filosofía Analítica (CIFA), that not only nurtured the
intellectual interests of an entire generation of young philosophers but also greatly expanded their
professional prospects. He was my primary mentor during my final years in Peru and remained a
tireless guide since then and an invaluable collaborator on countless projects. I owe him so much.
Among my colleagues and friends, I want to especially thank Edvard Avilés, Cristian Barturén,
Nurit Matuk-Blaustein, and Rodrigo Garro-Rivero.
Edvard has been there since day one, when we were both discovering analytic philosophy in
the meetings of CIFA. Many years later, it was long conversations with him that culminated in the
creation of Theorema. Edvard is someone I can always count on, who is always willing to talk
about philosophy and to embark on new ventures. Cristian has been a crucial presence in our recent
work at San Marcos. I admire his passion for advancing Peruvian philosophy and I thank him for
the trust he placed in me and the patience he showed me throughout this project. Nurit is the person
I have collaborated with the most in recent years. Together, we have written an article, co-taught
courses and organized events. I cannot overstate how much I value her presence in Theorema and
San Marcos. She is truly indispensable. I especially value her tireless dedication to her work and
thank her for teaching me, despite my stubbornness, the value of constant effort and organization.
In this regard, she and Rodrigo share credit. The long hours of joint work in the Grad Student
Lounge are among my most cherished memories from my time at USC. Although the three of us
only overlapped for one semester, what they taught me will last a lifetime. With the two of them
there, I know that Peruvian representation at USC is in good hands.
viii
I also extend my gratitude to my other colleagues and friends in Theorema—Andrés Abugattas,
Diego Arana, Víctor Carranza, Gabi Dumet, Alonso R. Molina, Gian Franco Sandoval, and Pyro
Suárez—as well as many Peruvian philosophers I have met at various stages of my academic life
and from whom I have learned so much: Ángela Almonte, Erik Alvarado, Luis Felipe Bartolo,
Pamela Lastres, Karl Palomino, Rubén Quiroz, Gonzalo Ricaldi, Marlon Rivas, Jorge Secada,
Mario Sheing, Julio Silva, Sandro D’Onofrio, and Abigail Terán. I also include in this list an
exceptional group of friends who have accompanied me since my first year as an undergraduate.
Infinite thanks to Isaías León, Raymond Ocampo, Rodrigo Rojas, and Jaime Vera. Though I don’t
see them as often as I would like, I carry their friendship with me wherever I go.
I want to conclude by thanking Clayde Vasallo, who was my partner for many years; Tirso
Vásquez, an incomparable friend and a philosopher at heart; and my family, especially my mother,
Miriam Gamboa, my aunt, Yolanda Gamboa, and my uncle, Jaime Gamboa. Words will never
suffice to express my gratitude for all they have done for me.
ix
Table of Contents
Dedication....................................................................................................................................... ii
Acknowledgments ......................................................................................................................... iii
Abstract.......................................................................................................................................... xi
Introduction..................................................................................................................................... 1
Chapter 1: The Argument from Vagueness..................................................................................... 4
1. Preliminaries.......................................................................................................................... 6
2. The argument ........................................................................................................................11
3. Responses to the argument................................................................................................... 17
Chapter 2: The Argument from Vagueness Revisited................................................................... 22
1. Understanding vague existence............................................................................................ 22
2. Against negative vague existence ........................................................................................ 29
3. Against negative vague composition ................................................................................... 33
4. Against restricted composition ............................................................................................ 38
5. Indeterminism and ontology ................................................................................................ 44
Chapter 3: Vagueness and Social Ontology.................................................................................. 46
1. Problematic vagueness in the social world .......................................................................... 47
2. Rejecting problematic vagueness......................................................................................... 51
3. Moderate responses.............................................................................................................. 54
4. Radical responses................................................................................................................. 59
Chapter 4: Anchoring Plenitude.................................................................................................... 65
1. Introducing anchoring plenitude .......................................................................................... 66
2. Anchoring plenitude and problematic vagueness................................................................ 70
3. An objection......................................................................................................................... 72
4. Anchoring plenitude and maximalist ontology.................................................................... 77
References..................................................................................................................................... 79
x
List of Tables
Table 1: Distribution of cases ....................................................................................................... 53
Table 2: Distribution of cases (social reductionism)..................................................................... 57
Table 3: Distribution of cases (necessitism) ................................................................................. 63
Table 4: Distribution of cases (account plenitude) ....................................................................... 70
Table 5: Distribution of negative cases (account plenitude)......................................................... 74
xi
Abstract
Social objects—such as bank accounts, linguistic expressions, governments, football teams, and
laws—exist as a result of our social practices. For instance, a bank account exists as a result of a
customer signing a contract with a bank. Most of the recent literature on the metaphysics of social
objects assumes that a moderate conception of which social objects exist is more or less accurate:
there are ordinary social objects and just those objects. Consequently, radical approaches that either
deny the existence of ordinary social objects or postulate “strange” social objects are seldom
explored. This dissertation challenges this assumption by motivating and defending a radical
approach to the metaphysics of social objects. Chapter 1 offers a detailed presentation of a highly
influential argument in the metaphysics of material objects—the argument from vagueness.
Chapter 2 introduces a new conception of vague existence, a concept central to the argument, and
uses it to develop new versions of the argument while establishing important results concerning
vagueness and ontology. Chapter 3 builds on these results to criticize several ontological positions
about the social world, including various versions of the moderate account. Chapter 4 presents
anchoring plenitude, a novel radical approach to social objects according to which ordinary social
objects coexist with a multitude of other objects whose existence is intertwined with our social
practices in unexpected ways.
1
Introduction
Ontologists are in the business of providing an inventory of the world. For any kind of entity, they
ask whether we should accept that there are entities of that kind or not. Given this characterization
of their task, ontologists can be classified into three classes: moderates, minimalists and
maximalists. A moderate ontologist typically accepts that there are entities of the kinds recognized
by common sense and only of those kinds. Minimalists and maximalists depart substantially from
this conception, respectively by rejecting entities of kinds moderates accept or by accepting entities
of kinds moderates reject.1
A paradigmatic example of this dialectic can be found in debates about the metaphysics of the
material world. While material moderates believe that the material world contains such things as
chairs, tables, people, dogs, planets, etc., and just those things, material minimalists deny that there
are any such things, and material maximalists claim that there are things of those kinds and also
of many others. For instance, they accept that there is a thing that has my laptop, your chair, and
the Moon as parts, or a thing that coincides with a car but goes out of existence once the car leaves
its garage.2
Given the prominence of this dialectic in the metaphysics literature, it is surprising that it is
virtually absent in a domain of inquiry that has gained considerable attention in recent decades,
namely the metaphysics of the social world. Most of the work on social metaphysics seems to
proceed under the assumption that a moderate conception of the social world is correct. Thus, it is
taken for granted that the kinds of entities that populate the social world are roughly only those
1
I am roughly assuming a standard, Quinean picture of ontology. See Quine 1948 and van Inwagen 1998.
For an alternative conception, see Schaffer 2009.
2 For moderate approaches to the material world, see Markosian 1998, Korman 2015 and Carmichael 2015.
Minimalists include Sider 2013 and Dorr 2005. Sider 2001, van Cleve 2008, Fairchild 2019 and Dorr et al
2021 defend different versions of maximalism.
2
accepted by common sense. While there are clubs, songs, borders, bank accounts, constitutions,
governments, contracts, laws, etc., there are no entities belonging to “strange” social kinds, i.e.,
kinds analogous to the strange kinds accepted by the maximalist about the material world.3
This dissertation is an attempt to fill this gap. My main goal is to motivate a departure from a
moderate approach to the social world and to articulate a maximalist alternative. I do so via a
detailed study of one of the most influential challenges to material moderatism: the argument from
vagueness.4 This is the focus of the first two chapters. After identifying the main lesson from the
argument from vagueness, I argue that it can be used to challenge the moderate approach to the
social world and to motivate a maximalist alternative, a view I label anchoring plenitude. The last
two chapters take up these tasks. In what follows, I offer an overview of the content of each chapter.
Chapter 1: The argument from vagueness. I start by introducing some logical and philosophical
assumptions that I make throughout the dissertation. Then, I offer a detailed presentation of the
argument from vagueness, which targets a crucial component of material moderatism, namely the
thesis that composition is restricted. The argument goes, roughly, as follows:
P1 If composition is restricted, then there are cases of vague composition.
P2 If there are cases of vague composition, then existence is vague.
P3 Existence is not vague.
C Therefore, composition is not restricted.
The chapter concludes with a summary of the main response strategies in the literature. I pay
special attention to the strategy I call indeterminism, which proceeds by rejecting P3.
3 Some notable exceptions to this rule are the discussions of eliminativism in Mason & Ritchie 2020 and
Passinsky 2020, 2021, forthcoming b, and the plenitudinous approach to social groups developed in
Uzquiano 2018.
4 Lewis 1986: 212-213; Sider 1997: 214-222, 2001: 120-132.
3
Chapter 2: The argument from vagueness revisited. In this chapter, I argue that indeterminism
is ineffective as a response to the argument from vagueness. I start by presenting a new
understanding of vague existence and by distinguishing two varieties: positive and negative. Then,
I provide an argument against negative vague existence that relies only on assumptions
indeterminists should be happy to accept. After that, I use this result to formulate two new versions
of the argument from vagueness that are immune to the indeterminist attack. The chapter concludes
by considering the implications of the failure of indeterminism for ontological debates.
Chapter 3: Vagueness and social ontology. From my discussion of the argument from
vagueness, we learn that, given any predicate ‘𝑃’, the following three claims cannot all be true: (i)
it is vague whether there is something that is 𝑃, (ii) nothing is such that is it vague whether it is 𝑃,
and (iii) nothing is 𝑃. When this happens, I say that there is problematic vagueness with respect to
‘𝑃’. In this chapter, I argue that a moderate conception of the social world entails many instances
of problematic vagueness. Then, I argue that the moderate social ontologist lacks the resources to
avoid this problem. I conclude by discussing some non-moderate alternatives and arguing that they
are also unsatisfactory.
Chapter 4: Anchoring plenitude. This chapter introduces anchoring plenitude, a novel,
maximalist approach to the social world, and my preferred strategy for avoiding problematic
vagueness. According to this view, the conventions that govern the existence of ordinary social
objects also guarantee the existence of many other objects. After introducing anchoring plenitude
in some detail, I respond to an objection arising from higher-order vagueness. I conclude by
reflecting on the connections between anchoring plenitude and other maximalist views.
4
Chapter 1: The Argument from Vagueness
If we look around ourselves, we encounter objects that have many parts. Take, for instance, your
kitchen table. The following pieces of wood are parts of it: the four legs and the top. Moreover,
they seem to make up for all of it. In such a scenario, it is said that the table is a fusion of those
pieces or that the pieces compose the table.5 So, there is something that is a fusion of the pieces.
To abbreviate: the pieces have a fusion. Other pluralities of objects lack this feature. For instance,
consider my laptop, your chair and the Moon. It is natural to think that nothing is a fusion of these
objects. In other words, they don’t have a fusion.
These judgments lead to the following thesis:
Restricted Composition Some pluralities have a fusion, and some pluralities don’t have a
fusion.6
The thesis that composition is restricted is a crucial component of a moderate conception of the
material world. However, many metaphysicians reject it due to the difficult challenges it faces and
adopt either maximalist or minimalist alternatives.
Among the challenges against restricted composition, one that has received considerable
attention is the argument from vagueness (Lewis 1986: 212-213; Sider 1997: 214-222, 2001: 120-
132). It goes, roughly, as follows:7,8
5 More precisely, an object 𝑦 is a fusion of some objects 𝑥𝑥 (alternatively, 𝑥𝑥 compose 𝑦) just in case (i)
each of 𝑥𝑥 is a part of 𝑦 and (ii) anything that overlaps 𝑦 also overlaps one of 𝑥𝑥.
6 Strictly speaking, we should limit our attention to pluralities of more than one object. Otherwise, it would
be trivial to say that some pluralities have a fusion and plainly false to say that no plurality has a fusion. I
leave this clarification implicit in what follows.
7 For an overview of discussions about the argument from vagueness, see Korman 2010 and Korman &
Carmichael 2016 (sections 3 and 4).
8 Two caveats about my presentation of the argument. First, the argument is sometimes presented as an
argument for compositional universalism. To get such an argument from the one I am presenting, we just
need to add the premise that some pluralities have fusions, i.e., compositional nihilism is false. Second, it
5
P1 If composition is restricted, then there are cases of vague composition.
P2 If there are cases of vague composition, then existence is vague.
P3 Existence is not vague.
C Composition is not restricted.
If successful, this argument refutes Restricted Composition and forces us to accept one of the
following views:
Compositional Nihilism No plurality has a fusion.
Compositional Universalism Every plurality has a fusion.
Both composition nihilism and compositional universalism lead to radical pictures of the material
world. Given the assumption that ordinary material objects (chairs, tables, people, etc.) are fusions
of smaller objects, compositional nihilism entails that there are no such things. On the other hand,
compositional universalism allows for the existence of ordinary objects but at the cost of also
accepting that there are such things as a fusion of your table, my laptop and the Moon.
To avoid these radical results, friends of moderatism about the material world have proposed
different strategies for rejecting the argument from vagueness. In this chapter, I provide a detailed
reconstruction of Sider’s version of the argument (section 2) and outline the main lines of response
(section 3). Before doing go, I introduce some philosophical and logical assumptions that I will
make throughout this dissertation.
is not clear that Lewis’ version of the argument from vagueness follows the template above. On the one
hand, he seems to suggest that composition is not vague because mereological vocabulary is not vague
(what he calls “the words … for the partial identity of overlap”), which suggests that he, unlike Sider,
doesn’t derive the non-vagueness of composition from the non-vagueness of existence. On the other hand,
he expresses some discomfort with (some version of) the idea of vague existence: “There is such a thing as
the sum, or there isn’t. It cannot be said that … there sort of is and sort of isn’t. What is this thing such that
it sort of is so, and sort of isn’t, that there is any such thing?”.
6
1. Preliminaries
Most of the topics I discuss in this dissertation are related to vagueness. Here are three crucial
assumptions I will make about this phenomenon:
(A) Some linguistic expressions from different syntactic categories are vague. This includes,
among others, predicates (‘… is tall’, ‘… is a cat’, etc.), quantifiers (‘many things …’,
‘most things …’, etc.), and sentences (‘John is tall’, ‘Many dogs are happy’, etc.). Vague
expressions typically have borderline cases. For instance, some people are borderline
cases of ‘… is tall’. In those cases, I say that it is vague whether they are tall, which is
equivalent to saying that (i) it is not determinate that they are tall and (ii) it is not
determinate that they are not tall.
(B) When a linguistic expression is vague, its meaning can be made precise in multiple ways.
In other words, many worldly items can be associated with it in a way that is compatible
with at least a subset of our linguistic practices. Each of those items is called a
precisification. The precisifications of a vague predicate are properties. For instance, it
is natural to take precisifications of ‘… is tall’ to be properties of the form being at least
𝑛 cm tall.
9 The precisifications of a quantifier are second-order properties (i.e., properties
of properties of objects) and those of a sentence are propositions.
(C) Vague predicates have cut-off points. Consider, for instance, a series of people 𝑎150-𝑎200
such that, for any 𝑖, 𝑎𝑖
is 𝑖 cm tall. Call it the height series. Clearly, 𝑎151 is not tall and
𝑎200 is tall. The cut-off assumption tells us that there is cut-off point for ‘… is tall’ in the
9 The adjective ‘tall’ is context sensitive. Thus, its precisifications vary across contexts. I leave this
clarification implicit throughout the dissertation.
7
height series. That is, someone who is not tall and who is immediately followed by
someone who is tall. In addition, I assume that ‘… is tall’ has what we may call a
privileged precisification. That is, there is a property 𝑇 such that (i) 𝑇 is a precisification
of ‘… is tall’ and (ii) something is tall if and only if it instantiates 𝑇.
I take it that (A) is uncontroversial. Just to be thorough, I shall explicitly formulate the connection
it assumes between ‘it is vague whether …’ and ‘it is determinate that …’:
Vag-Det it is vague whether 𝑝 iff [it is not determinate that 𝑝 and it is not determinate that
not 𝑝]
It is also helpful to notice that vagueness operators can appear under the scope of quantifiers. That
is, we can say things like “there is someone such that it is vague whether they are tall”, which I
shall abbreviate as “there is someone who is borderline tall”. In general, 𝑥 is a borderline 𝑃 just in
case it is vague whether 𝑥 is 𝑃.
Let’s now consider (B). The ideology of precisifications is typically associated with
supervaluationist accounts of vagueness (Fine 1975, McGee & McLaughlin 1995, Keefe 2000).
As a result, it is sometimes thought that accepting something like (B) requires a commitment to
such theories. However, the characterization of a precisification I have offered is neutral with
respect to which theory of vagueness one accepts. First, it is neutral with respect to what kind of
connection exists between a precisification and the linguistic item it precisifies. Thus, an
epistemicist like Williamson (1994) can say that some properties are the precisifications of a
linguistic expression in virtue of them being epistemically indiscernible. Second, it is neutral with
respect to the question whether vagueness is a property only of linguistic items (predicates,
sentences, etc.) or also of worldly items (properties, propositions, etc.). Defenders of worldly
8
vagueness such as Barnes & Williams (2011) and Bacon (2018) can take a precisification of a
predicate ‘𝑃’ to be a precise property that precisifies (in a sense that would have to be specified)
the vague property expressed by 𝑃.
Besides accepting that vague expressions have multiple precisifications, I accept the following
principles connecting vagueness operators and precisifications:
Vag-Prec It is vague whether 𝑝 iff [some precisification of ‘𝑝’ is true and some
precisification of ‘𝑝’ is false].
Det-Prec It is determinate that 𝑝 iff every precisification of ‘𝑝’ is true.
My third claim, (C), also tends to be regarded as controversial. I won’t deny that. However, it is
important to clarify the extent to which it is controversial. Crucially, we should notice that the
claim that the there is a cut-off point for ‘… is tall’ in the height series follows, given classical
propositional logic, from the assumptions that 𝑎150 is not tall and 𝑎200 is.10 Thus, rejecting the
existence of cut-off points requires rejecting classical propositional logic. Contrary to what one
might have thought, a substantial portion of philosophers of vagueness accept classical
propositional logic. This includes supervaluationists, epistemicists and proponents of ontic
vagueness. Throughout this dissertation, I also assume classical propositional logic.
A related result is that vague expressions have privileged precisifications. For concreteness,
we can assume that the precisifications of ‘… is tall’ are properties of the form being at least 𝑛 cm
tall (for short, 𝑇𝑛). Among those, there is exactly one that distinguishes correctly between those
10 Assume that 𝑎151 is not tall. By classical propositional logic, either 𝑎152 is tall or 𝑎152 is not tall. If 𝑎152
is tall, then 𝑎151 is a cut-off point. What if 𝑎152 is not tall? Again, by classical propositional logic, either
𝑎153 is tall or 𝑎153 is not tall. If 𝑎153 is tall, then 𝑎152 is a cut-off point. What if 𝑎152 is not tall? We repeat
this reasoning for every person in the series. Since 𝑎200 is tall, we are guaranteed to find a cut-off point
somewhere in the series. For a detailed discussion of this point and other proves of the same result in weaker
logics, see Bacon 2018: ch 1.
