GLOP THEORY: A NEW TROPE ONTOLOGY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHILOSOPHY

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Daniel G. Giberman

May 2010

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/mg102tn9485

© 2010 by Daniel Gary Giberman. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

John Perry, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Mark Crimmins, Co-Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Thomas Ryckman

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

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Abstract

The dissertation addresses three issues at the center of contemporary naturalistic

metaphysics, where ‘naturalistic’ is intended in David Armstrong’s ontological sense,

according to which nothing exists outside of spacetime. The first issue concerns the

underlying ontology of property exemplification. I argue that properties are

resemblance classes of particular features—tropes—and that material objects are trope

bundles. The second concerns the nature of ontological fundamentality. I defend a new

version of the familiar view that the fundamental properties are those from whose

local exemplifications the rest of the world is “built,” much as a language is “built”

from primitive terms. The third issue concerns persistence through time. I develop and

defend a novel approach from within the framework of four-dimensionalism, the view

that material objects have proper temporal parts much as they have proper spatial

parts.

In my positive treatment of each issue, I invoke a new underlying metaphysic

that I call ‘Glop Theory’. ‘Glop’ functions as an acronym for Grounding Local

Ontological Primitive. This grounding primitive is a special property, markedness.

Tropes of this property, mark tropes, are not like familiar qualitative tropes. They do

not confer color or texture or charge. Rather, mark tropes are mere fillers of spacetime.

They serve the role of marking certain locations from others, thus furnishing a sort of

binary code for fundamental ontology. The key hypothesis of glop theory is that at any

given world there are (law-like) regularities between the spatiotemporal arrangements

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of mark tropes, on the one hand, and the distribution of all qualitative tropes like color

or charge, on the other.

Part 1

The three chapters in this part of the dissertation motivate glop theory by

arguing against its naturalistic competitors for various philosophical jobs: bare

particulars and immanent universals in the case of property exemplification, zero-

dimensional material objects and extended simples in the case of fundamentality, and

three-dimensionalism in the case of persistence.

In Chapter 1, Against Zero-Dimensional Material Objects, I reject bare

particulars because they seem to have the problematic feature of being such that they

might have existed without exemplifying any natural properties. I argue that zero-

dimensional material objects cannot be treated convincingly by the only competitive

theory of material property exemplification that is not committed to bare particulars,

namely, bundle theory. I conclude that there are no zero-dimensional material objects.

In Chapter 2, T-Gunk and Exact Occupation, I address the worry that since

most physicists presuppose that spacetime decomposes into points, we should be

suspicious of any metaphysics that banishes zero-dimensional material objects. One

way to address this worry is to develop a theory of non-pointy spacetime that is

equipped for doing physics. Such theories, however, are highly controversial. As an

alternative response to the worry, I argue that we can allow for zero-dimensional

spacetime points without being committed to zero-dimensional material objects.

Specifically, I defend the thesis that all material objects are extended in each

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dimension (T-theory) against thought experiments set to show that, in pointy space,

certain points are exactly occupied by zero-dimensional material objects.

The final chapter in this section, Problems from Whole Self-Distance, presents

an argument against extended simples, immanent universals, and three-dimensionally-

persisting objects. The key premise is that endorsing any of these items commits one

to the implausible claim that something might exist at a non-zero distance (whether

spatial or temporal) from its whole self. Attempts to defend these items without being

so committed are considered and shown to be independently implausible.

Part 2

The three chapters in this part of the dissertation use glop theory to develop

positive accounts of exemplification, fundamentality, and persistence. In Chapter 4,

Gloppy Trope Bundles, I develop a trope bundle theory of property exemplification

that uses mark tropes as the means of bundling other tropes into material objects. The

upshot is a theory of objects and properties that remains faithful to the austere

ontology of traditional trope theory without being committed to spacetime

substantivalism, a primitive ‘compresence’ relation, or essential connections among

distinct, qualitative properties—commitments which plague competing trope bundle

theories. The chapter contains a discussion of why these commitments are best

avoided.

In Chapter 5, Reconciling Sparse Fundamentality with Infinite Complexity and

Emergence, I defend the “builder” approach to ontological fundamentality against the

objection that it fails to account for worlds that contain atomless gunk, infinite

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qualitative complexity, or emergent properties. The key to the defense in the first two

cases is the idea that mark tropes can serve as ultimate supervenience bases even if

they are not the smallest or mereologically simplest items at a given world. Those who

have argued against the “builder” approach from the possibilities of gunk and infinite

complexity have not adequately considered this idea. The defense in the emergence

case consists of (i) an argument against the clarity of putative examples of emergent

mentality and (ii) a sketch of how glop theory furnishes a “builder”-friendly

interpretation of a putative example of emergence from physics: quantum

entanglement.

Finally, in Chapter 6, Gloppy Four-Dimensionalism, I use glop theory to

develop a version of four-dimensionalism that can treat the puzzles of fission and

fusion without being committed to either (i) instances of simultaneous co-location of a

spatial region by more than one material object or (ii) instantaneous objects. The

former commitment, as stage theorists have emphasized, is counterintuitive. Yet the

latter commitment is problematic as well since there might be temporally atomless

gunk. Historically, those versions of four-dimensionalism that avoid the former

commitment bear the latter, and vice versa. The chapter constitutes a step forward for

four-dimensionalism by furnishing a theory that avoids both commitments.

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“I should have liked to be a piecemeal, unsystematic philosopher, offering independent proposals on a variety of topics. It was not to be."

- David Lewis

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Acknowledgements Without comparison my deepest debts are to my parents, my brother, and my amazing wife. I could never ask for more support than these loved ones have provided. I would like to thank my advisor John Perry and my other committee members, Mark Crimmins and Tom Ryckman, for helping my professional development in a great many ways, not least of which has been the contribution of substantial improvement to the present project. I owe special thanks to Jonathan Schaffer for extraordinarily generous involvement with the project (and for inviting me to visit Australia!). Thank you to Thomas Hofweber, Ted Sider, and Jessica Wilson for engaging in very helpful email exchanges with me at one stage or another. Thank you to Stephen Schiffer and Peter Unger for helping me so much as an undergraduate and for encouraging my pursuit of graduate study in philosophy. Thank you to Russ Heller, Scott Arnold, and Chris Niebrand for helping me so much as a wee high school student and for first turning me on to the discipline of philosophy. Thank you to all of the past Stanford philosophy PhDs, whose names grace the cardinal spines of the dissertations in Tanner Library, for showing me what to work toward. The word tokens comprising this sentence, and all that follow, are delighted to be among such company in their final resting place. I have benefited a great deal from audiences and friends at Stanford, Oxford, Boulder, UMass Amherst, and the 2009 and 2010 APA Pacific division meetings. Particular thanks are extended to Lanier Anderson, Alexei Angelides, Ralf Bader, Jim Binkoski, Einar Bohn, Alexis Burgess, Chad Carmichael, Sam Cowler, Tal Glezer, David Hills, Alistair Isaac, Mary Krizan, Krista Lawlor, Micah Lewin, Dustin Locke, Trenton Merricks, Alexander Paseau, Rob Rupert, Raul Saucedo, Wolfgang Schwartz, Brian Skyrms, Quayshawn Spencer, Elanor Taylor, Ken Taylor, Johanna Wolff, and Ben Wolfson. Finally, thank you to Stella Giberman for reading with me on the couch over the last five years, and especially to Georgie Giberman for being my inspiration.

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Table of Contents

Introduction .......................................................................................................................1 1. Overview of the Project ......................................................................................................... 1 2. Presuppositions ...................................................................................................................... 2 3. Not Your Father’s Trope Theory ......................................................................................... 4 4. Glop Theory............................................................................................................................ 7 5. How the Pieces Fit Together ................................................................................................. 7

PART ONE.......................................................................................................................12 Chapter 1: Against Zero-Dimensional Material Objects (and Other Bare Particulars).......................................................................................................................13

1. Introduction.......................................................................................................................... 13 2. Property Exemplification .................................................................................................... 16 3. Argument for Premise 1 ...................................................................................................... 20 4. Argument for Premise 2 ...................................................................................................... 29

Chapter 2: T-Gunk and Exact Occupation...................................................................39 1. Introduction.......................................................................................................................... 39 2. T-Theory and T-Gunk......................................................................................................... 42 3. The Weak Supplementation Argument Against T-Theory.............................................. 47 4. Parts and Exact Occupation ............................................................................................... 52 5. Prospects for Defending Multi-Regional Exact Occupation............................................ 55 6. Taking Stock......................................................................................................................... 63 7. Conclusion ............................................................................................................................ 63

Chapter 3: Problems from Whole-Self-Distance ..........................................................65 1. Introduction.......................................................................................................................... 65 2. Argument Against WSDP ................................................................................................... 67 3. Being Committed to WSDP ................................................................................................ 70 4. Entension .............................................................................................................................. 76 5. Running Without Hiding, Part 1: Immanent Universals ................................................. 77 6. Running Without Hiding, Part 2: Three-Dimensionalism............................................... 79 7. Conclusion ............................................................................................................................ 87

PART TWO .....................................................................................................................88 Chapter 4: Gloppy Trope Bundles.................................................................................89

1. Introduction.......................................................................................................................... 89 2. Terminology and Motivation .............................................................................................. 92 3. Tropes in Spacetime............................................................................................................. 94 4. The Bundling Problem ...................................................................................................... 101 5. Worries for Prior Approaches.......................................................................................... 106 6. Markedness and Glop Theory .......................................................................................... 111 7. Costs and Benefits .............................................................................................................. 124

Chapter 5: Reconciling “Sparse” Fundamentality with Infinite Complexity and Emergence ......................................................................................................................128

1. The Builder and the Heir .................................................................................................. 128

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2. Fundamentality* and Modal Generality ......................................................................... 139 3. De-Motivating the Global: Ontological Economy and MKJ ........................................... 144 4. Markedness......................................................................................................................... 147 5. Emergence .......................................................................................................................... 159 6. Conclusion .......................................................................................................................... 163