9
who are tall and those who are not tall. Simplifying things a little bit, we can say that, if the cutoff point for ‘… is tall’ is 𝑎𝑘, then the privileged precisification for such a predicate is 𝑇𝑘+1, since
𝑘 + 1 is the minimum height of a tall person.
It is important to distinguish the concept of a cut-off point from that of a determinate cut-off
point. Whereas a cut-off point for ‘… is tall’ is a non-tall person followed by tall person, a
determinate cut-off point for such a predicate is a person who is determinately not tall person and
who is followed by someone who is determinately tall. Classical propositional logic doesn’t entail
that there are determinate cut-off points. Indeed, such a commitment is inconsistent with the
existence of borderline cases.11 As we shall see in section 3, some metaphysicians posit
determinate cut-off points as a response to the argument from vagueness. However, they don’t take
them to be a result of logic, but a substantial philosophical posit.
A similar distinction applies to the concept of a privileged precisification. Whereas I assume
that vague predicates have privileged precisifications, I deny that they have determinately
privileged precisifications. For instance, whereas ‘… is tall’ has a privileged precisification, it
doesn’t have a determinately privileged precisification, since that would mean that there is a
property that is determinately instantiated by all and only the tall people, which clearly is not the
case.
Before moving on from classical propositional logic, let me mention another result that might
cause some surprise. It is sometimes said that both 𝑝 and not 𝑝 are incompatible with it being
vague whether 𝑝. Consider, however, the following reasoning. Suppose Bob is borderline tall.
11 Suppose 𝑎𝑖
is a determinate cut-off point. So, 𝑎𝑖
is determinately not tall and 𝑎𝑖+1 is determinately tall.
Plausibly, everyone shorter than a determinately not tall person is determinately not tall. So, 𝑎150-𝑎𝑖 are
determinately not tall. Also, it is plausible that everyone taller than a determinately tall person is
determinately tall. So, 𝑎𝑖+1-𝑎190 are determinately tall. Therefore, everyone in the series is either
determinately tall or determinately not tall. That is, there are no borderline cases.
10
Given classical propositional logic, either Bob is tall or he is not. So, either Bob is borderline tall
and tall, or he is borderline tall and not tall. In either case, we have a violation of the idea under
discussion. Much like the existence of cut-off points, this is something we have to accept if we
want to accept classical logic.
It is important to clarify that, even though neither 𝑝 nor not 𝑝 is incompatible with it being
vague whether 𝑝, neither [𝑝 and it is vague whether 𝑝] nor [not 𝑝 and it is vague whether 𝑝] can
be determinately true. Suppose it is determinate that [𝑝 and it is vague whether 𝑝]. That entails that
(i) it is determinate that 𝑝 and (ii) it is determinate that it is vague whether 𝑝 (see K-Det below).
But (ii) entails that it is vague whether 𝑝 (see T-Det below), which is inconsistent with (i). A similar
reasoning applies to the possibility of it being determinate that [not 𝑝 and it is vague whether 𝑝].
Since these claims can’t be determinately true, they can’t be felicitously asserted, which make
explain our sense that they are inconsistent.
Besides classical propositional logic, I assume two standard principles about the logic of
vagueness, which are reminiscent of similarly named principles in modal logic:
K-Det If it is determinate that [if 𝑝, then 𝑞], then [if it is determinate that 𝑝, then it is
determinate that 𝑞].
T-Det If it is determinate that 𝑝, then 𝑝.
I also assume that logical truths are closed under what we may call “determination”. That is, every
logical truth is determinately true, determinately determinately true, etc.
Some of the debates we will be concerned with involve the interaction between vagueness and
existence. Thus, we should be careful with the assumptions we make about quantifiers and identity.
11
For now, I will refrain from making any such assumptions, introducing them only as they become
relevant.
This concludes my presentation of my logical and philosophical assumptions about vagueness.
2. The argument
In this section, I offer a detailed reconstruction of Sider’s version of the argument from vagueness.
While most of the content is drawn from his texts, I have slightly modified the structure of the
argument and have also filled some gaps in his argumentation. Let me start by clarifying what is
meant by “vague composition” and “vague existence” in the argument.
2.1. Vague composition and vague existence
Consider the following case:
Building
A few minutes ago, you took some pieces of wood 𝑎𝑎 and started building a chair. The current
arrangement of 𝑎𝑎 is such that it is not determinate that they have a fusion and it is not
determinate that they don’t have a fusion. Equivalently, it is vague whether 𝑎𝑎 have a fusion.
Two features of 𝑎𝑎 make them a case of vague composition in the intended sense. I have already
stated the first: it is vague whether 𝑎𝑎 have a fusion. To understand the second, it is helpful to
consider a different case:
Separation
Yesterday you built a table from some pieces of wood 𝑏𝑏. A few seconds ago, you started
separating one of 𝑏𝑏 from the table. Call that piece 𝑏1. The current arrangement of 𝑏𝑏 is such
12
that it is vague whether 𝑏1 is a part of the table. Since 𝑏1 is one of 𝑏𝑏, this entails that it is
vague whether the table is a fusion 𝑏𝑏. That is, the table is a borderline fusion of 𝑏𝑏. In this
scenario, it is also vague whether 𝑏𝑏 have a fusion.
What is the difference between 𝑎𝑎 in Building and 𝑏𝑏 in Separation? According to Sider, whereas
there is a determinate connection between 𝑎𝑎 having a fusion and how many concrete objects there
are, the same cannot be said about 𝑏𝑏. In Separation, whether 𝑏𝑏 have a fusion or not doesn’t
affect the number of concrete objects. Such number is the number of 𝑏𝑏 plus one (i.e., the table)
regardless of whether 𝑏𝑏 have a fusion. However, in Building, such number is the number of 𝑎𝑎
if they don’t have a fusion, and the number of 𝑎𝑎 plus one if they do.
On Sider’s approach, then, 𝑥𝑥 are a case of vague composition just in case (i) it is vague
whether 𝑥𝑥 have a fusion and (ii) for some number 𝑛, it is determinate that [𝑥𝑥 have a fusion iff
there are exactly 𝑛 concrete objects]. To abbreviate, when (ii) is true about 𝑥𝑥, I shall say that 𝑥𝑥
are numerically relevant. As we will see, the notion of numerical relevance will play an important
role in Sider’s argument for the connection between vague composition and vague existence.12
Let’s now consider vague existence. On Sider’s original presentation of the argument as well
as in most of the subsequent literature, the phrase “existence is vague” is interpreted as the claim
12 Two clarifications are in order. First, Sider stipulates that, for the purposes of the argument, to be concrete
is to not belong to such categories as sets, classes, numbers, properties, etc. (2001: 127) Thus, the notion of
concreteness deployed in the argument from vagueness is closer to Williamson’s notion of non-abstractness
(2013: ch 1) than to common-sense concreteness. Second, what Sider actually assumes is that there are
possible worlds that contain numerically relevant pluralities. This is required in order to avoid the worry
that the number of concrete objects in the actual world might be infinite, in which case there would be no
numerically relevant pluralities (2001: 127). Taking this into account would make my presentation of
Sider’s argument even more convoluted, so I will skip it. As we will see in Chapter 2, my preferred way of
understanding vague composition doesn’t appeal to the notion of numerical relevance and hence, makes the
appeal to possible worlds unnecessary.
13
that the unrestricted existential quantifier (‘∃’, from now on) has multiple precisifications. This
conception will be revisited in Chapter 2.
Now that we have specified what is meant by “vague composition” and “vague existence” in
the argument from vagueness, we can reformulate it as follows:
P1 If composition is restricted, then, for some 𝑥𝑥, (i) it is vague whether 𝑥𝑥 have a fusion
and (ii) 𝑥𝑥 are numerically relevant.
P2 For any 𝑥𝑥, if (i) it is vague whether 𝑥𝑥 have a fusion and (ii) 𝑥𝑥 are numerically
relevant, then ‘∃’ has multiple precisifications.
P3 ‘∃’ doesn’t have multiple precisifications.
C Composition is not restricted.
The argument is clearly valid. Let’s now consider Sider’s defense of each of the premises.
2.2. Sider on restricted composition and vague composition
According to P1, restricted composition entails that there are cases of vague composition. Sider
starts his defense of P1 by claiming that restricted composition gives us a picture of the material
world where there are enough numerically relevant pluralities to form a series 𝑥𝑥1-𝑥𝑥𝑛 satisfying
the following conditions:
(1) It is determinate that 𝑥𝑥1 don’t have a fusion.
(2) It is determinate that 𝑥𝑥𝑛 have a fusion.
(3) For any 𝑖, 𝑥𝑥𝑖 differ from 𝑥𝑥𝑖+1 very slightly with respect to the features intuitively
relevant for determining whether some objects have a fusion (e.g., qualitative
homogeneity, spatial proximity, unity of action, etc.).
14
Sider calls this a continuous series. Here is an example of such a series. Take the pieces of wood
𝑎𝑎 from Building. Suppose that the process whereby they are assembled into a chair goes from 𝑡1
to 𝑡𝑛. Now consider a series of pluralities 𝑐𝑐1-𝑐𝑐𝑛 such that, for any 𝑖, 𝑐𝑐𝑖
resemble 𝑎𝑎 at 𝑡𝑖 with
respect to the features intuitively relevant for composition. Call this series the chair series. The
chair series is an example of a continuous series.13
Now we can state Sider’s argument as a reductio. Support that P1 is false. So:
(A) Composition is restricted.
(B) For any 𝑥𝑥, either (i) it is not vague whether 𝑥𝑥 have a fusion or (ii) 𝑥𝑥 are not
numerically relevant.
As we said before, (A) gives us continuous series of numerically relevant pluralities. Given (B),
any 𝑥𝑥 in those series must be such that it is not vague whether 𝑥𝑥 have a fusion. As we know, in
order for that to be the case, those series must contain a determinate cut-off point. That is, some
𝑥𝑥𝑖
such that, it is determinate that 𝑥𝑥𝑖 don’t have a fusion and it is determinate that 𝑥𝑥𝑖+1 have a
fusion. Sider argues that that would constitute a case of metaphysical arbitrariness, something he
believes should be avoided. Therefore, we should accept P1.
2.3. Sider on vague composition and vague existence
13 Two clarifications about continuous series are in order. First, notice that accepting the existence of
continuous series of numerically relevant pluralities doesn’t require accepting that such factors as
qualitative homogeneity or spatial proximity are indeed relevant for composition. This is because condition
(3) only requires that each member of a continuous series resemble the adjacent member with respect to
features that are intuitively relevant for composition. Thus, this shouldn’t alarm defenders of restricted
composition who endorse non-conservative theories of composition (e.g., van Inwagen 1990, Merricks
2001). Second, since those philosophers deny that objects arranged chair-wise have a fusion, they wouldn’t
accept the chair series as an example of a continuous series. They should feel free to adopt a different
example. For instance, one involving objects arranged person-wise.
15
Sider’s argument for P2 proceeds in two steps, the first one links cases of vague composition with
vagueness in numerical claims while the second one links vagueness in such claims with vague
existence. Let’s consider each step.
For the first step, take some arbitrary 𝑥𝑥 that are a case of vague composition. That is, (i) it is
vague whether 𝑥𝑥 have a fusion and (ii) for some number 𝑛, it is determinate that [𝑥𝑥 have a fusion
iff there are exactly 𝑛 concrete objects]. Given K-Det, (ii) entails that, for some number 𝑛, if it is
vague whether 𝑥𝑥 have a fusion, then it is vague whether there are exactly 𝑛 concrete objects.
From this claim and (i), it follows that, for some number 𝑛, it is vague whether there are exactly 𝑛
concrete objects.
The second step relies on a principle connecting the vagueness of a claim with the vagueness
of its constituents. The principle can be formulated as follows (where ‘{𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}’ stands for
a sentence that has only 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
as constituents):
Precisifications If it is vague whether {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} and each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
lacks
multiple precisifications, then ‘∃’ has multiple precisifications.
Sider doesn’t say much in defense of Precisifications. Since this principle will play an important
role in my discussion in Chapter 2, it is helpful to fill that gap. We can argue for Precisifications
as follows. Assume it is vague whether {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}. As explained above, if it is vague
whether 𝑝, then ‘𝑝’ has a true precisification and a false one. So, ‘{∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}’ has a true
precisification and a false one. We also know that the precisifications of a sentence result from
precisifications of each the sentence’s constituent. So, every precisification of ‘{∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}’
results from a precisification of each of ‘∃’, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
. We have then two propositions that
result from precisifications of each of ‘∃’, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
, one of them is true and the other false.
16
Since each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
lacks multiple precisifications, both propositions result from the unique
precisification of each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛 plus a precisification of ‘∃’. Since they differ in truthvalue, they must result from different precisifications of ‘∃’. Therefore, ‘∃’ must have multiple
precisifications.
With Precisifications at his disposal, Sider completes the second step as follows. Suppose that,
for some number 𝑛, it is vague whether there are exactly 𝑛 concrete objects. The sentence ‘there
are exactly 𝑛 concrete objects’ can be expressed using only ‘∃’, the identity sign, logical
connectives and the concreteness predicate.14 Sider assumes that the identity sign, the logical
connectives and the concreteness predicate lack multiple precisifications.15 So, given
Precisifications, the vagueness of the numerical sentence entails that ‘∃’ has multiple
precisifications.
To sum up, if 𝑥𝑥 are a case of vague composition, then some numerical sentence is vague. If
some numerical sentence is vague, then ‘∃’ has multiple precisifications. So, if 𝑥𝑥 are a case of
vague composition, then ‘∃’ has multiple precisifications. That is, P2 is true.
2.4. Sider against vague existence
Let’s now consider Sider’s argument for P3. Here is Sider:
Imagine there are two second-order properties, ∃1 and ∃2, which allegedly are precisifications
of ‘∃’. ∃1 and ∃2 need to differ in their domain. Thus, there must be some thing 𝑥 that is in the
domain of one but not the other. But in that case, whichever lacks 𝑥 in its domain will fail to
14 For instance, if the relevant number is 2, then the claim could be formulated as follows: ∃𝑥∃𝑦(𝐶𝑥 ∧ 𝐶𝑦 ∧
𝑥 ≠ 𝑦 ∧ ∀𝑧(𝐶𝑧 → (𝑧 = 𝑥 ∨ 𝑧 = 𝑦))).
15 Notice that Sider’s assumption that ‘… is concrete’ is precise is plausible given his understanding of
concreteness as non-abstractness. See note 12.
17
be an acceptable precisification of the unrestricted quantifier. It quite clearly is restricted since
there is something that fails to be in its domain. (adapted from 2001: 128-129)16
Sider’s argument can be understood as relying on two principles, both of which impose restrictions
on the kind of second-order properties that can be precisifications of ‘∃’:
Domains If ‘∃’ has two different precisifications, call them ∃1 and ∃2, then something
is in ∃1’s domain but not in ∃2’s domain or vice versa.
Unrestricted If a second-order property is a precisification of ‘∃’, then everything is in
its domain.
With Domains and Unrestricted at hand, Sider’s argument proceeds as follows. Suppose ‘∃’ has at
least two precisifications. Call them ∃1 and ∃2. Given Domains, something is in ∃1’s domain but
not in ∃2’s domain or vice versa. Let’s suppose, without loss of generality, that something is in
∃1’s domain but not in ∃2’s domain. So, something is not in ∃2’s domain. Call that thing 𝑒. Since
∃2 is a precisification of ‘∃’, Unrestricted tells us that everything is in ∃2’s domain. So, 𝑒 is both
in ∃2’s domain and not in ∃2’s domain. Since the assumption that ‘∃’ has multiple precisification
leads to a contradiction, we conclude that it is false. That is, P3 is true.
This concludes my presentation of Sider’s version of the argument from vagueness.
3. Responses to the argument
This final section outlines the main strategies metaphysicians have employed to resist the argument
from vagueness and avoid radical views about the material world. In what follows, I consider only
16 Here I take Sider’s argument as trying to establish that ‘∃’ can’t have multiple precisifications. The
argument is sometimes interpreted as trying to establish that we can’t describe such precisifications.
18
views that reject either P1 or P3. A quick discussion of the prospects of rejecting P2 can be found
in this footnote.17
3.1. Rejecting P1
Rejecting P1 amounts to saying that composition is restricted but there are no cases of vague
composition. That is, for any 𝑥𝑥, it is not the case that both (i) it is vague whether 𝑥𝑥 have a fusion
and (ii) 𝑥𝑥 are numerically relevant. Philosophers have resisted P1 either by saying that every
plurality fails to satisfy condition (i) or that some pluralities satisfy condition (i) but fail to satisfy
condition (ii).
Markosian (1998), Merricks (2005) and Hawthorne (2006: ch. 5) focus on (i) and explore
pictures of composition where series like the chair series don’t contain borderline cases of ‘… have
a fusion’. As I explained above, Sider’s argument for the existence of borderline cases depends on
the assumption that, if there were no borderline cases, we would have determinate cut-off points,
which would entail the existence of arbitrary lines dividing pluralities that have a fusion from those
that don’t. This metaphysical arbitrariness is supposed to come from the fact that adjacent
pluralities in the series are virtually identical from a microphysical point of view. Thus, there is no
difference in any fundamental feature of these pluralities that could explain why one of them
determinately lacks a fusion whereas the other determinately has one.
17 Even though Korman’s preferred response to the argument from vagueness is to endorse vague existence
(2015: ch 9), he also rejects Precisifications, which suggests that he would plausibly reject Sider’s argument
for P2 as well. His rejection of this principle relies, however, on a false assumption—namely, that accepting
it requires endorsing a linguistic conception of vagueness rather than a worldly one. Another point worth
noting is that the versions of the argument from vagueness I will discuss in the next chapter use principles
like Precisifications not in the second premise of the argument but in the third one, which would classify
Korman as an opponent of the latter, not of the former.
19
In response to this argument, philosophers have tried to posit such fundamental differences in
different ways. Let a plurality be causally relevant just in case, if it had a fusion, such an object
would have causal powers different from those of the members of the plurality. Merricks believes
that being causally relevant is what distinguishes a plurality that determinately lacks a fusion from
one that determinately has one. Thus, he rejects that making such a distinction is metaphysically
arbitrary. On the other hand, Markosian argues that having a fusion is itself a fundamental property
and its instantiation is always determinate.
Carmichael (2011) and Williamson (2013: ch 1, note 9) argue that when it is vague whether 𝑥𝑥
have a fusion, 𝑥𝑥 are not numerically relevant. Both claim that in such cases, there is a concrete
(i.e., non-abstract) object that is a borderline fusion of 𝑥𝑥. In that sense, all alleged cases of vague
composition are indeed like 𝑏𝑏 in Separating and hence, not numerically relevant.