Chapter 6: Gloppy Four-Dimensionalism (with a Pinch of Causation) ...................165 1. Introduction........................................................................................................................ 165 2. Some Background on Persistence..................................................................................... 166 3. Gloppy Persistence............................................................................................................. 171 4. Further Worries and Responses ....................................................................................... 177 5. Recovering Ordinary Objects: Adopting Casullo’s “Double Bundle” Strategy.......... 182 6. Stages and Worms ............................................................................................................. 184 7. Immanent Causation Avoided .......................................................................................... 192 8. Prospects for Trope-Based Singular Causation .............................................................. 200

References ......................................................................................................................204

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Introduction

1. Overview of the Project

This dissertation concerns material objects like planets, plants, platform shoes, and

particles. The first part, consisting of Chapters 1 through 3, carves out a territory

within the landscape of the metaphysics of such objects. The second part, consisting of

Chapters 4 through 6, builds a new settlement upon the tract cleared. The central

questions concern the ontology, mereology, and persistence of material objects. The

ontological questions concern (i) whether the existence of material objects entails the

existence of anything that outstrips their property instances and (ii) whether their

properties are universals or classes of tropes. The mereological questions concern (i)

whether any material objects are mereological atoms or ‘simples’ and (ii) whether any

non-simples are fundamental. The question of persistence asks whether material

objects persist through time by having (or being appropriately related to) distinct

temporal parts at distinct times. Chapters 1, 2, and 3 all have a hand in addressing the

mereological questions by trimming away mereological atoms of one sort or another.

Chapter 1 prepares the ontological landscape by clearing out bare particulars; and

Chapter 3 readies it by excising immanent universals. Chapter 3 also carves the

persistence countryside by barring theories that attempt to do away with temporal

parts. The territory that is left is friendly only to four-dimensional trope bundles,

including those that are not composed of simples. Chapters 4, 5, and 6 then build upon

the newly constrained real estate. Chapter 4 offers a theory of trope bundles; Chapter 5

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a theory of fundamentality that does not require simples; and Chapter 6 a theory of

four-dimensionalism that does not require temporally simple parts.

2. Presuppositions

Three chief assumptions that lurk in the background of the project are worth making

explicit. The first is that spacetime and the contents of spacetime are all that there is.

That is, naturalism is true, in David Armstrong’s ontological sense of the term. This is

not a methodological or epistemological sense of ‘naturalism’. I make no claim that

invokes some notion of a final physics or that turns on empirical methodology, nor do

I claim that we cannot make sense of the conceptual possibility that human knowledge

outstrips the contents of spacetime. Indeed, I make no claim that bars the metaphysical

possibility of abstracta. What my assumption of ontological naturalism bars is the

possibility that any of the objects (indeed, any instances of the kinds of objects) that

actually exist might have been abstract. If Plato’s heaven exists at any possible world,

it is a world that is alien with respect to ours. (This is inconsistent with what Platonists

say about their heaven but then, of course, Platonists reject ontological naturalism. It is

also not exactly what Armstrong would say, since Armstrong rejects alien objects and

worlds.)

I here use ‘spacetime’ to refer to whatever it may be that in fact underlies the

general systematic treatment of distance relations among parts of the external world,

which general treatment is common to all, or nearly all, of the various specific theories

of spacetime, both philosophical and physical. In short, I intend no serious

commitment to one treatment of spacetime over another, whether philosophical or

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physical. It may be the case that spacetime is just as fundamental as its contents

(substantivalism), “more” fundamental (supersubstantivalism), or “less” fundamental

(relationalism). I will not take sides. It may be the case that spacetime is in some sense

quantized. Again, I will not take sides. Indeed, I am inclined to think it possible that,

even if spacetime is quantized, it might not have been, and that even though it is

curved, it might not have been.

The second assumption is that there exists a mind-independent plurality of

qualitative diversity. There is actually very little that this assumption rules out with

respect to extant theories of property exemplification and material objects. It is

consistent even with so-called “ostrich” nominalism and property holism (though see

the next assumption). What it rules out is, for example, radical epistemic skepticism

according to which the qualitative variation in our experiences does not correspond

with any accuracy to the way the world is.

The third background assumption is that there exist individual material objects

with independent, “intrinsic” properties. The world does not reduce to mere relational

structure, nor does its qualitative variation reduce to one (or more) holistic super-

quality. There have been a number of interesting recent attempts to undermine this

assumption in the literature, and I plan in future research to address them directly. In

this dissertation, however, I simply register the assumption.

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3. Not Your Father’s Trope Theory

The trope ontology to be defended in part 2 differs in several important respects from

earlier such ontologies. In this section, I say in some detail just what these differences

come to.

Tropes are particular, non-repeatable features. Your jacket and mine may both

be the exact same shade of brown, but they have numerically distinct tropes of that

shade. Properties are primitive resemblance classes of tropes. The brown property of

our jackets just is the class of all tropes that are of that shade of brown.

(Companionship and Imperfect Community problems that have been raised for trope

resemblance classes are addressed in Chapter 4 by applying the new trope ontology

introduced there.)

Unlike some traditional trope theories, the trope theory to be defended here

locates tropes in spacetime. (This is not unprecedented. See (Campbell 1981), (Simons

1994), and (Schaffer 2001)). This should come as little surprise given the assumption

of ontological naturalism. Now, since tropes are located in spacetime, they have sizes

and shapes (which may be arbitrarily small and arbitrarily simple). Indeed, the view to

be defended makes a great deal of spatiotemporal properties like size and shape, and it

makes a great deal of the fact that tropes themselves exemplify these properties. To

highlight size, shape, and duration as special in this sense, I introduce the following

terminology. Size, shape, and duration are ‘non-qualitative’ properties. Mass, color,

and all of the other properties involved in the familiar, varied character of the world

are ‘qualitative’.

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The claim that tropes exemplify properties marks another important difference

from traditional trope theories. The brown trope of my jacket has a shape property, a

size property, a duration property, and a color property. One might then wonder what

the difference is between tropes and material objects. The difference is that only

material objects can possibly exemplify more than one qualitative property.

Now it may be asked, how, if tropes have sizes, can your jacket and mine have

tropes of the same brown property? After all, your jacket is a medium and mine a

large. So how can the two tropes resemble exactly when they differ in size? The

answer is that the language needed to express the notion of exact resemblance that

underwrites property classes invokes a pre-theoretical ideology of respects. The two

brown tropes exactly resemble in respect of color, but not in respect of size (compare

the fact that they do not exactly resemble with respect to whose jacket they confer

color upon). Respects are not universals and talk of respects does not carry ontological

commitment to universals (compare the fact that talk of a difference in whose jacket a

trope confers color upon does not carry commitment to three-place relations among

jackets, owners, and color tropes). Likewise, talk of respects does not equate to

property nominalism: leaving respects out of the final ontology does not require

leaving tropes out of the final ontology. Respect talk is merely useful, at a pre-

theoretical level, for expressing claims about the qualitative variation of the world.

Trope theory, realism about universals, and nominalism in its various forms are all

ontological theories of that variation. Pre-theoretical talk that is useful for expressing

facts about the variation does not in itself carry commitment to, or favor, any one of

the competing theories.

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Yet another way in which the trope theory defended here differs from some of

its traditional forbears is that it does not require the existence of material objects in

order for the existence of tropes. On the present view, material objects are not

ontologically prior to their tropes. Rather, tropes are ontologically prior to material

objects.

Finally, the trope theory to be defended does not presuppose that the

conventions of our predicative discourse furnish any very accurate guide for

determining which properties a given object exemplifies. On the view to be defended,

objects exemplify all and only the tropes in their respective bundles, and all of the

facts about what the objects are like intrinsically are given by facts about which tropes

are in their bundles. So, for example, the fact that a fire engine has silver bumpers

entails, on the view to be defended, that a silver trope is a member of the fire engine

bundle (the entailment stems from how the theory to be defended in Chapter 4 uses

location facts to determine bundling). But this does not link up with our ordinary

predicative talk, which licenses ‘the fire engine is red’ but not ‘the fire engine is

silver’. Chapter 4 includes a discussion of how to account for this rift between

language and world. This final novelty about the theory to be defended has the

following attractive feature: it builds in a reductive treatment of our polyadic

predicative talk. We may say that the fire engine has the polyadic property (relation)

of being the largest in the fleet without being committed to the existence of some

largest in the fleet relational trope that must be in the fire engine bundle. The

possession of a built-in way to circumvent worries about polyadic relations is a major

improvement over traditional trope theories.

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4. Glop Theory

The biggest novelty of the ontology defended here is that it invokes an unprecedented

perfectly natural, primitive property that I call ‘markedness’. Markedness primitively

marks certain locations from others. The places where tropes of markedness—mark

tropes—reside are thereby distinguished from surrounding places. As a heuristic, we

might say that markedness furnishes a sort of binary code for fundamental ontology.

Markedness is neither a property of material objects nor (necessarily) a property of the

regions that its tropes occupy. Mark tropes are ontologically fundamental.

I call the overarching ontology that markedness underlies ‘glop theory’, where

‘glop’ functions both as an acronym for ‘grounding local ontological primitive’ and as

a mass term for designating mark tropes. So a quantity of glop just is some collection

of mark tropes. That the term ‘glop’ serves this double duty, coupled with the fact that

it conveniently evokes some homogeneous stuff with a size and a shape, satisfied my

own criteria for licensing the use of it as a new technical term in addition to

‘markedness’. Still, for all that, some glop just is some collection of mark tropes.*

5. How the Pieces Fit Together

We have seen that Chapters 1 through 3 clear a tract upon which Chapters 4 through 6

build glop theory. In this final introductory section, I would like to say in a little more

detail how the first three chapters connect to the positive glop-theoretic project. There

* In the interest of full intellectual disclosure, I should also mention that some inspiration for my use of ‘glop’ came from David Lewis’s use of ‘gunk’ and Donald Williams’s use of ‘trope’. The latter first used ‘trope’ as a sort of joke on the literary meaning of the word, and the former first used ‘gunk’ with an element of tongue-in-cheek. ‘Glop’ too has a meaning upon which I take my use of it as something of a joke, namely, the meaning mawkishness. I confess some mawkishness in my admiration for Williams’s and Lewis’s senses of humor.