Whereas Carmichael’s view is designed specifically to deal with the argument from vagueness,
Williamson’s view is more general. On his necessitist picture, if 𝑥𝑥 could have had a fusion, then
something is such that it could have been a fusion of 𝑥𝑥. So, since every plurality in the series
could have had a fusion, each plurality has what we may call a possible fusion. In some cases, such
possible fusions are borderline cases of ‘… is a fusion of 𝑥𝑥’. In others, they are determinately a
fusion of 𝑥𝑥 or determinately not a fusion of 𝑥𝑥.
3.2. Rejecting P3
By far the most popular strategy for resisting the argument for vagueness is to reject P3. Call this
strategy indeterminism. What makes indeterminism particularly appealing is that, unlike the other
strategies we have considered, endorsing it doesn’t require adopting any substantial view about
20
composition but only very general claims about vagueness and existence. Thus, it can also be
applied to versions of the argument from vagueness in other domains.
The list of philosophers who have at least expressed some sympathy towards indeterminism
includes van Inwagen (1990: ch 19), Hawley (2002), Smith (2005), Koslicki (2008: ch 2), Båve
(2011), Woodward (2011), Barnes (2013), Korman (2015: ch 9), Torza (2017) and Russell (ms).
Each of these authors differ in their exact approach to the argument. However, they agree that the
right approach to the argument is to claim that it incorrectly rejects the kind of vague existence
entailed by restricted composition cannot.
I will not review every indeterminist proposal in detail. For my purposes, it will suffice to give
an example of how an indeterminist challenges each of the principles Sider uses in his argument
against vague existence.
Barnes’ challenge against Domains starts with a scenario where 𝑥 determinately exists and it
is vague whether 𝑦 exists. In such a scenario, we have two precisifications for ‘∃’, ∃1 and ∃2, such
that ∃1 quantifies over 𝑥 and ∃2 quantifies over 𝑥 and 𝑦. She argues that Domains fails in this
scenario. This is because, since it is vague whether 𝑦 exists, it is also vague whether ∃2 is even
associated with a domain, since domains contain only existing things. This in turn results in it
being vague whether ∃1 and ∃2 have different domains.
On the other hand, Russell argues that Unrestricted begs the question against the indeterminist.
To see why, consider an analogous principle about ‘… is red’:
Redness If a property is a precisification of ‘… is red’, then it is instantiated by every red
thing.
Given classical propositional logic, there is a unique set of red things. In order for different
properties to be precisifications of ‘… is red’, they must differ in their extension. Thus, there might
21
be a precisification of ‘… is red’ that is not instantiated by all the red things. Redness incorrectly
rules out such a possibility. Similarly, Unrestricted incorrectly rules out the possibility of there
being a precisification of ‘∃’ that leaves out some actually existing things.
I sympathize with Russell’s argument against Unrestricted and remain skeptical about Barnes’
argument against Domains.
18 However, for the purposes of this project, I grant that they are both
compelling. Thus, indeterminist remains a threat for the argument from vagueness. I defuse this
threat in the next chapter.
18 My main issue with Barnes’ argument is that, if sound, it establishes that ∃1 and ∃2 don’t determinately
differ in their domains. However, Domains doesn’t require that the difference be determinate, only that
there be such a difference. I will not pursue this criticism any further.
22
Chapter 2: The Argument from Vagueness Revisited
In Chapter 1, I discussed different strategies for resisting the argument from vagueness and
preserving a moderate conception of the material world. In this chapter, I criticize the most popular
among them: indeterminism. I argue that a clear understanding of the idea of vague existence
allows for improved versions of the argument that rely only on assumptions indeterminists should
be happy to accept. The main idea behind these new arguments is that restricted composition
entails a kind of vague existence that even indeterminists should reject.
The chapter is structured as follows. In section 1, I introduce a new conception of vague
existence and distinguish between two varieties: positive and negative. In section 2, I provide an
argument against negative vague existence that relies only on indeterminist friendly assumptions.
The claim that existence is not negatively vague becomes then the starting point of two new
versions of the argument from vagueness that are immune to the indeterminist response. The
remaining premises of these two arguments are defended in sections 3 and 4. Section 5 concludes
by discussing the implications of the failure of indeterminism for ontological debates, some of
which will be explored in more detail in chapters 3 and 4.
1. Understanding vague existence
As mentioned in Chapter 1, both Sider and his opponents understand the phrase “existence is
vague” as saying that ‘∃’ has multiple precisifications. Here I want to suggest a different approach.
My starting point is the intuitive gloss of vague existence as the claim that “it is vague which things
exist”. This claim hasn’t bee given a lot of attention, the main reason for which being the lack of
a precise formulation of it. My task here is to provide such a formulation.
23
What could it mean to say that it is vague which things exist? Let’s start with a more familiar
claim, namely that it is contingent which things exist. Plausibly, to say that it is contingent which
things exist is to say that either (i) there is something such that it is possible that it doesn’t exist or
(ii) it is possible that there is something such that actually it doesn’t exist. Say that existence is
positively contingent in the first case and negatively contingent in the second. Formally:19
Positive Contingent Existence ∃𝑥◆¬𝐸𝑥
Negative Contingent Existence ◆∃𝑥@¬𝐸𝑥
My strategy for understanding the claim that it is vague which things exist proceeds along similar
lines. This requires finding analogues of ‘◆’ and ‘@’ in the case of vagueness. As for the first task,
I shall introduce the operator ‘it is open that’, which is the dual of ‘it is determinate that’. Thus:
Op-Det it is open that 𝑝 iff it is not determinate that not 𝑝
Also, given the connection between vagueness and determinacy (see the principle Vag-Det in
Chapter 1, section 1):
Vag-Op it is vague whether 𝑝 iff [it is open that 𝑝 and it is open that not 𝑝]
To avoid convoluted sentences, from now on, these operators will be formalized as follows:
it is vague whether: ∇
it is determinate that: □
19 Two clarifications about the formalism. First, I use ‘∎’ and ‘◆’ to represent metaphysical modality and
reserve ‘□’ and ‘◇’ to represent determinacy and its dual, to be introduced shortly. Second, I use ‘𝐸𝑥’ to
abbreviate ‘∃𝑦 𝑦 = 𝑥’.
24
it is open that: ◇
With the openness operator at our disposal, we can say that it is vague which things exist just in
case either (i) there is something such that it is open that it doesn’t exist or (ii) it is open that there
is something such that “actually” it doesn’t’ exist. Say that existence is positively vague in the first
case and negatively vague in the second. For now, we have the resources to express positive vague
existence:
Positive Vague Existence ∃𝑥◇¬𝐸𝑥20
Introducing actuality in the context of vagueness is a more delicate matter. The desired effect of
‘@’ is to neutralize the effect of vagueness operators over some formula and to “take us back” to
that formula’s privileged precisification. If this were all we wanted, we could just borrow the ‘@’
from modal logic. However, things are a bit trickier. Given the matters under discussion, we want
to be able to reason about a sentence’s precisifications via principles like Precisifications (see
Chapter 1, §2.3). This requires being able to identify the constituents of the sentence and the kind
of precisifications they have. Unfortunately, it is not clear how such an analysis could proceed for
sentences involving ‘@’, since it is not clear what the precisifications of ‘@’ might be. To avoid
this issue, I shall pursue a different route.21
20 Everything exists. So, by T-Det, everything is such that it is open that it exists. Thus, Positive Vague
Existence entails that there is something such that it is vague whether it exists. In other words, there is
something such that it is vague whether it is something. This claim is usually regarded as incoherent: how
could it be that (i) 𝑥 is something and (ii) it is vague whether 𝑥 is something? This worry is an instance of
a more general worry: how could it be that (i) 𝑝 and (ii) it is vague whether 𝑝? As I explained in Chapter 1
(section 1), claims of this form are unassertible but not incoherent. A more detailed discussion of this issue
in connection with vague existence can be found in Sud 2023. Unfortunately, I became aware of that paper
only in the later stages of writing this dissertation, which is why I didn’t engage with it more thoroughly.
21 The same worry applies to backspace operators (‘↑’, ‘↓’), which are introduced as a way of increasing the
expressive power of modal languages. For discussion of backspace operators, Forbes 1989: 27-29, Bricker
1989 and Williamson 2010: 685ff.
25
Let’s start by introducing a new way of regimenting actuality in the modal context. Suppose
we want to say of 𝜙 that it is actually true. I propose we understand actual truth as follows: to say
of 𝜙 that it is actually true is to say that it is one of the truths.22 To make this idea more precise, I
shall work in a higher-order language that contains singular and plural propositional quantifiers
for both objects and propositions, as well as a higher-order predicate ‘[… < … ]’ that expresses
the propositional analogue of plural membership. The left argument of this predicate is occupied
by a sentence and its right argument, by the sentential analogue of a plural term. Given this
formalism, the claim that 𝜙 is actually true can be formalized as follows:23
∃𝑝𝑝(∀𝑝(𝑝 ↔ [𝑝 < 𝑝𝑝]) ∧ [𝜙 < 𝑝𝑝])
Informally, this says that there are some propositions such that (i) they are exactly those
propositions that are true and (ii) 𝜙 is one of them.
Let’s now try to express negative contingent existence given this new understanding of
actuality. To do that, we need to be able to say things like “possibly actually 𝜙”. One might think
that we can do so as follows: to say that 𝜙 is possibly actually true is to say that, possibly, the
truths are such that 𝜙 is one of them. Formally (where ‘𝑇𝑝𝑝’ abbreviates ‘∀𝑝(𝑝 ↔ [𝑝 < 𝑝𝑝])’):
◆∃𝑝𝑝(𝑇𝑝𝑝 ∧ [𝜙 < 𝑝𝑝])
However, this won’t do, as any contingent claim 𝜙 that is false would satisfy the formula above
without it being possibly actually true. Fortunately, this problem can be avoided by having the
propositional plural quantifier take wide scope over the modal operator:
22 My proposal draws inspiration from previous discussions of the connection between actuality and plural
quantification such as those in Bricker 1989 and Forbes 1989.
23 For a defense of the intelligibility of plural propositional quantification, see Fritz, Lederman & Uzquiano
2021, Fritz 2021, and Fritz 2024.
26
∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◆[𝜙 < 𝑝𝑝])
Informally, to say that 𝜙 is possibly actually true is to say that the truths are such that, possibly, 𝜙
is one of them.24
We now have the resources to say of 𝜙 that it is possibly actually true. Let’s now consider the
question of how to say of 𝑥 that it is actually 𝐹. To do so, we say of the truths that they are such
that the proposition that 𝑥 is 𝐹 is one of them:
∃𝑝𝑝(𝑇𝑝𝑝 ∧ [𝐹𝑥 < 𝑝𝑝])
To say that existence is negatively contingent is to say that possibly, something actually doesn’t
exist. First, we deal with the claim that something is such that actually it doesn’t exist. That should
be expressed as the claim that the truths are such that there is something such that the proposition
that it doesn’t exist (“its non-existence”, from now on) is one of them. Formally:
∃𝑝𝑝(𝑇𝑝𝑝 ∧ [¬𝐸𝑥 < 𝑝𝑝])
Finally, we add the modal operator under the scope of the propositional plural quantifier and say
that the truths are such that, possibly, there is something whose non-existence is one of them.
Formally:
24 Notice that, in order for this formalism to capture the intuitive sense of actuality, we need to assume that
pluralities of propositions have their members necessarily. Thus, we have to accept:
Necessary Plural Membership
∎∀𝑝𝑝∀𝑞(([𝑞 < 𝑝𝑝] → ∎[𝑞 < 𝑝𝑝]) ∧ (¬[𝑞 < 𝑝𝑝] → ∎¬[𝑞 < 𝑝𝑝]))
This principle is reminiscent of the principle that pluralities of objects have their members necessarily. For
discussion, see Linnebo 2016.
27
Negative Contingent Existence ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◆∃𝑥[¬𝐸𝑥 < 𝑝𝑝])
We can now make a similar move in the case of negative vague existence, the claim that it is open
that there is something such that actually it doesn’t exist. Such a claim will be understood as the
claim that the truths are such that it is open that there is something whose non-existence is one of
them. Formally:
Negative Vague Existence ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝])
25
We now have a precise formulation of both kinds of vague existence, the claim that it is vague
which things exist, i.e., our interpretation of the phrase “existence is vague”, can be formulated as
the claim that either existence is positively vague or it is negatively vague.
As we shall see, this conception of vague existence pairs nicely with a conception of vague
composition different from Sider’s. In Chapter 1, we saw that, on Sider’s version of the argument
from vagueness, 𝑥𝑥 are a case of vague composition just in case (i) it is vague whether 𝑥𝑥 have a
fusion and (ii) for some number 𝑛, it is determinate that [𝑥𝑥 have a fusion iff there are exactly 𝑛
concrete objects]. For our purposes, it is helpful to adopt an alternative approach, which has already
been suggested in the literature.26 Recall that the main point of condition (ii) was to distinguish
between pluralities like 𝑎𝑎 in Building, which satisfy (ii) and those like 𝑏𝑏 in Separation, which
don’t (see Chapter 1, §2.1). On Sider’s approach, the difference is cashed out in terms of the
25 For this regimentation to work, we need to assume the analogue of Necessary Plural Membership for the
vagueness case:
Determinate Plural Membership
□∀𝑝𝑝∀𝑞(([𝑞 < 𝑝𝑝] → □[𝑞 < 𝑝𝑝]) ∧ (¬[𝑞 < 𝑝𝑝] → □¬[𝑞 < 𝑝𝑝]))
26 For remarks in this direction, see, among others, van Inwagen 1990: ch 19, Hawley 2002, Donnelly 2009
and Carmichael 2011.
28
connection between the relevant objects having a fusion and the number of concrete objects. On
the new approach, we say that what distinguishes 𝑎𝑎 in Building from 𝑏𝑏 in Separation is that
only the latter are such that something is a borderline fusion of them. This leads to the following
understanding of vague composition: some 𝑥𝑥 are a case of vague composition just in case (i) it is
vague whether 𝑥𝑥 have a fusion and (ii) nothing is a borderline fusion of 𝑥𝑥.
The remainder of this chapter introduces and defends two new versions of the argument from
vagueness that deploy this conception of vague composition and vague existence. Let 𝑥𝑥 be a case
of negative vague composition just in case (i) 𝑥𝑥 are a case of vague composition and (ii) 𝑥𝑥 don’t
have a fusion, and a case of openly negative vague composition just in case it is open that 𝑥𝑥 are
a case of negative vague composition. The arguments proceed as follows:
The argument from negative vagueness
P1-n If composition is restricted, then, for some 𝑥𝑥, 𝑥𝑥 are a case of negative vague
composition.
P2-n For any 𝑥𝑥, if 𝑥𝑥 are a case of negative vague composition, then existence is negatively
vague.
P3-n Existence is not negatively vague.
C Composition is not restricted.
The argument from openly negative vagueness
P1-on If composition is restricted, then, for some 𝑥𝑥, 𝑥𝑥 are a case of openly negative vague
composition.
P2-on For any 𝑥𝑥, if it is open that 𝑥𝑥 are a case of negative vague composition, then it is
open that existence is negatively vague.
29
P3-on It is determinate that existence is not negatively vague.
C Composition is not restricted.
Both arguments are clearly valid. In next three sections, I defend their premises in pairs and in
reverse order, beginning with P3-n and P3-on, and concluding with P1-n and P1-on.
2. Against negative vague existence
Sider’s argument against vague existence relies on Unrestricted and Domains. Each of these
principles imposes a constraint on the kind of second-order properties that can be precisifications
of the unrestricted quantifier. As we saw in Chapter 1, indeterminists have challenged both. In this
section, I argue for a constraint on the precisifications of ‘∃’ that even indeterminists should be
happy to accept. Then, I show that negative vague existence violates this constraint. So, existence
is not negatively vague. That is, P3-n is true. After presenting my argument, I argue that it can be
strengthened to show that it is determinate that existence is not negatively vague. That is, P3-on is
true.
When discussing the possibility of there being multiple precisifications of ‘∃’, even
indeterminists accept that every such precisification must possess all the logical features of ‘∃’.
For instance, Båve speaks of an auxiliary logic as a list of axioms and inference rules that is
common to all precisifications of ‘∃’ (2011: 106ff). Plausibly, the logical features of ‘∃’ are those
that are expressed in the logical truths that feature ‘∃’. This leads to the following general
constraint on the precisifications of ‘∃’ (where ‘Φ’ is a higher-order predicate that takes a secondorder predicate as its argument and doesn’t contain any occurrence of ‘∃’):
Logical Determinism If it is a logical truth that Φ∃, then any precisification 𝑄 of ‘∃’ is
such that Φ𝑄.
30
Unlike Domains and Unrestricted, Logical Determinism is supported not just by specific thoughts
about ‘∃’ but by a plausible conception of the connection between logical truths and
precisifications.
I shall now show look at a specific logical truth and show that applying Logical Determinism
to it ledas to a substantial constraint on the precisifications of ‘∃’. Here is such a logical truth:
Universal Being ∀𝑥𝐸𝑥
Universal Being can also be formulated as ¬∃𝑥¬𝐸𝑥. Moreover, it is also logically equivalent to:
∃𝑝𝑝(𝑇𝑝𝑝 ∧ ¬∃𝑥[¬𝐸𝑥 < 𝑝𝑝]). Informally, this formula says that the truths are such that nothing
is such that its non-existence is one of them.
Given a second-order term ‘𝑄’, let ‘𝐸
𝑄𝑥’ abbreviate ‘𝑄𝑦 𝑦 = 𝑥’. We can then understand
Universal Being as predicating a higher-order property of ∃, namely (𝜆𝑄. ∃𝑝𝑝(𝑇𝑝𝑝 ∧
¬𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝])). Informally, what this higher-order predicate says about ∃ is that the truths
are such that nothing∃ is such that its non-existence∃ is one of them.
Universal Being is a logical truth. Given Logical Determinism, that entails that any
precisification of ‘∃’ satisfies the feature Universal Being attributes to ∃. That is, for every secondorder property 𝑄 that is a precisification of ‘∃’, the following is true: the truths are such that
nothingQ is such that its non-existenceQ is one of them. Formally (where ‘𝒫(𝑋, 𝑌)’ is to be read as
“𝑋 is a precisification of 𝑌”):
UniversalPrec ∀𝑄(𝒫(𝑄, ′∃′) → ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ¬𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝]))
31
I shall now argue that if existence is negatively vague, then ‘∃’ has precisifications that violate
UniversalPrec. Let me start by introducing a principle that will play an important role in the
argument. This principle resembles Sider’s Precisifications:
Precisificationsopen If is open that {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} and each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
lacks
multiple precisifications, then there is a precisification of ‘∃’, call it
∃1, such that {∃1, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}.