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are three principal ways that Chapters 1, 2, and 3 serve to motivate glop theory. The

first involves the mereological question, the second the ontological questions, and the

third the question of persistence. In each case, Part 1 of the dissertation motivates a

certain general type of answer to the question, Part 2 raises a puzzle for that type of

answer, and glop theory comes to the rescue by furnishing an instance of the relevant

type that discharges the relevant puzzle. So in each case, Part 1 indirectly motivates

glop theory by contributing to its dialectical importance.

Here is the first of the three instances of this general scheme. Chapter 1 argues

that, necessarily, there are no zero-dimensional material objects. If the argument is

sound, then the possibility of material gunk is thereby motivated. While it remains an

open question, for all that Chapter 1 has to say, whether extended simples are possible,

Chapter 3 provides an argument that they are not. Moreover, Chapter 2 argues that

giving up zero-dimensional material objects need not lead one to give up the

possibility that spacetime is composed of zero-dimensional points, which possibility is

attractive given the theories of spacetime invoked in standard physics. So Chapters 1

through 3 collectively serve to motivate and defend the rejection of material (spatial)

simples, whether zero-dimensional or not. This result bears upon the importance of the

puzzle about mereological infinite complexity rehearsed in Chapter 5, and thus upon

the importance of the solution offered therein. Since that solution invokes glop theory,

we have a fairly clear path of motivation for glop that weaves through Chapters 1, 2,

and 3.

The second instance of the general motivational scheme involves the fact that

Chapter 1 strongly motivates bundle theory by arguing against bare particular theory.

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Chapter 4 argues that bundle theory is problematic and offers a solution that requires

the truth of glop theory. So Chapter 1 indirectly bolsters the importance of glop.

Similar remarks apply with respect to the final motivation for glop from Part 1, which

involves the motivation of trope theory and four-dimensionalism by Chapter 3. Since

Chapter 3 argues against universals and three-dimensionalism, it motivates tropes and

four-dimensionalism. Since Chapter 4 raises problems for tropes (the bundling

problem and the problem of locating tropes in spacetime), for which glop theory

provides the most attractive solutions, and Chapter 6 raises a problem for four-

dimensionalism that only glop theory solves (the problem of how to account for the

possibility of gunky time), Chapter 3 serves to bolster the importance of glop. It is

worth mentioning, in connection with the persistence motivation, that the possibility

of gunky time is motivated by the rejection of material spatial simples that stems from

Part 1, given any significant analogy between space and time of the sort typically

presupposed by four-dimensionalists.

Having thus explained how the six chapters link up, I would like to make the

case that the resulting view presents a neat package deal that is situated

advantageously and uniquely in between the members of each of several extant pairs

of prominent, competing views: bare particular theory and bundle theory; “builder”

fundamentality (i.e. priority pluralism) and priority monism; and stage theory and

worm theory. The bundle theory developed in Chapter 4, like other bundle theories,

rejects bare particularity; but like bare particular theory and unlike (most) other bundle

theories, it uses a non-relational particular to unify properties into objects. The theory

of fundamentality developed in Chapter 5, like other theories that use small, local,

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fundamental items to build up the rest of the world, rejects priority monism; but like

priority monism and unlike other “builder” theories, it allows for worlds wherein the

smallest items are not the most fundamental. Finally, the theory of persistence

developed in Chapter 6, like stage theory, does not require material co-location; but,

like worm theory and unlike stage theory, it also resists being committed to

instantaneous objects.

Indeed, the three major families of theories defended—bundle theory,

“builder” fundamentality, and four-dimensionalism, are all instances of a general

“builder” approach to first order metaphysics. On this approach, puzzles are solved by

looking to local, particular items and then building up further items by telling a story

about how items of the first kind behave. Material objects are built up from tropes into

bundles; worlds are built up from local, fundamental property exemplifications;

continuants are built up from temporally local parts; and spatially extended objects are

composed of smaller proper parts. It is interesting to me that some philosophers have

worked very hard to defend one of these “builder” views while rejecting others. For

example: David Armstrong and Ted Sider defend four-dimensionalism while rejecting

bundle theory; Jonathan Schaffer defends four-dimensionalism while rejecting

“builder” fundamentality and keeping a cool distance from bundle theory; Douglas

Ehring defends trope theory while rejecting four-dimensionalism; Peter Simons

defends spatially extended simples while defending trope bundles; many others reject

priority monism while defending universals, bare particulars, three-dimensionalism, or

extended simples. I do not claim that any of these specific instances of the general

“builder” approach entails any other, and so I do not claim that any of the

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aforementioned theorists endorses an ultimately inconsistent metaphysic. But there is

something nice about the harmony of the view defended in this dissertation.

In concluding this introduction, I would like to make one final remark about

the continuity of the text that follows (or, perhaps better, the lack of continuity). The

six chapters contained here were written with an eye toward eventual publication as

self-contained journal articles. I would like to have attended graduate school in a time

where one was not likely to need much in the way of a publication record in order to

attain his first tenure-track position, but alas, I did not. As a consequence, I made the

pragmatic choice to divide my dissertation up into potential journal papers from the

get-go, so that their publication might come sooner than later. Though I have

attempted to incorporate the six into a relatively unified document, I have decided—

primarily as a result of time constraints—to keep them largely self-contained. As a

result, there is some overlap in content, particularly in expository content about glop

theory. While this may make for moderately repetitive reading in some instances, it

also makes for more thorough exposition. Since glop theory is new and rather

idiosyncratic, I trust that most readers will welcome, or at least tolerate, this result.

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Part One

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Chapter 1 Against Zero-Dimensional Material Objects (and Other Bare Particulars) ABSTRACT: A modus tollens against zero-dimensional material objects is presented from the premises (i) that if there are zero-dimensional material objects then there are bare particulars, and (ii) that there are no bare particulars. The argument for the first premise proceeds by elimination. First, bare particular theory and bundle theory are motivated as the most appealing theories of property exemplification. It is then argued that the bundle theorist’s Ockhamism ought to lead her to reject spatiotemporally located zero-dimensional property instances. Finally, it is argued that since she would need such instances in order to construct zero-dimensional material object bundles, she ought to reject the latter. This leaves bare particular theory as the default view of zero-dimensional material objects. The argument for the second premise invokes the thesis that the exemplification of at least one sparse property is a prerequisite for the existence of any particular. It is argued from Humean considerations that bare particulars fail this prerequisite.

1. Introduction

Material objects are concrete items that may bear multiple intrinsic features like

charge and mass.1 Examples include estuaries, elephants, éclairs, and electrons.

Concrete spacetime points, if there are any such entities, are zero-dimensional: they

have zero extension in every dimension. Might material objects, like points, be zero-

dimensional? Many philosophers accept this possibility. For example, David Lewis

(1986) maintains that sparse2 properties need nothing more than a point at which to be

exemplified; David Armstrong (1989) allows that atomic states of affairs may involve

point-sized individuals and point-sized instances of universals; Ted Sider (2006)

argues that points are either the fundamental bearers of fundamental properties or the

containers of such bearers; and Jonathan Schaffer (2009) contends that points

1 An object is concrete just in case it bears spatiotemporal relations. 2 I have in mind the notion of sparseness advanced by Lewis (1983, 1986), according to which a property is sparse if it is needed in order to completely characterize the world without redundancy. Sparse properties “carve at the joints,” ground resemblance, and have a place in the minimal supervenience base.

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exemplify point-values of sparse properties, even if for him the points themselves are

not fundamental. Indeed, some philosophers take seriously the suggestion that

ordinary dry goods and even human bodies are zero-dimensional (Hudson 2005). Pace

this impressive group of theorists, I will defend the following argument that material

objects cannot possibly be zero-dimensional.

The Main Argument:

1. Necessarily, if there are zero-dimensional material objects then there are bare

particulars.

2. Necessarily, there are no bare particulars; therefore:

3. Necessarily, there are no zero-dimensional material objects.

This modus tollens has direct bearing not only upon the ontology of zero-

dimensional objects but also upon the metaphysics of property exemplification, for

premise 2 contravenes bare particular theory. Moreover, there is an important

mereological consequence of rejecting zero-dimensional material objects, namely, that

if one also rejects extended simples then one is committed to material gunk.3 The

modal strength of one’s consequent commitment to gunk turns on the strength of one’s

rejection of extended simples. If one holds that extended simples are impossible, then

one will be committed to the very strong view that material objects are necessarily

gunky, for one will have denied the possibility of material simples whether extended

or not. If one holds only the more modest view that there are some possible material

3 An object is composed of gunk just in case all of its parts have proper parts (Lewis 1991).

15

objects that are neither extended simples nor complexes containing extended simples

as parts, then one will be committed only to the possibility of gunky material objects.

And, of course, if one holds the view that there are actual material objects that are

neither extended simples nor complexes containing extended simples, then one will be

committed to actual material gunk.

In the next section, I set the stage for premises 1 and 2 of the Main Argument

by giving some background on the metaphysics of property exemplification. Section 3

then presents my argument for premise 1 and section 4 my argument for premise 2.

The argument for premise 1 proceeds by elimination. I first argue that bare particular

theory and bundle theory are the most attractive theories of property exemplification

for material objects. I then argue that bundle theory, given its core Ockhamist

motivation, ought to reject zero-dimensional material objects. The thought is that

instead of accepting concrete zero-dimensional property instances, the bundle theorist

can account for the characters associated with point locations by recourse to ordered

pairs of points (at worlds) and appropriately related extended property instances.

Having thus done away with the concrete property instances out of which zero-

dimensional material objects must be bundled, she can resist commitment to such

objects without theoretical loss. That she ought to do so is secured by her antecedent

commitment to Ockham’s razor in arguing for bundle theory. Bare particular theory is

then left as the only suitable ontology of zero-dimensional material objects.

The argument for premise 2 turns on the claims (i) that no particular is such

that it might have existed even if no sparse properties were exemplified and (ii) that

bare particular theory violates (i). The argument for (ii) trades on the Humean thesis

16

that distinct fundamental kinds such as bare particulars and universals are open to free

modal recombination. I defend (i) and (ii) against counterarguments from Armstrong

(1989), Moreland and Pickavance (2003), and Sider (2006).

2. Property Exemplification

The realist about property exemplification holds that material objects exemplify

intrinsic properties. There are four principal realist theories corresponding to two

cross-cutting debates, universals vs. tropes and bare particulars vs. bundles.4

Universals are repeatable features of particulars. If your shirt and my shirt resemble

exactly with respect to color and if properties are universals, then there is just one

item—a blueness universal—that is directly responsible for the resemblance.