We can argue for Precisificationsopen as follows. Take the claim that {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} and assume
each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
lacks multiple precisifications. For any expression 𝑒, let 𝑒∗ be the unique
precisification of 𝑒, if 𝑒 has one. Each precisification of {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} then results from a
precisification of ‘∃’ plus 𝑐∗
1
, 𝑐∗
2
, …, 𝑐∗
𝑛
. Suppose it is open that {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}. Given the way
I introduced ‘it is open that’, the following is true:
Op-Prec It is open that 𝑝 iff some precisification of ‘𝑝’ is true.
So, one of the precisifications of {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} is true. So, there is a precisification of ‘∃’,
call it ∃1, such that the proposition that results from ∃1, 𝑐∗
1
, 𝑐∗
2
, …, 𝑐∗
𝑛
is true. Letting ‘∃1’
determinately express ∃1, that entails that the claim {∃1, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} is true. So, there is a
precisification of ‘∃’, call it ∃1, such that {∃1, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
}. Therefore, Precisificationsopen is
true.
With Precisificationsopen at my disposal, I shall show that negative vague existence is
inconsistent with UniversalPrec. Suppose existence is negatively vague. That is, suppose
∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]). Like Sider, I assume that the logical connectives and the identity
predicate lack multiple precisifications. In addition, I assume the same about the propositional
32
plural membership predicate. Now, given that assumption and Precisificationsopen, ∃𝑝𝑝(𝑇𝑝𝑝 ∧
◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]) entails ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ∃𝑄(𝒫(𝑄, ′∃′) ∧ 𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝])). Equivalently,
∃𝑄(𝒫(𝑄,
′ ∃
′
) ∧ ∃𝑝𝑝(𝑇𝑝𝑝 ∧ 𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝])). This result contradicts UniversalPrec. Informally,
it says that there is a precisification 𝑄 of ‘∃’ with the following feature: the truths are such that
somethingQ is such that its non-existenceQ is one of them. This is prohibited by UniversalPrec,
which says that no precisification of ‘∃’ has that feature. In light of this result, I conclude that
existence is not negatively vague. That is, P3-n is true.
In the remainder of this section, I show that this argument can be strengthened to show that it
is determinate that existence is not negatively vague. That is, P3-on is true. To do so, we need
stronger versions of the principles above:
Det-Logical Determinism It is determinate that [if it is a logical truth that Φ∃, then
any precisification 𝑄 of ‘∃’ is such that Φ𝑄].
Det- Precisificationsopen It is determinate that [if it is open that {∃, 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
} and
each of 𝑐
1
, 𝑐
2
, …, 𝑐
𝑛
lacks multiple precisifications, then
there is a precisification of ‘∃’, call it ∃1, such that {∃1, 𝑐
1
,
𝑐
2
, …, 𝑐
𝑛
}].
I take it that one should accept Det-Logical Determinism if one has already accepted Logical
Determinism. Also, I take it that we accept the principles involved in the argument for
Precisificationsopen because we know them. So, given the standard assumption that knowledge
33
requires determinacy, they must hold determinately.27 Therefore, said argument can be turned into
one for Det-Precisificationsopen.
To argue for P3-on, we start with the assumption that it is determinate that Universal Being is
a logical truth, which I take to be a harmless strengthening of the claim that it is a logical truth.
Given Det-Logical Determinism, that assumption entails that it is determinate that every
precisification of ‘∃’ instantiates the feature Universal Being attributes to ∃. This gives us a
stronger version of UniversalPrec:
Det-UniversalPrec □∀𝑄(𝒫(𝑄, ′∃′) → ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ¬𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝]))
I shall now argue that openly negative vague existence is inconsistent with Det-UniversalPrec.
Suppose that it is open that existence is negatively vague. That is, suppose ◇∃𝑝𝑝(𝑇𝑝𝑝 ∧
◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]). Given the assumption that logical connectives, the identity predicate, and the
propositional plural membership predicate determinately lack multiple precisifications and DetPrecisificationsopen, ◇∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]) entails ◇∃𝑝𝑝(𝑇𝑝𝑝 ∧ ∃𝑄(𝒫(𝑄, ′∃′) ∧
𝑄𝑥[¬𝐸
𝑄𝑥 < 𝑝𝑝])). Equivalently, ◇(∃𝑄(𝒫(𝑄, ′∃′) ∧ ∃𝑝𝑝(𝑇𝑝𝑝 ∧ 𝑄𝑥[¬𝐸^𝑄 𝑥 < 𝑝𝑝]))). This
result contradicts Det-UniversalPrec for the same reason that the result identified in my previous
argument contradicted UniversalPrec. Therefore, it is not open that existence is negatively vague.
Equivalently, it is determinate that existence is not negatively vague. That is, P3-on is true.
3. Against negative vague composition
This section shows that P2-n and P2-on follow from plausible logical assumptions. Here are P2-n
and P2-on again:
27 Most theorists of vagueness accept this assumption. For a dissenting opinion, see Dorr 2003. For a
response, see Bacon 2018: ch 5.
34
P2-n For any 𝑥𝑥, if 𝑥𝑥 are a case of negative vague composition, then existence is negatively
vague.
P2-on For any 𝑥𝑥, if 𝑥𝑥 are a case of openly negative vague composition, then it is open that
existence is negatively vague.
Before providing proofs for P2-n and P2-on, I shall do two things. First, I shall introduce a principle
that will be used in said proofs. Second, I shall deal with a complication in the formalization of the
claim that 𝑥𝑥 lack borderline fusions.
3.1. Indeterminist Barcan
The first principle is akin to a familiar principle in modal logic. We start with the Barcan formula:
Barcan ∀𝑥∎𝜙𝑥 → ∎∀𝑥𝜙𝑥
As is well-known, the Barcan formula rules out what I have called negative contingent existence.28
Thus, those who want to make room for the possibility of negative contingent existence cannot
accept such a principle. However, they can accept a weaker version:
Contingentist Barcan ∀𝑥∎𝜙𝑥 → ∎∀𝑥(¬𝜙𝑥 → @¬∃𝑦 𝑦 = 𝑥)
Instead of saying that everything being necessarily 𝜙 entails that necessarily everything is 𝜙,
Contingentist Barcan says that everything being necessarily 𝜙 entails that necessarily, if something
28 It is uncontroversial that everything necessarily actually exists. By the Barcan formula, that entails that
necessarily everything actually exists. In other words, existence is not negatively contingent. It is worth
noting that this reasoning goes through given my way of understanding actuality, as the Barcan formula
allows us to go from ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ∀𝑥∎[𝐸𝑥 < 𝑝𝑝]) to ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ∎∀𝑥[𝐸𝑥 < 𝑝𝑝]), which is the denial of
negative contingent existence.
35
is not 𝜙, then actually it doesn’t exist. Given our preferred way of expressing actuality, this
principle can be reformulated as follows:
Contingentist Barcan ∀𝑥∎𝜙𝑥 → ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ∎∀𝑥(¬𝜙𝑥 → [¬∃𝑦 𝑦 = 𝑥 < 𝑝𝑝]))
Informally, Contingentist Barcan says that, if everything is necessarily 𝜙, then the truths are such
that, necessarily, if something is not 𝜙, then its non-existence is one of them.
When it comes to vagueness, an analogue of the Barcan formula can be stated as follows:
Barcan-Det ∀𝑥□𝜙𝑥 → □∀𝑥𝜙𝑥
Barcan-Det rules out negative vague existence. Thus, indeterminists have every right to resist it.
The following principle, however, shouldn’t cause them any trouble:
Indeterminist Barcan ∀𝑥□𝜙𝑥 → ∃𝑝𝑝(𝑇𝑝𝑝 ∧ □∀𝑥(¬𝜙𝑥 → [¬∃𝑦 𝑦 = 𝑥 < 𝑝𝑝]))
Informally, Indeterminist Barcan says that, if everything is determinately 𝜙, then the truths are
such that, determinately, if something is not 𝜙, then its non-existence is one of them.
3.2. No borderline fusions
Given the formalism I have been using, it seems natural to formalize the claim that 𝑥𝑥 lack
borderline fusions as follows: ¬∃𝑥∇𝐹𝑥. Given basic quantification theory and Vag-Det, this entails
(indeed, is equivalent to) ∀𝑥(□𝐹𝑥 ∨ □¬𝐹𝑥). The problem is that this formalization might be seen
as begging the question against the indeterminist. Let me explain. If one is a contingentist, then
one cannot formalize the claim that being human is not contingent in such a way that it entails
∀𝑥(∎𝐻𝑥 ∨ ∎¬𝐻𝑥). For, assuming that being human entails existing, that formula entails that all
humans exist necessarily. Similarly, if one is an indeterminist, one cannot formalize the claim that
36
being a fusion of 𝑥𝑥 is not borderline in such a way that it entails ∀𝑥(□𝐹𝑥 ∨ □¬𝐹𝑥), where ‘𝐹𝑥’
is to be read as ‘it is a fusion of 𝑥𝑥’. For, assuming that being a fusion of 𝑥𝑥 entails existing, that
formula entails that all fusions of 𝑥𝑥 exist determinately, which might be an undesirable
consequence from an indeterminist perspective.
The solution to this problem is to adopt a different formalization of the no-borderline-fusions
claim, which mirrors a strategy deployed by contingentists to formalize the non-contingency of
certain properties. Contingentists formalize the claim that being human is not contingent as saying
that everything is such that either (i) necessarily, if it exists, it is human or (ii) necessarily, if it
exists, it is not human. Formally, ∀𝑥(∎(𝐸𝑥 → 𝐹𝑥) ∨ ∎(𝐸𝑥 → ¬𝐹𝑥)). Similarly, an indeterministfriendly way of understanding the claim that being a fusion of 𝑥𝑥 is not borderline is as saying that
everything is such that either (i) determinately, if it exists, it is a fusion of 𝑥𝑥 or (ii) determinately,
if it exists, it is not a fusion of 𝑥𝑥. Formally: ∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥)).
3.3. Defending P2-n and P2-on
I can now offer proofs for both P2-n and P2-on. Let’s start with P2-n. Assume that 𝑥𝑥 are a case
of negative vague composition. That is, (i) it is vague whether 𝑥𝑥 have a fusion, (ii) nothing is a
borderline fusion of 𝑥𝑥, and (iii) 𝑥𝑥 don’t have a fusion. Formally:
(i) ∇∃𝑥𝐹𝑥
(ii) ∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥))
(iii) ¬∃𝑥𝐹𝑥
First, we prove that (ii) and (iii) entail ∀𝑥□(𝐸𝑥 → ¬𝐹𝑥). Informally, everything is determinately
a non-fusion of 𝑥𝑥 if it exists.
37
1. ¬∃𝑥𝐹𝑥 assumption (iii)
2. ∀𝑥¬𝐹𝑥 (1)
3. ∀𝑥(𝐸𝑥 ∧ ¬𝐹𝑥) (2), by Universal Being
4. ∀𝑥¬(𝐸𝑥 → 𝐹𝑥) (3)
5. ∀𝑥¬□(𝐸𝑥 → 𝐹𝑥) (4), by T-Det
6. ∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥)) assumption (ii)
7. ∀𝑥□(𝐸𝑥 → ¬𝐹𝑥) (5,6)
We now prove that this result and (i) entail that existence is negatively vague:
1. ∇∃𝑥𝐹𝑥 assumption (i)
2. ◇∃𝑥𝐹𝑥 (1), Vag-Op
3. ∀𝑥□(𝐸𝑥 → ¬𝐹𝑥) previous result
4. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ □∀𝑥(¬(𝐸𝑥 → ¬𝐹𝑥) → [¬𝐸𝑥 < 𝑝𝑝])) (3), by Indet. Barcan
5. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ □∀𝑥((𝐸𝑥 ∧ 𝐹𝑥) → [¬𝐸𝑥 < 𝑝𝑝])) (4)
6. ◇∃𝑥(𝐸𝑥 ∧ 𝐹𝑥) (2), by Univ. Being
7. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ (◇∃𝑥(𝐸𝑥 ∧ 𝐹𝑥) ∧ □∀𝑥((𝐸𝑥 ∧ 𝐹𝑥) → [¬𝐸𝑥 < 𝑝𝑝]))) (5, 6)
8. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇(∃𝑥(𝐸𝑥 ∧ 𝐹𝑥) ∧ ∀𝑥((𝐸𝑥 ∧ 𝐹𝑥) → [¬𝐸𝑥 < 𝑝𝑝]))) (7), by K-Det
9. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥((𝐸𝑥 ∧ 𝐹𝑥) ∧ ((𝐸𝑥 ∧ 𝐹𝑥) → [¬𝐸𝑥 < 𝑝𝑝]))) (8)
10. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]) (9)
This completes the proof of P2-n.29
29 Notice that a similar derivation shows that (i) ◇∃𝑥𝐹𝑥 and (ii) ¬∃𝑥◇𝐹𝑥 entail that existence is negatively
vague:
1. ◇∃𝑥𝐹𝑥 assumption
2. ¬∃𝑥◇𝐹𝑥 assumption
3. ∀𝑥□¬𝐹𝑥 (2), Op-Det
4. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ □∀𝑥(𝐹𝑥 → [¬𝐸𝑥 < 𝑝𝑝])) (3), by Indeterminist Barcan
5. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ (◇∃𝑥𝐹𝑥 ∧ □∀𝑥(𝐹𝑥 → [¬𝐸𝑥 < 𝑝𝑝]))) (1, 4)
6. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇(∃𝑥𝐹𝑥 ∧ ∀𝑥(𝐹𝑥 → [¬𝐸𝑥 < 𝑝𝑝]))) (5), by K-Det
7. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥(𝐹𝑥 ∧ (𝐹𝑥 → [¬𝐸𝑥 < 𝑝𝑝]))) (6)
8. ∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝]) (7)
Thus, if existence is not negatively vague, then, if ◇∃𝑥𝐹𝑥, then ∃𝑥◇𝐹𝑥. That is, the rejection of negative
vague existence allows us to recover Barcan-Det. This result will be useful in Chapter 4.
38
I will now prove P2-on. Assume that 𝑥𝑥 are a case of openly negative vague composition. That
is, it is open that [(i) it is vague whether 𝑥𝑥 have a fusion, (ii) nothing is a borderline fusion of 𝑥𝑥,
and (iii) 𝑥𝑥 don’t have a fusion]. Formally:
1. ◇(∇∃𝑥𝐹𝑥 ∧ ∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥)) ∧ ¬∃𝑥𝐹𝑥)
We know from the first derivation above that the second and third conjunct under the scope of ‘◇’
entail ∀𝑥□(𝐸𝑥 → ¬𝐹𝑥). Since all the principles used in said derivation hold determinately, K-Det
allows us to conclude:
2. ◇(∇∃𝑥𝐹𝑥 ∧ ∀𝑥□(𝐸𝑥 → ¬𝐹𝑥))
We also know from the second derivation above that the two conjuncts under the scope of ‘◇’
entail that existence is negatively vague. Again, all the principles used in said derivation hold
determinately. Thus, K-Det allows us to conclude:
3. ◇∃𝑝𝑝(𝑇𝑝𝑝 ∧ ◇∃𝑥[¬𝐸𝑥 < 𝑝𝑝])
Informally, it is open that existence is negatively vague. This completes the proof of P2-on and my
defense of the second premise of each argument.
4. Against restricted composition
Let take stock. In the last two sections, I argued for P2-n, P3-n, P2-on and P3-on. From the first
two, it follows that no 𝑥𝑥 are a case of negative vague composition. From the last two, it follows
that no 𝑥𝑥 are such that it is open that 𝑥𝑥 are a case of negative vague composition. To argue
against restricted composition, then, we need only one of P1-n or P1-on to be true:
39
P1-n If composition is restricted, then, for some 𝑥𝑥, 𝑥𝑥 are a case of negative vague
composition.
P1-on If composition is restricted, then, for some 𝑥𝑥, 𝑥𝑥 are a case of openly negative vague
composition.
My goal in this section is to defend both premises. Crucially, unlike with the pairs P3-n/P3-on and
P2-n/P2-on, my arguments for P1-n and P1-on are independent. Thus, when it comes to the
connection between restricted composition and vague composition, the argument from negative
vagueness and the argument from openly negative vagueness constitute two distinct objections
against restricted composition and one can endorse one while rejecting the other.
4.1. Restricted composition and negative vague composition
My argument for P1-n relies on two premises:
P1-na If composition is restricted, then there are cases of vague composition.
P1-nb If there are cases of vague composition, then there are cases of negative vague
composition.
To argue for P1-na, I adapt Sider’s argument for P1 to make it suitable to the conception of vague
composition introduced in section 1. In his argument for P1, Sider assumes that, given restricted
composition, there is at least one continuous series containing only numerically relevant
pluralities. Given my understanding of vague composition, for my argument for P1-na, I assume
that, if composition is restricted, then there is at least a continuous series containing only pluralities
that lack borderline fusions. The chair series is an example of such a series. The argument then
proceeds as a reductio. Suppose P1-na is false. So:
40
(A) Composition is restricted.
(B) For any 𝑥𝑥, either (i) it is not vague whether 𝑥𝑥 have a fusion or (ii) 𝑥𝑥 have a
borderline fusion.
As I said in the previous paragraph, (A) entails that there are continuous series of pluralities that
lack borderline fusions. Thus, given (B), any 𝑥𝑥 in those series must be such that it is not vague
whether 𝑥𝑥 have a fusion. As we saw before (see Chapter 1, section 1), this requires that each such
series contain a determinate cut-off point, i.e., a plurality that determinately lacks a fusion followed
by one that determinately has one. As Sider argues, that can’t happen, as it would involve
metaphysical arbitrariness. Therefore, P1-na is true.
Let’s now consider P1-nb. Let 𝑥𝑥 be a case of positive vague composition just in case (i) 𝑥𝑥
are a case of vague composition and (ii) 𝑥𝑥 have a fusion. Classical propositional logic tells us
that, if 𝑥𝑥 are a case of vague composition, then 𝑥𝑥 are either a case of positive vague composition
or a case of negative vague composition. Thus, given P1-na, we know that restricted composition
entails that there are either cases of positive vague composition or cases of negative vague
composition. To that, P1-nb adds that there have to be cases of negative vague composition.
My starting point is the idea that, if there are cases of vague composition, then there can be a
continuous series whose members lack borderline fusions and differ only with respect to one of
the features intuitively relevant for composition. Say, for instance, that they differ only with respect
to qualitative homogeneity. In this scenario, we begin with a plurality of objects that are
qualitatively very similar and end with a plurality of objects that are qualitatively very different.
In between, we have pluralities of objects that are borderline cases of ‘… are similar enough’,
which are also borderline cases of ‘… have a fusion’.
41
Say that 𝑥 is a positive (negative) borderline case of ‘𝑃’ iff (i) it is vague whether 𝑥 is 𝑃 and
(ii) 𝑥 is (not) 𝑃. In the above scenario, some 𝑥𝑥 in the series are a case of negative vague
composition iff 𝑥𝑥 are a negative borderline case of ‘… are similar enough’. It strikes as very
plausible that a vague predicate like ‘… are similar enough’ has both positive and negative
borderline cases in the series. Thus, there are case of negative vague composition. What if ‘… are
similar enough’ happens to have only positive borderline cases in the series? I haven’t provided a
definitive argument against this possibility. However, notice that all I need for my argument to
work is that there be some vague predicate among those relevant for composition that has negative
borderline cases in some series. Thus, to resist my argument the object must claim such a scenario
is impossible. I find such a claim highly implausible.30
This completes my defense of P1-n and hence of the argument from negative vagueness. In a
nutshell, the arguments says that we should reject restricted composition because it leads to
negative vague composition, which leads to negative vague existence, which is in itself an
undesirable result.
4.2. Restricted composition and openly negative vague composition
Let’s now consider P1-on and the argument from openly negative vagueness. Let 𝑥𝑥 be a case of
determinately vague composition just in case it is determinate that 𝑥𝑥 are a case of vague
composition. The argument for P1-on relies on the following premises:
P1-ona If composition is restricted, then there are cases of determinately vague
composition.
30 A more detailed argument for a similar conclusion can be found in Sud 2023. See note 20 for a disclaimer
about that paper.
42
P1-onb For any 𝑥𝑥, if 𝑥𝑥 are a case of determinately vague composition, then it is open that
𝑥𝑥 are a case of negative vague composition.
To argue for P1-ona, I start by considering our reasons for accepting that, given restricted
composition, there are continuous series of pluralities that lack borderline fusions. It is plausible
to assume that we accept such a claim on the basis of our knowledge of specific cases (e.g.,
Building). That is, we know of those pluralities that they lack borderline fusions. Given the
assumption that knowledge requires determinacy, this entails that there are continuous series
whose elements determinately lack borderline fusions.
We know that, if composition is restricted, then continuous series can’t contain determinate
cut-off points. So, they must contain borderline cases. Given what was established in the previous
paragraph, this entails that there are continuous series with borderline cases that also lack
borderline fusions determinately. In other words, for some 𝑥𝑥, (i) it is vague whether 𝑥𝑥 have a
fusion and (ii) it is determinate that 𝑥𝑥 don’t have borderline fusions.
Suppose now that P1-ona is false. This entails that, for any 𝑥𝑥, it is not determinate that [(i) it
is vague whether 𝑥𝑥 have a fusion and (ii) nothing is a borderline fusion of 𝑥𝑥]. This entails that,
for any 𝑥𝑥, either (i) it is not determinate that it is vague whether 𝑥𝑥 have a fusion or (ii) it is not
determinate that 𝑥𝑥 don’t have a borderline fusion. So, since we have at least some 𝑥𝑥 for which
condition (ii) fails, they would have to satisfy condition (i). That is, even though it is vague whether
those 𝑥𝑥 have a fusion, it is vague whether it is vague whether that is so.31 In other words, all
borderline cases are borderline borderline cases.
31 It is not determinate that they are borderline cases. Since they are borderline cases, it is not determinate
that they are not borderline cases. So, it is vague whether they are borderline cases.
43
Though I lack a knock-down argument against this picture, I found it implausible and
unmotivated. First, as I mentioned before, it is natural to assume that knowledge requires
determinacy. So, this picture entails that there are borderline cases of the relevant kind, but we are
unable to know that they are borderline cases. Even though we have accepted that there are
borderline cases on the basis of an argument against determinate cut-off points, it also seems
reasonable to accept it on the basis of specific pluralities (e.g., Building) of which we know that
they are borderline cases, which would require that it be determinate that they are borderline cases.
On the other hand, there doesn’t seem to be anything particularly attractive about this picture
besides the fact that it allows us to resist P1-ona while retaining P1-na. In that sense, it is different
from the kind of picture suggested by those who oppose the argument for P1-na either by accepting
determinate cut-off points or by positing borderline fusions. For these reasons, I conclude that we
should accept P1-ona.
The argument for P1-onb is more straightforward. Assume 𝑥𝑥 are a case of determinately
vague composition. So, it is determinate that [(i) it is vague whether 𝑥𝑥 have a fusion and (ii) 𝑥𝑥
lack a borderline fusion]. Given K-Det, this entails that it is determinate that it is vague whether
𝑥𝑥 have a fusion (formally: (A) □∇∃𝑥𝐹𝑥) and that it is determinate that 𝑥𝑥 lack a borderline fusion
(formally: (B) □∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥))). Now, given T-Det, (A) entails ∇∃𝑥𝐹𝑥,
which further entails (C) ◇¬∃𝑥𝐹𝑥. We then have:
(A) □∇∃𝑥𝐹𝑥
(B) □∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥))
(C) ◇¬∃𝑥𝐹𝑥
44
Given K-Det, (A-C) entail ◇(∇∃𝑥𝐹𝑥 ∧ ∀𝑥(□(𝐸𝑥 → 𝐹𝑥) ∨ □(𝐸𝑥 → ¬𝐹𝑥)) ∧ ¬∃𝑥𝐹𝑥). That is, it
is open that 𝑥𝑥 are a case of negative vague composition. So, P1-onb is true.
This completes my defense of P1-on and hence of the argument from openly negative
vagueness. In a nutshell, the argument says that we should reject restricted composition because it
leads to openly negative vague composition, which leads to openly negative vague existence,
which is in itself an undesirable result.
5. Indeterminism and ontology
My goal in this chapter has been to argue that indeterminism is ineffective as a response to the
argument from vagueness. When confronted with said argument, indeterminists resist Sider’s
attack on vague existence by rejecting either Domains or Unrestricted. Since my rejection of
negative vague existence relies only on indeterminist friendly assumptions, a similar move is
unavailable to them in the case of the arguments from negative vagueness.
This doesn’t suffice for establishing the success of the argument from vagueness. As I
explained in Chapter 1, one can also resist the argument by positing determinate cut-off points or
by accepting borderline fusions. Such strategies can also be appealed to in response to the
arguments from negative vagueness. The failure of indeterminism, however, teaches us an
important lesson about the dialectical power of the argument from vagueness. An attractive feature
of indeterminism is that it allows us to respond to this without making any substantial claims about
the metaphysics of composition. When confronted with the argument, indeterminists just reject the
third premise, endorse vague existence and keep a moderate metaphysical view about composition.
In fact, the same strategy can be used to deal with the analogue of the argument from vagueness
in other domains. Indeterminism appears thus to be a very convenient way of defending moderate
metaphysics from its critics.
45
Nevertheless, what we learn from the arguments from negative vagueness is that this is not a
viable alternative. If one wants to keep a moderate picture about a certain domain D, it is not
enough to make general claims about vagueness and existence. Rather, one has to engage in
substantial metaphysical debates about D and claim that the moderate picture about D doesn’t lead
to the kind of vagueness that entails the wrong kind of vague existence. The next two chapters take
up such a task in the context of debates about the metaphysics of social objects.
46
Chapter 3: Vagueness and Social Ontology
In Chapter 2, I offered two arguments against restricted composition. According to the first
argument, the argument from negative vagueness, we should reject restricted composition because
it entails that there are cases of negative vague composition. That is:
Negative Vague Composition For some 𝑥𝑥, (i) it is vague whether there is something that
is a fusion of 𝑥𝑥, (ii) nothing is a borderline fusion of 𝑥𝑥,
and (iii) nothing is a fusion of 𝑥𝑥.
Cases of negative vague composition are problematic because they entail that existence is
negatively vague, which, I argued, is an undesirable result.
An important feature of the argument against negative vague composition is that it doesn’t rely
on any assumptions specific to the predicate ‘… is a fusion of …’. Thus, it can be generalized to
any predicate. That is, it entails that, given any predicate ‘𝑃’, the following can’t be the case:
Problematic Vagueness (i) It is vague whether there is something that is a 𝑃, (ii)
nothing is a borderline 𝑃, and (iii) nothing is a 𝑃.
This chapter argues that a moderate ontological picture of the social world leads to instances of
problematic vagueness and explores different strategies to deal with this phenomenon. In section
1, I introduce cases involving social objects that lead to instances of problematic vagueness. In
section 2, I outline two strategies for rejecting problematic vagueness. In section 3, I argue that a
moderate social ontologist can’t appeal to any of them. In section 4, I explore and criticize some
radical implementations of these strategies. The discussion of these views will pave the way for
47
my presentation in Chapter 4 of anchoring plenitude, a maximalist ontology of the social world
that avoids problematic vagueness in a satisfactory manner.
1. Problematic vagueness in the social world
Let me start by considering a rather understudied kind of social object: bank accounts.32 Imagine
a bank whose regulations say: “Once such and such conditions are met, an account is created for
a customer if and only if they have signed form 1A”. Call that bank Sorites Bank. Now consider:
The account series A series of people 𝑐1-𝑐100, each sitting at one of Sorites Bank’s
offices and who are such that, for any 𝑖, 𝑐𝑖 have inscribed exactly 𝑖%
of their signature in their copy of form 1A.
Clearly, for some people in the account series, it is vague whether they have signed form 1A. Given
Sorites Bank’s regulations, that entails that, for those people, it is vague whether there is something
that is an account of theirs at Sorites Bank (for short, ‘whether they have an account’). Call these
people borderline account holders.
In addition, considering the borderline account holders, there doesn’t seem to be anything that
is a borderline bank account of theirs at Sorites Bank (for short ‘they don’t have a borderline
account’). To make this intuition more vivid, consider a different case. Suppose that the regulations
of Sorites Bank also establish that a client transfers their account to someone else if and only both
sign form 1B. Now consider:
32 Not much discussion can be found on the metaphysics bank accounts. Some brief comments in the context
of discussions about the metaphysics of money can be found in Smith & Searle 2003 and Larue 2024.
48
The transfer series A series of people 𝑐1
∗
-𝑐100
∗ with the following features: (i) for any 𝑖,
𝑐𝑖
∗
’s mother wants to transfer her account to 𝑐𝑖
∗
and has inscribed
100% of her signature in her copy of form 1B (ii) for any 𝑖, 𝑐𝑖
∗
has
inscribed exactly 𝑖% of their signature in their copy of form 1B.
Clearly, for some people in the transfer series, it is vague whether both they and their mother have
signed form 1B. Given the regulations of Sorites Bank, that entails that, for those people, it is
vague whether they have an account. However, each of them has a borderline account, i.e., their
mother’s account. By contrast, nothing seems to be a good candidate for playing an analogous role
in the case of borderline account holders in the account series. This is certainly true of the objects
involved in the signing process (the borderline account holders themselves, the piece of paper, the
pen, the bank, etc.) and of other objects both concrete (tables, dogs, etc.) and abstract (properties,
states of affairs, etc.). It is reasonable then to conclude that no borderline account holder has a
borderline account.
So far, we have people in the account series for whom the following two conditions obtain: (i)
it is vague whether they have an account and (ii) they don’t have a borderline account. That is, we
have the first two components of an instance of problematic vagueness involving the predicate ‘…
is an account of … at Sorites Bank’. Before discussing the third component, let me consider
another kind of social object.
Take the case of linguistic expressions.33 Imagine a community, call it 𝑐, whose conventions
dictate that a proper name is created for a baby if and only if one of their parents utters an
appropriate sentence of the form “We hereby name you …”. Now consider:
33 A significant portion of the current debates about the metaphysics of linguistic expression comes from
Kaplan 1990. For a summary, see Miller 2020.
49
The name series A series of babies 𝑑1-𝑑100 with the following features: (i) they are
members of 𝑐 who are in the process of being given a name and (ii)
for any 𝑖, 𝑑𝑖
’s parents have uttered 𝑖% of the relevant sentence of
the form “We hereby name you …”.
Clearly, for some babies in the name series, it is vague whether their parents have uttered the
relevant sentence. Given 𝑐’s conventions, that entails that it is vague whether there is something
that is a name for them 𝑐 (for short, ‘whether they have a name’). Call these people the borderline
name bearers. Borderline name bearers exemplify the first component of an instance of
problematic vagueness involving the predicate ‘… is a name for …’.
What about the second component? My sense is that the intuition that no borderline name
bearer is such that there is something that is a borderline name for them in 𝑐 (for short, ‘has a
borderline name’) is weaker than the intuition no borderline account holders has a borderline
account. In the latter case, no objects suggest themselves as candidates for being their borderline
accounts. By contrast, one might be tempted to say that, when someone is given a name, a certain
sound type becomes their name. Thus, when a baby is a borderline name bearer, there is a sound
type that is in the process of becoming their name and that sound is their borderline name. In §
3.2., I will come back to views like this one. For now, I will say that, modulo such views, borderline
name bearers don’t have borderline names. That is, they exemplify the second component of an
instance of problematic vagueness involving ‘… is a name for …’.
I shall now argue that some borderline account holders and some borderline name bearers also
exemplify the third component of an instance of problematic vagueness involving the relevant
predicates. My first argument is analogous to the one I used in Chapter 2 (§ 4.1) to argue for the
existence of cases of negative vague composition given the existence of cases of vague
50
composition. Recall that 𝑥 is a positive (negative) borderline case of ‘𝑃’ just in case (i) it is vague
whether 𝑥 is 𝑃 and (ii) 𝑥 is (not) 𝑃. It strikes me as very plausible that a vague predicate like ‘…
has signed form 1A’ has both positive and negative borderline cases. Given the determinate
connection between ‘… has signed form 1A’ and ‘… has an account’, that entails that the latter
also has both positive and negative borderline cases. Call the negative borderline cases negative
borderline account holders. Since every borderline account holder lacks a borderline account,
negative borderline account holders exemplify the three components of an instance of problematic
vagueness involving the predicate ‘… is an account of …’. A parallel argument shows that negative
borderline name bearers exemplify the three components of an instance of problematic vagueness
involving the predicate ‘… is a name for …’.
As before, to resist this argument, the objector would need to claim that any predicate that can
be determinately correlated with ‘… has an account’ or ‘… has a name’ has only positive borderline
cases. The situation here is worse than in the case of composition, since virtually any vague
predicate can play such a role. To make things worse, one can imagine peculiar scenarios where
‘… has an account’ is determinately correlated not with ‘… has signed form 1A’, but with its
negation, i.e., ‘… hasn’t signed form 1A’. Given their prior commitment, the objector would then
have to say that every borderline account holder in the new series is a negative borderline account
holder. Thus, their position is not only implausible but self-defeating.
We have seen that a moderate conception of bank accounts and linguistic expressions leads to
instances of problematic vagueness. The problem can be generalized in a straightforward way to
other social objects. For some of them, like for accounts, our intuitions will be very strong. For
others, like for linguistic expressions, less so. I’ll call objects of the first kind type A social objects
51
and objects of the second kind type B social objects. Here are more examples of both kinds of
objects (for type B objects, I include the objects with which they could in principle be identified):
Type A: corporations, governments, fictional characters, email accounts, positions (e.g.,
manager, president, etc.)
Type B: social groups [sets], musical works [sound types], dollar bills [pieces of paper], laws
[propositions], borders [space regions]
In sum, a moderate conception of social objects leads to widespread problematic vagueness. In the
next section, I outline two different strategies for dealing with this phenomenon. After that, I
discuss various ways of implementing those strategies.
2. Rejecting problematic vagueness
Let me distinguish two different strategies for dealing with a given instance of problematic
vagueness. Whereas the determinate instantiation strategy rejects the first component of
problematic vagueness, e.g., that there are borderline account holders/name bearers, the borderline
instances strategy rejects the second component and claims that borderline account holders/name
bearers have borderline accounts/names.
To explain the determinate instantiation strategy, let’s consider the account series. The
determinate instantiator claims that there are no borderline account holders. Our judgment that
some people in the account series are borderline account holders is motivated by the judgment that
those people are borderline cases of ‘… has signed form 1A’ and some principle like the following:
52
Account Existence For any 𝑥, it is determinate that [𝑥 has an account iff 𝑥 has signed form
1A].34
Our belief in Account Existence derives from what is stated at the regulation of Sorites Bank.
Since it is uncontroversial that there are borderline cases of ‘… has signed form 1A’, to deny
the existence of borderline account holders, the determinate instantiator must reject Account
Existence. Given the plausibility of this principle, he owes us a good explanation for why he rejects
it. This explanatory debt extends to analogous principles governing other social objects. I will
discuss some possible explanations when I look at specific versions of this strategy.
Let’s now move to the borderline instances strategy. Take now the name series. The friend of
borderline instances accepts that there are borderline name bearers in the series and claims that
they have borderline names. Plausibly, she wants to preserve the analogue of Account Existence
for the case of names (where ‘… has been baptized’ abbreviates ‘…’s parents have uttered the
relevant sentence’) :
Name Existence For any 𝑥, it is determinate that [𝑥 has a name iff 𝑥 has been baptized].
To further illustrate the commitments of the borderline instances strategy, let’s assume that the
precisifications of ‘… has been baptized’ are properties of the form your parents have uttered at
least 𝑛% percent of the relevant sentence (for short, ‘𝑈𝑛’). For concreteness, let’s say that such
precisifications are 𝑈60-𝑈80 and that 𝑈70 is the privileged precisification. I make a similar
stipulation about ‘… has signed form 1A’, whose precisifications are properties of the form having
completed at least 𝑛% of your signature (for short, ‘𝑆𝑛’). More specifically, such precisifications
34 This reasoning is valid given K-Det.
53
are 𝑆60-𝑆80. Among them, 𝑆70 is the privileged precisification. This allows for the following
classification of people in the account and name series:
Table 1: Distribution of cases
Negative cases 𝑐1-𝑐69
𝑑1-𝑑69
𝑐1-𝑐59
𝑑1-𝑑69
Negative determinate
cases
𝑐60-𝑐69
𝑑60-𝑑69
Negative borderline
cases
Positive cases 𝑐70-𝑐100
𝑑70-𝑑100
𝑐70-𝑐79
𝑑70-𝑑79
Positive borderline cases
𝑐80-𝑐100
𝑑80-𝑑100
Positive determinate
cases
The friend of borderline instances claims that 𝑑60-𝑑79, the borderline cases of ‘… has been
baptized’ in the name series, (i) are borderline name bearers and (ii) have borderline names. Among
them, 𝑑70-𝑑79 are are borderline name bearers who have borderline names but don’t have names.
For 𝑦 to be a borderline name for 𝑥 is for it to be vague whether 𝑦 is a name for 𝑥. Given the
way I have introduced precisifications (Chapter 1, section 1), in order for that to be the case, the
predicate ‘… is a name for …’ must have multiple precisifications. A natural way of developing
the borderline instances strategy is to claim that these precisifications are connected to
precisifications of ‘… has been baptized’, i.e, to 𝑈60-𝑈80. The following notion will help clarify
this connection:35
35 I borrow the term ‘anchoring’ from Epstein 2015.
54
Anchoring A property 𝐹 anchors a relation 𝑅 just in case, for any 𝑥, there is something
that is an 𝑅 of 𝑥 iff 𝑥 is 𝐹.
The friend of borderline instances believes that each of 𝑈60-𝑈80 anchors a relation and that each
of those relations, call them 𝑁60-𝑁80, is a precisification of ‘… is a name for …’. In addition, since
𝑈70 is the privileged precisification of ‘… has been baptized’, 𝑁70 is the privileged precisification
of ‘… is a name for …’.