Transcendent universals (sometimes called ‘Platonic’) differ from immanent

universals (sometimes called ‘Aristotelian’) primarily in that only the latter are located

in spacetime.

Tropes, in contrast to universals, are particular, non-repeatable features. If

property exemplification involves tropes then at least two items—your shirt’s blueness

trope and my shirt’s blueness trope—are directly responsible for the resemblance in

color between our shirts. The trope theorist holds that properties like blueness are

primitive resemblance classes of tropes and that tropes are concrete.5

4 There are, of course, other debates in the metaphysics of properties, for example, the dispositional/categorical debate. However, my concern is with debates over what properties in general are, not with debates over sub-species of properties. 5 Reference to tropes as ‘abstract particulars’ (Campbell 1990) derives from the sense of ‘abstract’ that involves mentally separating one item from another, not the sense that opposes concreteness.

17

According to bare particular theory (sometimes called ‘substance/attribute’ or

‘substratum’ theory), property exemplification involves two different kinds of entities

at the fundamental level, properties and non-qualitative or ‘bare’ particulars.6 On a

first approximation of this view, material object a exemplifies property F just in case a

special primitive relation, instantiation, obtains between a, a (bare) particular, and F, a

universal. There are three ways in which this characterization is merely approximate.

First, most bare particular theorists hold that single bare particulars suffice to

underwrite exemplification for fundamental objects only. Exemplification facts about

my car, for example, involve not one but a great many bare particulars. Second, bare

particular theorists disagree among themselves about whether exemplification or

rather some more abstract ‘tying’ relation (Moreland and Pickavance 2003) obtains

between bare particulars and properties. Finally, not all bare particular theorists treat

properties as universals, though those who do not are exceptions (Martin 1980).

Notice, then, that bare particulars bear at least some minimal tie to properties

despite being called ‘bare’. The sense in which bare particulars deserve to be called

‘bare’ is not always made clear. I propose two related senses. First, bare particulars—

unlike the particulars of the trope theorist or the nominalist—are not responsible for

their own character, if they have any at all; they do no characterizing. Second, bare

particulars are supposed to be “what is left” when we mentally abstract away all of a

material object’s properties, where to ‘abstract x away from y’ is to consider y in the

absence of x. If anything corresponds to our conception in such a case, it is bare. Some

bare particular theorists introduce bare particulars via a mereological sense of 6 Defenders of bare particular theory include Allaire (1963), Bergmann (1967), Armstrong (1978, 1989, 1997, 2004), Martin (1980), Moreland and Pickavance (2003), and Sider (2006).

18

‘abstracting away’ (Sider 2006). On this mereological sense, bare particulars still

instantiate properties even when the properties are ‘abstracted away’, for instantiation

is not mereological. But this is merely a terminological glitch. The fact remains, even

for Sider, that we can consider bare particulars without attributing to them any

property instantiations. When we do, we are considering bare particulars in accord

with the second of my proposed senses of ‘bare’. Following Sider, one might call bare

particulars so considered ‘truly bare’ particulars.7

According to bundle theory, property exemplification involves only one kind

of fundamental entity, properties, whether understood in terms of universals or tropes.

Material object a exemplifies property F just in case a either just is or is such that its

existence is derivative upon either (i) a ‘bundle’ of universals, one of which just is F,

or (ii) a ‘bundle’ of tropes, one of which is a member of F. Typically, bundles are

constructed via compresence, a primitive relation on either universals or tropes,

respectively.8 Whether or not the object in question is best thought of as numerically

identical to the bundle in question is a matter of some controversy (Hawthorne and

Sider 2002), but it will not hinder the arguments to be given here if we assume

numerical identity for simplicity. Notice that bundle theory so understood fits the

criteria for being a realist theory of exemplification since it posits a relationship

(bundle-membership) that connects objects and intrinsic properties, even though it

does not take objects to be fundamental.

7 Some readers will be familiar with a further terminological distinction for bare particulars: the ‘thin’/’thick’ distinction. I prefer to set this distinction aside because of its vast potential for confusion in the wake of the variety of ways that it has been employed by different theorists. I refer the curious reader to (Armstrong 1989), (Robinson 2004), and (Sider 2006) for a sampling of the variety. 8 For a defense of bundle theory, see (Campbell 1990). For alternatives to primitive compresence bundling, see (Simons 1994), (Denkel 2000), (Schaffer 2001), and (Paul 2002).

19

Of course, not all theories of property exemplification fit the realist mold.