Our judgment that the borderline name bearers don’t have borderline names comes from our
difficulty in finding an object that could play that role. Thus, the friend of borderline instances
should be able to tell a good story about what kind of object is a borderline name and about what
kind of relations are 𝑁60-𝑁80. As I anticipated, this is not an impossible challenge. Things will get
trickier with type A social objects.
Let’s now look at specific implementations of these strategies.
3. Moderate responses
A moderate approach to the social world accepts the existence of both type A and type B social
objects, but denies that of objects belonging to “strange” social kinds, i.e., objects that would be
the analogue of the arbitrary fusions accepted by the compositional universalist. This section
discusses three views that result from combining this approach with one of the strategies just
discussed. I start with a view based on the determinate instantiation strategy.
3.1. Determinate moderatism
Consider the account series. When confronted with such a series, a moderate social ontologist
accepts that some people lack an account, and some have one. He says, for instance, that 𝑐1-𝑐69
55
lack an account and 𝑐70-𝑐100 have one. Since he wants to adopt the determinate instantiation
strategy, the moderate social ontologist can’t admit borderline account holders, so he must claim
that 𝑐1-𝑐69 determinately lack an account and 𝑐70-𝑐100 determinately have one. Let’s call this view
determinate moderatism.
As I mentioned before, a determinate moderate must reject Account Existence. So, on his view,
whether someone has an account doesn’t determinately depend on whether they sign form 1A. We
can ask the determinate moderate then what the conditions under which someone has an account
are and why they are always determinate. I claim that he cannot offer a plausible answer to this
question.
To see why, compare the case of the determinate moderate about accounts with that of the
proponent of determinate cut-off points in the chair series we encounter in Chapter 1 (§ 3.1). When
it comes to composition, instances of problematic vagueness have the following form: (i) it is
vague whether 𝑥𝑥 have a fusion, (ii) nothing is a borderline fusion of 𝑥𝑥, and (iii) 𝑥𝑥 don’t have a
fusion. As I explained, philosophers like Merricks, Markosian and Hawthorne reject the first
component of these claims and maintain that whether 𝑥𝑥 have a fusion is always a determinate
matter. In addition, they claim that our intuition that some 𝑥𝑥 are a borderline case of ‘… have a
fusion’ comes from our ignorance about the instantiation of some fundamental property that
distinguishes pluralities that have fusions from pluralities that don’t.
Whatever the merits of such views are for the composition case, there doesn’t seem to be too
much hope for their social counterpart, i.e., determinate moderatism. To replicate their move, a
determinate moderate would have to claim that our intuition that some people in the account series
are borderline cases of ‘… has an account’ is due to our ignorance about the instantiation of a
fundamental property that distinguishes those who have accounts from those who don’t. It is highly
56
implausible that there is such a fundamental property. This result carries over to determinate
moderatism about other social objects, since it is equally implausible to claim that the relevant
predicates are appropriately connected to fundamental properties.
I take this to be a compelling reason against determinate moderatism. Therefore, I conclude
that the moderate social ontologist cannot reject instances of problematic vagueness by adopting
the determinate instantiation strategy.
3.2. Social reductionism
The moderate social ontologist cannot be a determinate instantiator. So, she must accept the first
and third components of problematic vagueness and reject the second. As I explained before, if
one does so, one must provide a plausible story about the kind of objects these borderline instances
are and about the kind of properties that make them borderline instances. To remain a moderate
ontologist, the friend of borderline instances must identify these borderline social objects with
familiar objects. Since, as we shall see, such objects are non-social objects, I shall call this
approach social reductionism.
Consider the name series. The social reductionist claims that borderline name bearers have
borderline names. In answering the question what kind of object this borderline is, the social
reductionist can appeal to metaphysical theories according to which linguistic expressions are
abstract types that exist independently of our social practices.36 On such a picture, the process
whereby someone goes from not having a name to having one is the process whereby an abstract
type goes from not being a name for them to being one.
36 For views along these lines, see Katz 1980 and Wetzel 2009.
57
For any 𝑖, let 𝑇𝑖 be the abstract type that is in the process of becoming a name for 𝑑𝑖
. If 𝑑𝑖
is a
borderline name bearer, then it is vague whether 𝑇𝑖
is a name for 𝑑𝑖
, which amounts to saying that
some but not all the precisifications of ‘… is a name for ..’, 𝑁60-𝑁80, obtain between 𝑇𝑖 and 𝑑𝑖
. In
general, the social reductionist classifies people in the name series as follows (where ‘𝑗’ is a
variable that ranges over numbers between 60 and 80):
Table 2: Distribution of cases (social reductionism)
As we can see, 𝑇60-𝑇79 are borderline names of 𝑑60-𝑑79, the borderline name bearers. Among
them, 𝑇70-𝑇79 are indeed names of 𝑑70-𝑑79, the positive borderline name bearers, whereas 𝑇60-𝑇69
are not names of 𝑑60-𝑑69, the negative borderline name bearers. This is because the privileged
precisification 𝑁70 obtains only among the former, not among the latter.
Without attempting to provide a full theory about 𝑁60-𝑁80, it is helpful to at least say a few
words about what kind of relations they might be. Suppose that whether a sound type is a name
for a person in 𝑐 is a matter of how many people in 𝑐 have a disposition to utter a token of that
Negative determinate cases 𝑑1-𝑑59
For any 𝑗, 𝑇𝑖
is not an 𝑁𝑗
for 𝑑𝑖
.
It is determinate that 𝑇𝑖
is not a name for 𝑑𝑖
.
Negative borderline cases 𝑑60-𝑑69
𝑇𝑖
is not a 𝑁70 for 𝑑𝑖
. For some 𝑗 and some 𝑗
′
, 𝑇𝑖
is
an 𝑁𝑗
for 𝑑𝑖 and 𝑇𝑖
is not an 𝑁𝑗
′ for 𝑑𝑖
.
𝑇𝑖
is not a name for 𝑑𝑖 and it is vague whether 𝑇𝑖
is
a name for 𝑑𝑖
.
Positive borderline cases 𝑑70-𝑑79
𝑇𝑖
is a 𝑁70 for 𝑑𝑖
. For some 𝑗 and some 𝑗
′
, 𝑇𝑖
is an
𝑁𝑗
for 𝑑𝑖 and 𝑇𝑖
is not an 𝑁𝑗
′ for 𝑑𝑖
.
𝑇𝑖
is a name for 𝑑𝑖 and it is vague whether 𝑇𝑖
is a
name for 𝑑𝑖
.
Positive determinate cases 𝑑80-𝑑100
For any 𝑗, 𝑇𝑖
is an 𝑁𝑗
for 𝑑𝑖
.
It is determinate that 𝑇𝑖
is a name for 𝑑𝑖
.
58
type to refer to that person. Then, each of 𝑁60-𝑁80 can be identified with relations of the form
being a sound type 𝑠 such that 𝑘% of 𝑐 are disposed to use 𝑠 to refer to.
Reductionist views can and have been articulated for other type B social objects. For instance,
philosophers have argued that social groups are sets of people, that musical works are sound types,
etc.37 When confronted with the relevant instance of problematic vagueness, these authors describe
the case as one where a certain non-social object is a borderline case of a socially relevant
predicate, such as ‘… is a group whose members are …’ or ‘… is a song composed by …’.
The first problem for social reductionism is that it is not clear that such a view could be
extended to type A social objects. For instance, there doesn’t seem to be any non-social object
sufficiently similar to an account so as to claim that they are identical. Thus, the moderate social
ontologist would have to find another way for dealing with instances of problematic vagueness
involving type A social objects. The second problem for this view is that, even if we find suitable
non-social objects to deal with type A social objects, it predicts that no object is social in the sense
of having existence conditions related to our social practices. Indeed, John Searle (1995, 2009) is
usually associated with a view like this. On his account, everything we consider a social object
just is a non-social object that has acquired a certain social status. His classic example is that of a
piece of paper that acquires the status of being a $1 bill.
I don’t take these problems to be fatal. However, I think they give us enough reason to look for
alternative approaches. This concludes my discussion of moderate versions of the borderline
instances strategy. Since we had similar results for the determinate instantiation strategy, I
37 For the view that groups are sets, see Effingham 2010. Dodd 2007 adopts a reductionist view about
musical works.
59
conclude that moderate social ontologists lack the resources to deal with problematic vagueness,
which gives us good reason to explore radical alternatives.
4. Radical responses
This section introduces some radical ontological pictures of the social world. I start with two
versions of the determinate instantiation strategy and then move to a version of the borderline
instances strategy.
4.1. Determinate radicalisms
The determinate instantiator claims that there are no borderline account holders, no borderline
name bearers, etc. In § 3.1, we encountered a moderate version of such a character. Given his
commitment to moderate ontology, he claimed that the account series contains both positive
determinate cases and negative determinate cases. Absent such a commitment, a determinate
instantiator is free to adopt other alternatives. In particular, he can say either that every person in
the account series determinately lacks an account (account nihilism) or that all of them
determinately have one (account universalism).
Let’s start with account nihilism. Unless it is argued that cases involving other banks are
relevantly different, it is natural to construe account nihilism as the view that there are no bank
accounts in general. Similar views can be formulated about other social objects, both type A and
type B. Social nihilism is the conjunction of all such views. That is, the claim that there are no such
things as bank accounts, governments, linguistic expressions, songs, etc.
Social nihilism faces the same challenges as compositional nihilism, namely that it seems to
predict that we are massively mistaken about the world. Compositional nihilists (e.g., Dorr 2005,
Sider 2013) have developed different strategies to deal with this challenge. As far as I can tell,
60
social nihilists should have no problem adopting any of them. Thus, social nihilism is at least as
plausible as compositional nihilism.38 I regard a nihilist approach to social objects as a last
resource. Thus, I shall continue looking for a plausible, realist alternative.
Consider now account universalism. Such a view entails that even people who haven’t signed
form 1A have accounts. One can generalize this view and claim that there are such things as
linguistic expressions, governments, constitutions, etc., even when their intuitive existence
conditions are not satisfied. Call the resulting view social universalism.
Universalist views in the social domain are non-starters. Their main problem, to be clear, is not
that they are committed to too many objects. After all, we already know that a moderate
metaphysical picture can’t avoid problematic vagueness. Rather, their problem lies in the kind of
objects they are committed to. In the compositional case, universalists accept the existence of
unfamiliar objects, i.e., arbitrary fusions like a fusion of my laptop and your coffee mug. However,
they are also in a position to theorize about them, for instance, by formulating mereological
systems. It is, to say the least, unclear whether the social universalist can do something like that.
This concludes my discussion of radical implementations of the determinate instantiation
strategy.
4.2. Necessitism
I turn now to radical implementations of the borderline instance strategy. Unlike their moderate
counterparts, these views don’t identify borderline social objects with familiar objects. Rather, they
38 Social nihilism might actually be more plausible than compositional nihilism. In his recent defense of
compositional nihilism, Sider considers the apparent indispensability of composite objects for science as
the biggest challenge for compositional nihilism (Sider 2013: §11). No analogous challenge arises for social
nihilism.
61
invoke objects belonging to “strange” kinds. The first view we will explore identifies borderline
social objects with “possible” objects.
Necessitism is a view in the metaphysics of modality that holds that which objects exist is not
a contingent matter. Consequently, if there could have been an 𝐹, there is something that could
have been an 𝐹. This is a particularly striking claim when 𝐹 is replaced by something like being a
biological child of Freddie Mercury (for short, ‘being a Mercury child’). Mercury never had
children, but he could have. So, there could have been a Mercury child. On a necessitist picture,
this entails that there is an object that could have been a Mercury child, even though it is not a
Mercury child.39
Necessitism has featured occasionally in discussion of the argument from vagueness
(Williamson 2013: ch 1, fn 9; Korman & Carmichael 2016: § 4.2). I here attempt to articulate a
necessitist approach to problematic vagueness in the social world. The initial thought goes as
follows. Take the borderline account holders. Regardless of whether they have an account or not,
they could have had one. So, given necessitism, for each of them, there is something that could
have been an account of theirs. Call an object like that a possible account of theirs.40 The idea is
then to identify borderline accounts with possible accounts. When it comes to 𝑐70-𝑐79 (i.e., the
positive borderline account holders), a possible account can just be an account. However, when it
comes to 𝑐60-𝑐69 (i.e., the negative borderline account holders), we have what we may call merely
possible accounts. That is, objects that are not accounts but could have been accounts.
39 Williamson 2013 defends necessitism at length. Goodman 2016 provides a different argument.
40 I avoid locutions of the form “their possible account” because there typically are more than one. Plausibly,
I could have had more than one account. Given the necessity of identity, necessitism entails that I have at
least two possible accounts.
62
As I mentioned before, a friend of the borderline instances strategy owes us a story about what
kind of objects these borderline accounts are and what makes them borderline accounts. Let’s see
how a necessitist might provide such a story.
Start with a merely possible Mercury child. When confronted with the question of what kind
of object this is, necessitists claim that it is a non-concrete object. They also clarify that being nonconcrete doesn’t entail being abstract. Abstract objects such as numbers, they say, “play specific
theoretically defined roles” (Williamson 2013: 8), whereas a merely possible Mercury child has
no theoretically defined role to play. Things are slightly more complicated when it comes to merely
possible accounts. Accounts seem to be non-concrete objects already. Some may even argue that
they are abstract objects in Williamson’s sense. So, the question arises what distinguishes a merely
possible account from an actual account. To answer this question, necessitists introduce the
concept of a chunky object. A chunky object is an object that exists in the intuitive sense, whether
it is concrete or abstract. Every other object the necessitist posits (merely possible Mercury
children, merely possible accounts, etc.) is non-chunky.
With the chunky/non-chunky distinction at hand, a necessitist might tell the following story.
For simplicity, I assume that, for each member of the series, there is an object that either is or will
be its account, whether it is currently chunky or not. When someone doesn’t have an account, this
object is non-chunky, whereas when they have an account, this object is chunky. When it is vague
whether someone has an account, it is vague whether such an object is chunky. In slogan form, it
is not the border between non-existence and existence that is vague, but the border between nonchunkiness and chunkiness. A similar story can be told about determinate cases, both positive and
negative (where 𝛽𝑖
is the thing that either is or will be 𝑐𝑖
’s account):
63
Table 3: Distribution of cases (necessitism)
Negative
determinate cases 𝑐1-𝑐59
For any 𝑗, 𝑐𝑖 doesn’t have a 𝐵𝑗
.
For any 𝑗, 𝛽𝑖
is a not 𝐵𝑗 of 𝑐𝑖
.
Negative borderline
cases
𝑐60-𝑐𝑐9
𝑐𝑖 doesn’t have a 𝐵70. For some 𝑗 and some
𝑗
′
, 𝑐𝑖 has a 𝐵𝑗 and 𝑐𝑖 doesn’t have a 𝐵𝑗
′ .
𝛽𝑖
is not a 𝐵70 of 𝑐𝑖
. For some 𝑗 and some 𝑗
′
,
𝛽𝑖
is a 𝐵𝑗 of 𝑐𝑖 and 𝛽𝑖
is not a 𝐵𝑗
′ of 𝑐𝑖
.
Positive borderline
cases
𝑐70-𝑐79
𝑐𝑖 has a 𝐵70. For some 𝑗 and some 𝑗
′
, 𝑐𝑖 has a
𝐵𝑗 and 𝑐𝑖 doesn’t have a 𝐵𝑗
′ .
𝛽𝑖
is a 𝐵70 of 𝑐𝑖
. For some 𝑗 and some 𝑗
′
, 𝛽𝑖
is a 𝐵𝑗 of 𝑐𝑖 and 𝛽𝑖
is not a 𝐵𝑗
′ of 𝑐𝑖
.
Positive determinate
cases
𝑐80-𝑐100
For any 𝑗, 𝑐𝑖 has a 𝐵𝑗
.
For any 𝑗, 𝛽𝑖
is a 𝐵𝑗 of 𝑐𝑖
.
Among the borderline account holders, the necessitist can distinguish between positive (𝑐70-𝑐79)
and negative (𝑐60-𝑐69). If 𝑐𝑖
is a negative borderline account holder, then 𝛽𝑖
is only a merely
possible account of 𝑐𝑖
’s.
There are different reasons why one might find the necessitist approach to problematic
vagueness unattractive. For one thing, one might think that the necessitist commitment to nonchunky objects is a hard pill to swallow. These criticisms target the core necessitist claim. Thus,
whether they succeed or not will depend to a great extent on how strong the motivations for
necessitism are. Though I am not entirely convinced by the arguments in favor of necessitism, I
don’t want to rely on this kind of objection.
A better motivation to explore alternative pictures comes from the fact that the necessitist
approach to problematic vagueness requires that there be borderline cases of ‘... is chunky’.
64
Chunkiness is the necessitist replacement for our intuitive notion of existence. So, to say that
chunkiness is vague is to say that intuitive existence is vague. Though it is not clear that the main
arguments against vague existence carry over to vague chunkiness, it would be preferable to have
an approach to problematic vagueness that doesn’t require chunkiness to be vague.41
This concludes my discussion of various ontological pictures that attempt to deal with
problematic vagueness in the social world. In the next chapter, I present a maximalist version of
the borderline instances strategy.
41 There are two kinds of arguments against vague existence. One relies on the fact that existence is a
fundamental property and hence shouldn’t be vague. This line of argument would carry over to chunkiness.
The other relies on logical features of the unrestricted existential quantifier. It is not clear that that would
carry over to chunkiness.
65
Chapter 4: Anchoring Plenitude
In Chapter 3, I argued that a moderate conception of the social world leads to widespread
problematic vagueness. For example, it entails that some members of the account series exemplify
problematic vagueness involving the predicate ‘… is an account of …’. As we saw, we have good
reason to accept the first and third components of the relevant instances of problematic vagueness.
Thus, we must reject the second component. For example, we must say that some people in the
account series are borderline account holders who don’t have accounts but have borderline
accounts. This is what I called the borderline instances strategy.
So far, I have discussed versions of the borderline instances strategy that identify borderline
social objects with familiar non-social objects (social reductionism) and with merely possible
social objects (necessitism). For instance, account reductionists identify borderline accounts with
abstract objects and necessitists with merely possible accounts. As we saw, these views have their
shortcomings. In this chapter, I introduce my preferred ontology of the social world. I call this
view anchoring plenitude. Anchoring plenitude is a radical version of the borderline instances
strategy that countenances myriads of non-ordinary social objects, some of which can be identified
with borderline social objects.
The chapter proceeds as follows. In section 1, I introduce the main commitments of anchoring
plenitude. I use the case of bank accounts as an illustration and then present a more general version
of the view. In section 2, I show that anchoring plenitude avoids instances of problematic
vagueness in a satisfactory way. In section 3, I respond to an objection that arises from the
observation that predicates such as ‘… has signed form 1A’ are higher-order vague. Section 4
concludes with some reflections on the connections between anchoring plenitude and other
maximalist ontologies.
66
1. Introducing anchoring plenitude
This section is divided into two subsections. In the first one, I illustrate the main commitments of
anchoring plenitude by considering the case of bank accounts. In the second one, I generalize this
approach and present anchoring plenitude as a general ontology of the social world.
1.1. Account plenitude
In Chapter 3 (section 2), I introduced the concept of anchoring. A property 𝐹 anchors a relation 𝑅
just in case, for any 𝑥, 𝑥 has an 𝑅 iff 𝑥 instantiates 𝐹. I also claimed that friends of the borderline
instances strategy must believe that every precisification of ‘… has signed form 1A’, 𝑆𝑖
, anchors a
relation, 𝐵𝑖
, which is a precisification of ‘… is an account of …’. Since ‘… has signed form 1A’
has 𝑆60-𝑆80 as precisifications, ‘… is an account of …’ has 𝐵60-𝐵80 as precisifications.
The application of anchoring plenitude to the case of bank accounts, call it account plenitude,
shares this commitment. What is distinctive about account plenitude is its conception of 𝐵60-𝐵80.
Let me explain. From an intuitive perspective, if someone in the account series has an account, the
existence of such an object “depends” on their having signed form 1A. One way of understanding
this idea goes as follows: if we keep fixed the regulations of Sorites Bank, then the object that is
your account wouldn’t have existed if you hadn’t signed form 1A. In this scenario, I say that the
relation being an account of is generative with respect to the account series. Here is a more precise
definition of the notion of a generative relation (where “necessarilySB, 𝑝” must be read as
“necessarily, if the regulations of Sorites Bank are the same, then 𝑝”):
Generative relations A relation 𝑅 is generative with respect to the account series just in
case, for any 𝑥 in the account series, for any 𝑦, if 𝑦 is an 𝑅 of 𝑥, then,
necessarilySB, 𝑦 exists iff 𝑦 is an 𝑅 of 𝑥.
67
It will be helpful to also introduce the following terminology. Say that 𝑥 is 𝑅-generated for 𝑦 or
that 𝑅 generates 𝑥 for 𝑦 just in case (i) 𝑅 is a generative relation and (ii) 𝑥 is an 𝑅 of 𝑦. Also, say
that an object 𝑥 is generated when, given a contextually relevant class of generative relations and
a contextually relevant object 𝑦, 𝑥 bears some 𝑅 in that class to 𝑦.
On both social reductionism and necessitism, 𝐵60-𝐵80 fail to be generative relations with
respect to the account series. Here is why. Consider 𝑐65, who has completed 65% of their signature.
They instantiate 𝑆65. So, they have a 𝐵65. A social reductionist would identify such an object with
some kind of non-social abstract object. Call that abstract object 𝐴. To see that 𝐵65 is not a
generative relation, consider a possibleSB world where 𝑐65 has only completed 40% of their
signature. In such a world, 𝐴 would still exist. However, 𝐴 wouldn’t be a 𝐵65 of 𝑐65, since 𝑐65
wouldn’t have a 𝐵65 in that world. So, 𝐵65 is not a generative relation on the social reductionist
picture. The same holds for all of 𝐵60-𝐵80. The same conclusion holds on a necessitist picture.
42
Things look different from the perspective of account plenitude. On this picture, 𝐵60-𝐵80 are
generative relations. Thus, any of 𝐵60-𝐵80 can generate an object for someone in the series. Here
is an example. Consider again 𝑐65 and their 𝐵65-generated object. Call such an object 𝑏. Since 𝐵65
is a generative relation, necessarilySB, 𝑏 exists iff 𝑏 is a 𝐵65 of 𝑐65. The same goes for all other
relations in 𝐵60-𝐵80. Since 𝑐65 also instantiates 𝑆60-𝑆64, they also have something 𝐵60-generated,
something 𝐵61-generated, …, and something 𝐵64-generated. All such objects, necessarilySB, exist
iff they are, respectively, a 𝐵60, a 𝐵61, …, and a 𝐵64 of 𝑐65.
What makes this approach plenitudinous is the fact that, for any 𝑖 and 𝑗, if 𝑖 ≠ 𝑗, then nothing
can be both 𝐵𝑖
-generated and 𝐵𝑗
-generated for the same person in the account series. Here is why.
42 Indeed, given necessitism, only relations that are necessarily instantiated by the relevant objects could be
generative. Otherwise, at least some objects would exist only contingently, a result that is unacceptable by
necessitist lights.
68
Assume, without loss of generality, that 𝑖 > 𝑗. Suppose for reductio that 𝑏 is both 𝐵𝑖
-generated and
𝐵𝑗
-generated for 𝑎. Since 𝐵𝑖 and a 𝐵𝑗 are generative relations with respect to the account series:
1. NecessarilySB, 𝑏 exists iff 𝑏 is a 𝐵𝑖 of 𝑎.
2. NecessarilySB, 𝑏 exists iff 𝑏 is a 𝐵𝑗 of 𝑎.
Given what the bank’s regulations say, we know that:
3. NecessarilySB, 𝑏 is a 𝐵𝑖 of 𝑎 iff 𝑎 instantiates 𝑆𝑖
.
4. NecessarilySB, 𝑏 is a 𝐵𝑗 of 𝑎 iff 𝑎 instantiates 𝑆𝑗
.
From (1) and (3), we can derive (5), and from (2) and (4), we can derive (6):
5. NecessarilySB, 𝑏 exists iff 𝑎 instantiates 𝑆𝑖
.
6. NecessarilySB, 𝑏 exists iff 𝑎 instantiates 𝑆𝑗
.
From (5) and (6), it follows:
7. NecessarilySB, 𝑎 instantiates 𝑆𝑖
iff 𝑎 instantiates 𝑆𝑗
.
This is an undesirable conclusion, since it clearly is possibleSB for 𝑎 to instantiate 𝑆𝑗 without
instantiating 𝑆𝑖
. That is, it is possibleSB for 𝑎 to have completed 𝑗% of their signature without
having completed 𝑖% of their signature. Thus, I conclude that, if 𝑖 ≠ 𝑗, then nothing is both 𝐵𝑖
-
generate and 𝐵𝑗
-generated for someone in the account series.
Thus, according to account plenitude, each of 𝑆60-𝑆80 anchors a different generative relation.
The further we move along the account series the more objects we can find that have been
69
generated for the relevant person. Each of these objects has been generated by exactly one of 𝐵60-
𝐵80.
1.2. Anchoring plenitude
The existence of such things as bank accounts, clubs, governments, linguistic expressions, etc.
depends on our social practices. One way of understanding this dependence is via principles like
Account Existence, which give conditions under which there are things of the relevant.
Call such principles existence principles. Two features of existence principles are of crucial
importance. First, existence principles are true in virtue of our social practices. In other words, we
make it so that they are true. To communicate this idea, I will say that we put in place existence
principles. Second, in existence principles, we can distinguish between an anchoring predicate and
an anchored predicate. For instance, in Account Existence, the anchoring predicate is ‘… has
signed form 1A’ and the anchored predicate is ‘… has an account’.
With this terminology at hand, we can begin to articulate a general version of anchoring
plenitude. First, we identify a general form for existence principles:
For any 𝑣1…𝑣𝑛, there is a 𝐾 of 𝑣1…𝑣𝑛 iff 𝜙𝑣1…𝑣𝑛
In the specific case of Account Existence, only one individual variable is needed. However, other
cases might differ. For instance, one can have an existence principle for groups where the relevant
variables are plural variables, or one for borders, where we need two individual variables, one for
each side of the border.
Given an existence principle of that form, anchoring plenitude tells us that, if the anchoring
predicate ‘𝜙’ is vague, then each precisification 𝜙𝑖 of ‘𝜙’ anchors a generative relation 𝐾𝑖 which
is a precisification of the anchored predicate ‘𝐾’. As I explained above, when the relevant entities
70
instantiate more than one precisification of ‘𝐾’, then anchoring plenitude predicts that many
objects have been generated for them, one for each such precisification.
2. Anchoring plenitude and problematic vagueness
I shall now discuss how anchoring plenitude deals with problematic vagueness. I shall use the bank
account case as an example, but my remarks can be easily generalized to other cases.
Being a version of the borderline instances strategy, account plenitude accepts Account
Existence. Thus, the borderline cases of ‘… has signed form 1A’ in the account series are also
borderline cases of ‘… has an account’, i.e., borderline account holders. The same goes for the
determinate cases both positive and negative.
What is peculiar about anchoring plenitude is the inclusion of generated objects in both
borderline cases and determinate positive cases. Here is the distribution of people in the account
series according to anchoring plenitude:
Table 4: Distribution of cases (account plenitude)
Determinate
negative cases 𝑐1-𝑐59
For any 𝑗, nothing has been 𝐵𝑗
-generated for 𝑐𝑖
For any 𝑗, 𝑐𝑖 doesn’t have a 𝐵𝑗
.
Borderline
cases
𝑐60-𝑐79
For some 𝑗 and some 𝑗
′
, something has been 𝐵𝑗
-generated
for 𝑐𝑖 and nothing has been 𝐵𝑗
′ -generated for 𝑐𝑖
.
For some 𝑗 and some 𝑗
′
, 𝑐𝑖 has a 𝐵𝑗 and 𝑐𝑖 doesn’t have a
𝐵𝑗
′ .
Positive
determinate
cases
𝑐80-
𝑐100
For any 𝑗, something has been 𝐵𝑗
-generated for 𝑐𝑖
.
For any 𝑗, 𝑐𝑖 has a 𝐵𝑗
.
Let’s focus on the borderline cases. Take, for instance, 𝑐65. In this case, let 𝑗 = 65 and 𝑗
′ = 66.
Something has been 𝐵65-generated for 𝑐65, but nothing has been 𝐵66-generated for 𝑐65. So, 𝑐65
71
has a 𝐵65 but not a 𝐵66. Since both relations are precisifications of ‘… is an account of’. We can
conclude that it is vague whether 𝑐65 has an account.
Since 𝑐65 is a borderline case, it must have borderline accounts. To see that this is indeed the
case on the anchoring plenitude picture, take any of 𝑐65’s generated objects. Call it 𝑏. 𝑏 has been
generated by exactly one of 𝐵60-𝐵80. So, for some 𝑗, 𝑏 is a 𝐵𝑗 of 𝑐65. For any other 𝑗
′
, 𝑏 is not a
𝐵𝑗
′ of 𝑐65. So, 𝑏 is a borderline account of 𝑐65. The same goes for any borderline case in the
account series. They all have generated objects, which are borderline accounts. So, they have
borderline accounts. Unlike the borderline accounts of social reductionism or necessitism, the
borderline accounts countenanced by anchoring plenitude are social objects in the sense that their
existence is dependent on our social practices.
Let me conclude this section by identifying a surprising consequence of anchoring plenitude:
even though some people in the series determinately have an account, no one has a determinate
account. Recall that no object can bear more than one of 𝐵60-𝐵80 to a person in the series. Now,
since 𝐵60-𝐵80 are the precisifications of ‘… is an account of …’, that means that no object bears
every precisification to a person in the series. So, for any object 𝑥 and any person in the series 𝑦,
it is not determinate that 𝑥 is an account of 𝑦. This result seems to conflict with our intuitive
judgments. After I have completed my signature, it seems that I can say things like “That is my
account”, which seem to require that there be an object that is a determinate account.
A similar challenge arises for some approaches to the problem of the many that admit that, for
instance, many aggregates of water droplets are borderline clouds but none of them is a determinate
cloud. The problem for these approaches is that it seems intuitive that we can say things like “That
is a cloud”, which seems to require that there be a determinate cloud.
72
The solution in both cases is to postulate penumbral connections between the use of the
demonstrative ‘that’ and that of the relevant predicate. On this picture, the sentence “that is a
cloud/my account” is determinately true, not because there is one single object that instantiates
every precisification of ‘… is a cloud’/‘… is my account’. Rather, the precisifications that make
the sentence determinately true are propositions that result from predicating each precisification
of the relevant predicate of an object that indeed instantiates that precisification. Thus, the sentence
is guaranteed to be determinately true.
3. An objection
I turn now to an objection against anchoring plenitude. In his discussion of the argument from
vagueness for abstract artifacts (Korman 2014), Korman discusses a view in the same spirit as
anchoring plenitude and formulates an objection against it. In this section, I articulate a version of
Korman’s objection that directly targets anchoring plenitude and respond to it. As we shall see,
this objection will require a slight revision of the original formulation of anchoring plenitude.
Let’s focus on account plenitude for concreteness. Account plenitude entails that 𝑐60-𝑐80 have
𝐵𝑗
-generated objects and 𝑐1-𝑐59 don’t. The first claim is a key component of the view and the
second follows from the fact that 𝐵60-𝐵80 are anchored by 𝑆60-𝑆80, which are not instantiated by
𝑐1-𝑐59. Korman argues that this result is problematic. In particular, he argues that the friend of
account plenitude faces a dilemma:
So, assuming […] the plenitude of [generated objects] gets started somewhere in [the account
series], either it is indeterminate when it gets started or it gets started at some unremarkable
exact point. If on the one hand the plenitude gets started at some unremarkable exact point in
[the series], then there will be metaphysical arbitrariness: arbitrariness with respect to why a
[generated object] first comes into existence at that point, and yet no [generated object] comes
into existence at the nearly indistinguishable point […] earlier. If on the other hand it is
indeterminate when the plenitude gets started, then that means that there are [elements in the
73
series such that] it is indeterminate whether there exists something in addition to [the person
and the bank] (namely, a [generated object]). (adapted from 2014: 67-68)
The two horns of the dilemma are:
(H1) There is someone in the account series such that (i) they don’t have a generated
object and (ii) they are followed by someone who has a generated object.
(H2) There is someone in the account series such that it is vague whether they have a
generated object.
Despite what Korman suggests, (H1) and (H2) are not mutually exclusive. Indeed, I accept (H1)
and a version of (H2). As for (H1), as I explained in Chapter 1 (section 1), the existence of a cutoff point is just a result of classical propositional logic, which I assume throughout this dissertation.
So, I accept that there is a cut-off point in the account series. However, that doesn’t force me to
accept that there is a determinate cut-off point, which I take to be the source of metaphysical
arbitrariness, as also explained in Chapter 1 (section 2).
My acceptance of (H2) is a more delicate matter. Let me start by reformulating (H2). Things
could get tricky if we start reasoning about borderline cases of having a generated object.
Fortunately for Korman, there is another way of making the same point. More specifically, the
objection will be that the second-order vagueness of the predicate ‘… has signed form 1A’ leads
to instances of problematic vagueness that are not covered by my initial formulation of account
plenitude. However, I will later argue, this can be fixed by modifying the letter of the view while
retaining its spirit.
That there is second-order vagueness with respect to ‘… has signed form 1A’ means that there
are borderline borderline cases. That is, people such that it is vague whether it is vague whether
they have signed form 1A. Such people are located both in the limit between determinate negative
74
cases and negative borderline cases and in the limit between positive borderline cases and
determinate positive cases. For reasons that I will explain soon, I will focus only on the first group.
Among them, I assume that some of them are actually borderline cases and others are actually
determinate negative cases. For concreteness, suppose the distribution goes like this:43
Table 5: Distribution of negative cases (account plenitude)
Negative
cases
Determinate
negative cases
Det Det Neg cases 𝑐1-𝑐54
Bord Bord cases 𝑐55-𝑐59
Negative
borderline cases
Bord Bord cases 𝑐60-𝑐65
Det Bord cases 𝑐66-𝑐69
So, the borderline borderline cases are 𝑐55-𝑐65. Again, for those same reasons that I will explain
soon, I will only focus on those that are actually determinate negative cases, namely 𝑐55-𝑐59.
To avoid convoluted sentences, I will express the argument in a formal language where ‘𝑆’
must be read as ‘… has signed forim 1A’ and ‘𝐵’ as ‘… is an account of …’. The formalizations
of the operators are those from Chapter 2. I will use ‘𝑐’ as a placeholder for any of 𝑐55-𝑐59. The
argument starts with the assumption that these people are borderline borderline cases:
1. ∇□¬𝑆𝑐
Recall the operator ‘it is open that …’ introduced in Chapter 2. Here are two important principles
I assumed about this operator:
Op-Det It is open that 𝑝 iff it is not determinate that not 𝑝
43 Stricty speaking, the borderline boderline cases on this side of the series can also be called borderline
determinate negative cases.
75
Vag-Op it is vague whether 𝑝 iff [it is open that 𝑝 and it is open that not 𝑝]
By Vag-Op, (1) entails:
2. ◇¬□¬𝑆𝑐
By the determinacy of Op-Det, (2) entails:
3. ◇◇𝑆𝑐
The following strengthening of Account Existence seems plausible:
Det-Account Existence ∀𝑥 □□(∃𝑦𝐵𝑦𝑥 ↔ 𝑆𝑥)
From (3) and Det-Account Existence, it follows:
4. ◇◇∃𝑥𝐵𝑥𝑐
The determinate rejection of negative vague existence, which I defended in Chapter 2 (section 2)
entails (see Chapter 2, note 29):
Barcan-Det □(◇∃𝑥𝜙𝑥 → ∃𝑥◇𝜙𝑥)
Given Barcan-Det, (4) entails:
5. ◇∃𝑥◇𝐵𝑥𝑐
By the same principle, (5) entails:
6. ∃𝑥◇◇𝐵𝑥𝑐
76
Let’s say that something is a quasi-account when it is open that it is an account, a quasi2-account
when it is open that it is open that it is an account, etc. The reasoning above shows that the fact
that ‘… has signed form 1A’ is second-order vague entails that people who are borderline
borderline (“borderline2”, from now on) cases have quasi2-accounts. Indeed, one can prove a more
general result: that if ‘… has signed form 1A’ is 𝑛th-order vague, then people who are borderlinen
cases have quasin-accounts.44
This is bad news for account plenitude. On the one hand, quasin-accounts are not familiar
objects. Thus, much like we were inclined to say that nothing was a quasi-account of a negative
borderline case, we are also inclined to say that nothing is a quasin-account for a negative
borderlinen case. On the other hand, account plenitude is unable to provide suitable objects to play
the role of quasi2-accounts for negative borderline2 cases, which are, recall people who actually
are not borderline cases. This is because account plenitude only provides generated objects for
relations that are indeed precisifications of ‘… is an account of …’ and a borderline2 case that is
not a borderline case doesn’t instantiate bear any such relation to anything. Since the existence
quasin-accounts follows from the 𝑛-th order vagueness of ‘… has signed form 1A’ given the anti
vague existence principles that motivate views like account plenitude, this at least turn account
plenitude and, consequently, anchoring plenitude more generally into unstable positions.
44 Assume that Account Existence is at least 𝑛th-order determinate and that Barcan-Det is at least (𝑛 − 1)thorder determinate. Letting ‘□
𝑛
’ stand for a chain of 𝑛 ‘□’s and ‘◇𝑛
’ for a chain of 𝑛 ‘◇’s, we can formulate
those assumptions as follows: ∀𝑥□
𝑛
(∃𝑦𝐵𝑦𝑥 ↔ 𝑆𝑥) and □
𝑛−1
(◇∃𝑥𝜙𝑥 → ∃𝑥◇𝜙𝑥). Suppose now that ‘…
has signed form 1A’ is 𝑛th-order vague and that 𝑐 is a borderlinen case of that predicate. That is, ◇𝑛𝑆𝑐.
Given the 𝑛th-determinacy of Account Existence, ◇𝑛𝑆𝑐 entails ◇𝑛∃𝑥𝐵𝑥𝑐, which we can rewrite as
◇𝑛−1◇∃𝑥𝐵𝑥𝑐. Now, given the (𝑛 − 1)th-order determinacy of Barcan-Det, ◇𝑛−1◇∃𝑥𝐵𝑥𝑐 entails
◇𝑛−1∃𝑥◇𝐵𝑥𝑐 , which we can rewrite that as ◇𝑛−2◇∃𝑥◇𝐵𝑥𝑐. Now, given the (𝑛 − 2)th-order
determinacy of Barcan-Det, ◇𝑛−2◇∃𝑥◇𝐵𝑥𝑐 entails ◇𝑛−2∃𝑥◇◇𝐵𝑥𝑐. If we apply this reasoning (𝑛 − 2)
more times, we reach ∃𝑥◇𝑛𝐵𝑥𝑐. That is, anyone who is a borderlinen case of ‘… has signed form 1A’ has
a quasin-account.
77
In order to solve this problem, I suggest that we reformulate account plenitude in such a way
that the properties that anchor a generative relation are not just 𝑆60-𝑆80, which are the actual
precisifications of ‘… has signed form 1A’, but also other properties. To deal with the existence of
quasi2-accounts, we say that every property such that it is open that it is a precisification of ‘… has
signed form 1A’ anchors a generative relation. Of that relation, we can say that that it is open that
it is a precisification of ‘… is an account of …’. Say that 𝑥 is a quasi-precisification of 𝑦 when it
is open that 𝑥 is a precisification of 𝑦, that 𝑥 is a quasi2-precisification of 𝑦 when it is open that it
is open that 𝑥 is a precisification of 𝑦, etc. By having quasi-precisifications anchor generative
relations, we guarantee the existence of quasi2-accounts. Indeed, we can generalize this result: if
quasin-1-precisifications of ‘… has signed form 1A’ anchor generative relations, which are quasin1-precisifications of ‘… is an account of …’, then there are quasin-accounts.45 So, the new version
of account plenitude says that, for any 𝑛, if a property is a quasin-precisification of ‘… has signed
form 1A’, then that property anchors a generative relation, which ends up being a quasinprecisification of ‘… is an account of …’. Any view that is an instance of anchoring plenitude
should be modified accordingly.
4. Anchoring plenitude and maximalist ontology
Maximalist views are widespread in discussions about the metaphysics of the material world.
Besides compositional universalism, we encounter material plenitude, a view that posits a
45 Let 𝑐 be a borderlinen case of ‘… has signed form 1A’. That is, ◇𝑛𝑆𝑐. So, 𝑐 instantiates some quasin-1
precisification of ‘… has signed form 1A’. Let 𝑆𝑘 be such a precisification. The new version of account
plenitude tells us that every quasin-1 precisification of ‘… has signed form 1A’ anchors a generative relation,
which is a quasin-1 precisification of ‘… is an account of …’. Let 𝑆𝑘 anchor 𝐵𝑘 and let 𝑏 be an object 𝐵𝑘-
generated for 𝑐. Since 𝐵𝑘 is instantiated by 𝑐 and 𝑏, and is a quasin-1 precisification of ‘… is an account of
…’, we can say that ◇𝑛𝐵𝑏𝑐. That is, 𝑏 is a quasin-account of 𝑐. So, 𝑐 has a quasin-account.
78
multitude of coincident objects which differ from one another in their modal properties.46 Such a
view is also motivated by systematic considerations such as the argument from vagueness.
An important contrast between material plenitude and anchoring plenitude deserves
discussion. Material plenitude principles are usually formulated in terms of such notions as
parthood, location, etc. That is, they are principles that apply only to material objects. Since it is
plausible that not every social object is a material object, there is not much hope for material
plenitude principles to guarantee the existence of the kind of objects required to deal with every
instance of problematic vagueness in the social world. This motivates an approach like anchoring
plenitude, which doesn’t require that the generated objects be located, have parts etc.
An alternative path would be to formulate a more general principle in the image of anchoring
plenitude. Let me explain. Material plenitude principles usually take a base (e.g., a material object)
and claim that there are multitudes of other objects that bear a relation to it (e.g., they coincide
with it) but differ in other respects. Anchoring plenitude takes a different approach. Given a
restricted domain of objects (e.g., people) and a restricted domain of properties (e.g., signing
properties), it guarantees that there are objects that exist insofar as one of those objects instantiate
one of those properties. One can consider the possibility of a less restrictive plenitude principle,
one that guarantees, for a wider domain of propositions, that there are objects that exist insofar as
those propositions are true.47 I leave discussion of those ideas for future work.
46 For early discussion, see Yablo 1987, Bennett 2004, Hawthorne 2006 (ch 3) and Leslie 2011. For recent
discussion, see Fairchild 2019 and Dorr, Hawthorne & Yli-Vakkuri 2021 (ch 11).
47 Sider (ms) contains some discussion of such principles.
79
References
Bacon, A. (2018). Vagueness and Thought. Oxford University Press.
Barnes, E. (2013). Metaphysically Indeterminate Existence. Philosophical Studies, 166(3), 495–
510. https://doi.org/10.1007/s11098-012-9979-3
Barnes, E., & Williams, J. R. G. (2011). A Theory of Metaphysical Indeterminacy. In K. Bennett
& D. W. Zimmerman (Eds.), Oxford Studies in Metaphysics Volume 6 (pp. 103–148). Oxford
University Press.
Båve, A. (2011). How to Precisify Quantifiers. Journal of Philosophical Logic, 40(1), 103–111.
https://doi.org/10.1007/s10992-010-9152-4
Bennett, K. (2004). Spatio-Temporal Coincidence and the Grounding Problem. Philosophical
Studies, 118(3), 339–371. https://doi.org/10.1023/b:phil.0000026471.20355.54
Bricker, P. (1989). Quantified Modal Logic and the Plural de Re. Midwest Studies in Philosophy,
14(1), 372–394. https://doi.org/10.1111/j.1475-4975.1989.tb00198.x
Carmichael, C. (2011). Vague Composition Without Vague Existence. Noûs, 45(2), 315–327.
https://doi.org/10.1111/j.1468-0068.2010.00807.x
Carmichael, C. (2015). Toward a Commonsense Answer to the Special Composition Question.
Australasian Journal of Philosophy, 93(3), 475–490.
https://doi.org/10.1080/00048402.2014.989397
Cleve, J. van. (2008). The Moon and Sixpence: A Defense of Mereological Universalism. In T.
Sider, J. P. Hawthorne, & D. W. Zimmerman (Eds.), Contemporary debates in metaphysics.
Blackwell.
Dodd, J. (2007). Works of Music: An Essay in Ontology. Oxford University Press.
Donnelly, M. (2009). Mereological Vagueness and Existential Vagueness. Synthese, 168(1), 53–
79. https://doi.org/10.1007/s11229-008-9312-z
Dorr, C. (2003). Vagueness Without Ignorance. Philosophical Perspectives, 17(1), 83–113.
https://doi.org/10.1111/j.1520-8583.2003.00004.x
Dorr, C. (2005). What We Disagree About When We Disagree About Ontology. In M. E. Kalderon
(Ed.), Fictionalism in Metaphysics (pp. 234–286). Oxford University Press UK.
Dorr, C., Hawthorne, J., & Yli-Vakkuri, J. (2021). The Bounds of Possibility: Puzzles of Modal
Variation (J. Hawthorne & J. Yli-Vakkuri, Eds.). Oxford University Press.
Effingham, N. (2010). The Metaphysics of Groups. Philosophical Studies, 149(2), 251–267.
https://doi.org/10.1007/s11098-009-9335-4
80
Epstein, B. (2015). The Ant Trap: Rebuilding the Foundations of the Social Sciences. Oxford
University Press.
Fairchild, M. (2019). The Barest Flutter of the Smallest Leaf: Understanding Material Plenitude.
Philosophical Review, 128(2), 143–178. https://doi.org/10.1215/00318108-7374932
Fine, K. (1975). Vagueness, Truth and Logic. Synthese, 30(3–4), 265–300.
https://doi.org/10.1007/bf00485047
Forbes, G. (1989). Languages of Possibility: An Essay in Philosophical Logic. Blackwell.
Fritz, P. (2021). Ground and Grain. Philosophy and Phenomenological Research, 105(2), 299–
330. https://doi.org/10.1111/phpr.12822
Fritz, P. (2024). The Foundations of Modality: From Propositions to Possible Worlds. Oxford
University Press.
Fritz, P., Lederman, H., & Uzquiano, G. (2021). Closed Structure. Journal of Philosophical Logic,
50(6), 1249–1291. https://doi.org/10.1007/s10992-021-09598-5
Goodman, J. (2016). An Argument for Necessitism. Philosophical Perspectives, 30(1), 160–182.
https://doi.org/10.1111/phpe.12086
Hawley, K. (2002). Vagueness and Existence. Proceedings of the Aristotelian Society, 102(1), 125–
140. https://doi.org/10.1111/j.0066-7372.2003.00046.x
Hawthorne, J. (2006). Metaphysical Essays (J. Hawthorne, Ed.). Clarendon Press.
Inwagen, P. V. (1990). Material Beings. Cornell University Press.
Inwagen, P. V. (1998). Meta-Ontology. Erkenntnis, 48(2–3), 233–250.
https://doi.org/10.1023/a:1005323618026
Kaplan, D. (1990). Words. Aristotelian Society Supplementary Volume, 64(1), 93–119.
https://doi.org/10.1093/aristoteliansupp/64.1.93
Katz, J. J. (1980). Language and Other Abstract Objects. Rowman & Littlefield Publishers.
Keefe, R. (2000). Theories of Vagueness. Cambridge University Press.
Korman, D. Z. (2010). The Argument From Vagueness. Philosophy Compass, 5(10), 891–901.
https://doi.org/10.1111/j.1747-9991.2010.00327.x
Korman, D. Z. (2014). The Vagueness Argument Against Abstract Artifacts. Philosophical Studies,
167(1), 57–71. https://doi.org/10.1007/s11098-013-0232-5
Korman, D. Z. (2015). Objects: Nothing Out of the Ordinary (D. Zemack, Ed.). Oxford University
Press UK.
81
Korman, D. Z., & Carmichael, C. (2016). Composition. In Oxford Handbooks Online.
Koslicki, K. (2008). The Structure of Objects. Oxford University Press.
Larue, L. (2024). John Searle?s Ontology of Money, and its Critics (1st edition). In J. J. Tinguely
(Ed.), The Palgrave Handbook of Philosophy and Money: Volume 2: Modern Thought (pp. 721–
741). Palgrave-Macmillan.
Leslie, S.-J. (2011). Essence, Plenitude, and Paradox. Philosophical Perspectives, 25(1), 277–296.
https://doi.org/10.1111/j.1520-8583.2011.00216.x
Lewis, D. K. (1986). On the Plurality of Worlds. Wiley-Blackwell.
Linnebo, Ø. (2016). Plurals and Modals. Canadian Journal of Philosophy, 46(4–5), 654–676.
https://doi.org/10.1080/00455091.2015.1132975
Markosian, N. (1998). Brutal Composition. Philosophical Studies, 92(3), 211–249.
https://doi.org/10.1023/a:1004267523392
Mason, R., & Ritchie, K. (2020). Social Ontology. In R. Bliss & J. Miller (Eds.), The Routledge
Handbook of Metametaphysics. Routledge.
McGee, V., & McLaughlin, B. (1995). Distinctions Without a Difference. Southern Journal of
Philosophy, 33(S1), 203–251. https://doi.org/10.1111/j.2041-6962.1995.tb00771.x
Merricks, T. (2001). Objects and Persons. Oxford University Press.
Merricks, T. (2005). Composition and Vagueness. Mind, 114(455), 615–637.
https://doi.org/10.1093/mind/fzi615
Miller, J. T. M. (2020). The Ontology of Words: Realism, Nominalism, and Eliminativism.
Philosophy Compass, 15(7), e12691. https://doi.org/10.1111/phc3.12691
Passinsky, A. (2020). Social Objects, Response-Dependence, and Realism. Journal of the
American Philosophical Association, 6(4), 431–443. https://doi.org/10.1017/apa.2019.51
Passinsky, A. (2021). Norm and Object: A Normative Hylomorphic Theory of Social Objects.
Philosophers’ Imprint, 21(25), 1–21.
Passinsky, A. (forthcoming). Metaphysics of social objects. Philosophy Compass.
Quine, W. V. O. (1948). On What There Is. Review of Metaphysics, 2(5), 21–38.
Russell, J. (n.d.). Vague Existence.
Schaffer, J. (2009). On What Grounds What. In R. Wasserman, D. Manley, & D. Chalmers (Eds.),
Metametaphysics: New Essays on the Foundations of Ontology (pp. 347–383). Oxford
University Press.
82
Searle, J. (1995). The Construction of Social Reality. Free Press.
Searle, J. R. (Ed.). (2009). Making the Social World: The Structure of Human Civilization. Oxford
University Press.
Sider, T. (n.d.). Plenitude and derivative ontology.
Sider, T. (1997). Four Dimensionalism. Philosophical Review, 106(2), 197–231.
https://doi.org/10.2307/2998357
Sider, T. (2001). Four Dimensionalism: An Ontology of Persistence and Time. Oxford University
Press.
Sider, T. (2013). Against Parthood. Oxford Studies in Metaphysics, 8, 237–293.
Smith, B., & Searle, J. (2003). The Construction of Social Reality: An Exchange. American
Journal of Economics and Sociology, 62(2), 285–309.
Smith, N. J. J. (2005). A Plea for Things That Are Not Quite All There: Or, is There a Problem
About Vague Composition and Vague Existence? Journal of Philosophy, 102(8), 381–421.
https://doi.org/10.5840/jphil2005102816
Sud, R. (2023). Quantifier Variance, Vague Existence, and Metaphysical Vagueness. Journal of
Philosophy, 120(4), 173–219. https://doi.org/10.5840/jphil202312048
Torza, A. (2017). Vague Existence. In D. W. Zimmerman (Ed.), Oxford Studies in Metaphysics.
Oxford University Press.
Uzquiano, G. (2018). Groups: Toward a Theory of Plural Embodiment. Journal of Philosophy,
115(8), 423–452. https://doi.org/10.5840/jphil2018115825
Wetzel, L. (2009). Types and Tokens: On Abstract Objects. MIT Press.
Williamson, T. (1994). Vagueness. Routledge.
Williamson, T. (2010). Necessitism, Contingentism, and Plural Quantification. Mind, 119(475),
657–748. https://doi.org/10.1093/mind/fzq042
Williamson, T. (2013). Modal Logic as Metaphysics. Oxford University Press.
Woodward, R. (2011). Metaphysical Indeterminacy and Vague Existence. Oxford Studies in
Metaphysics, 6, 183–197.
Yablo, S. (1987). Identity, Essence, and Indiscernibility. Journal of Philosophy, 84(6), 293.
https://doi.org/10.2307/2026781
Abstract (if available)
Abstract
Social objects—such as bank accounts, linguistic expressions, governments, football teams, and laws—exist as a result of our social practices. For instance, a bank account exists as a result of a customer signing a contract with a bank. Most of the recent literature on the metaphysics of social objects assumes that a moderate conception of which social objects exist is more or less accurate: there are ordinary social objects and just those objects. Consequently, radical approaches that either deny the existence of ordinary social objects or postulate “strange” social objects are seldom explored. This dissertation challenges this assumption by motivating and defending a radical approach to the metaphysics of social objects. Chapter 1 offers a detailed presentation of a highly influential argument in the metaphysics of material objects—the argument from vagueness. Chapter 2 introduces a new conception of vague existence, a concept central to the argument, and uses it to develop new versions of the argument while establishing important results concerning vagueness and ontology. Chapter 3 builds on these results to criticize several ontological positions about the social world, including various versions of the moderate account. Chapter 4 presents anchoring plenitude, a novel radical approach to social objects according to which ordinary social objects coexist with a multitude of other objects whose existence is intertwined with our social practices in unexpected ways.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
The metaphysics of social groups: structure, location, and change
PDF
Unlimited ontology and its limits
PDF
What it is to be located
PDF
Dimensional analysis: essays on the metaphysics and epistemology of quantities
PDF
The grammar of individuation, number and measurement
PDF
Reasons, obligations, and the structure of good reasoning
PDF
Reproducibility and management of big data in brain MRI studies
PDF
The constitution of action
PDF
A statistical ontology-based approach to ranking for multi-word search
PDF
Essays on wellbeing disparities in the United States and their social determinants
PDF
Mechanisms of recovery in rotator cuff tendinopathy
PDF
Instantial terms, donkey anaphora, and individual concepts
PDF
Social environment, self-concept, and social conduct: sense of self as mediator of the relationship between life-change and marital interaction
PDF
Processing the dynamicity of events in language
PDF
High risk, low rewards: online casinos have found a way to make gambling private, accessible, and more addicting – all without the same regulations of their physical counterparts
PDF
Examining the effects of personalized feedback about ALDH2*2, alcohol use, and associated health risks on drinking intentions and consumption: the role of self-efficacy and perceived threat
PDF
Data modeling approaches for continuous neuroimaging genetics
PDF
Six walks in digital worlds: walking simulators, neuroaesthetics, video games, and virtual reality
PDF
Exploiting web tables and knowledge graphs for creating semantic descriptions of data sources
PDF
Investigations on the muscular dystrophy protein emerin: from nuclear mechanotransduction to molecular interactions
Asset Metadata
Creator
Castillo Gamboa, Edgardo Jaime David
(author)
Core Title
Vagueness, ontology, and the social world
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Philosophy
Degree Conferral Date
2024-12
Publication Date
01/12/2025
Defense Date
10/21/2024
Publisher
Los Angeles, California
(original),
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
OAI-PMH Harvest,plenitude,social objects,social ontology,vague existence,vagueness
Format
theses
(aat)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Uzquiano, Gabriel (
committee chair
), Hawthorne, John (
committee member
), Russell, Jeffrey (
committee member
), Schein, Barry (
committee member
), Wellwood, Alexis (
committee member
)
Creator Email
jaime.castillo.gamboa@gmail.com,jcastillog@usc.edu
Unique identifier
UC11399FAGW
Identifier
etd-CastilloGa-13743.pdf (filename)
Legacy Identifier
etd-CastilloGa-13743
Document Type
Dissertation
Format
theses (aat)
Rights
Castillo Gamboa, Edgardo Jaime David
Internet Media Type
application/pdf
Type
texts
Source
20250113-usctheses-batch-1234
(batch),
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Repository Email
cisadmin@lib.usc.edu
Tags
plenitude
social objects
social ontology
vague existence
vagueness