Among those that do not are (i) “ostrich” nominalism (Quine 1980, Devitt 1980), the

view that the underwriting ontology of true utterances of arbitrary predicative claim ‘a

is F’—where ‘a’ purports to designate a material object—involves a alone, not any

universals or tropes; (ii) resemblance nominalism (Rodriguez-Pereyra 2002), the view

that for arbitrary a to exemplify F is for a to appropriately resemble a certain class of

objects; (iii) ontological structuralism, which comes in various forms (Ladyman 2007,

Maudlin 2007), the common feature among which is that the world fundamentally

exemplifies global relational structures and that individual objects, if they exist at all,

are merely derivative upon patterns in the global structure; and finally, ontological

nihilism (Hawthorne and Cortens 1995), the view that there are no individual

propertied objects, only located properties. Most proponents of each of the latter two

views consider the realist’s search for the exemplification relation between material

object a and intrinsic property F to be misguided, preferring instead to paraphrase

away talk of objects by recourse to patterns in fundamental relational structures or

quality locations.9

This survey of the metaphysics of property exemplification for material objects

is neither exhaustive nor entirely uncontroversial, but it will suffice for present

purposes.10 It is intended merely to depict the landscape well enough to be able to

9 Their proponents do not always make clear whether these theories reject objects full stop or only at the fundamental level. In the latter case, it is not always clear in what ways the theories in question differ from bundle theory. I mention these subtleties only to set them aside. 10 Among the theories left out are various alternative forms of nominalism. In defense of their omission, I refer the reader to the arguments against them in (Armstrong 1978). Also omitted are neo-Aristotelian views of substance ala Michael Loux (1997). While such views do not fit neatly into either the bundle or bare particular families, their idiosyncrasies do not render them immune to the arguments in this

20

work within it when arguing for premises 1 and 2 of the Main Argument in the

following two sections.

3. Argument for Premise 1

In this section, I argue that the existence of zero-dimensional material objects entails

the existence of bare particulars. The argument deploys three controversial premises.

Argument for Premise 1 of the Main Argument

1.1 Necessarily, either bundle theory or bare particular theory is the correct theory of

property exemplification for material objects.

1.2 Necessarily, if bundle theory is true and if there are zero-dimensional material

objects, then there are concrete zero-dimensional property instances from which

the objects are bundled.

1.3 Necessarily, there are no concrete zero-dimensional property instances available to

the bundle theorist, therefore:

1.4 Necessarily, either bundle theory is not true or there are no zero-dimensional

material objects.

From 1.4 and 1.1 it follows that:

1.5 Necessarily, if there are zero-dimensional material objects then there are bare

particulars.

The remainder of this section will be devoted to defending 1.1-1.3. paper. The neo-Aristotelian who favors zero-dimensional material objects must still accept either bare particulars or concrete, zero-dimensional instances of universals.

21

According to 1.1, the existence of material objects necessarily requires the

existence of either (a) bare particulars or (b) bundles of either (i) transcendent

universals, (ii) immanent universals, or (iii) tropes. The major views opposing 1.1,

then, are ostrich and resemblance nominalism, ontological structuralism, and

ontological nihilism. Now, there are rather powerful (if less than knockdown)

arguments against ostrich nominalism in the literature (Armstrong 1978, 1989, and

Rodriguez-Pereyra 2002); and resemblance nominalism carries what many would

consider an unattractive commitment to Lewisian modal realism (in order to

circumvent coextension worries, for example).11 However, I will not attempt to refute

either form of nominalism here. In the end, to those who are steadfast ostrich or

resemblance nominalists, I say only that the question as to the truth of nominalism is

sufficiently open as not to have rendered the Main Argument otiose for relying on 1.1;

the argument would still wield substantial force if resisting it required commitment to

nominalism. As for structuralism and nihilism, the versions that are clearly distinct

from bundle theory deny that there are any individual objects with intrinsic properties.

Consequently, they a fortiori deny that there are zero-dimensional material objects. As

such, they are off limits to those who would seek to avoid the conclusion of the Main

Argument by rejecting 1.1. If none of the considered alternatives holds at any possible

world, and if no further alternatives are forthcoming, then it is necessary that either 11 Moreover, it is unclear to me whether Rodriguez-Pereyra’s resemblance nominalism is committed to bare particulars. The thought here is that, whatever the individuals are like that stand in the relevant primitive resemblance relations, it seems possible that at least one of them might have failed to stand in any non-trivial instance of resemblance (the trivial instance that I have in mind is reflexive resemblance; but to whatever extent such a relation makes any sense in this case, it would amount to the property of being a bare particular). But this is just to say that it is possible that one of them lacks any properties whatsoever, save for the trivial properties of being a particular, being an object, being self-identical, etc. If Rodriguez-Pereyra’s view is indeed committed to bare particulars, then it is not in tension with 1.1.

22

bare particular theory or bundle theory is the correct theory of zero-dimensional

material objects.

To assess 1.2, let us return to the three above options for endorsing bundle

theory. Option (b)/(i) (bundling transcendent universals) may be ruled out

straightaway. Material objects stand in spatiotemporal relations. No exclusive

collection of transcendent universals can stand in spatiotemporal relations since

transcendent universals are not located in spacetime. (Notice that even a being located

at region r transcendent universal is not itself located in spacetime.) To add or connect

something that is not a transcendent universal (for example, a spacetime region) to

some collection of transcendent universals in order to yield a bundle that (at least in

part) does stand in spatiotemporal relations would be to abandon bundle theory, for

universal bundle theorists hold that objects are bundled exclusively from universals.

Material objects cannot be strictly exclusive bundles of transcendent universals.

Options (b)/(ii) and (b)/(iii) cannot be so easily dispensed with since both

immanent universals and tropes stand in spatiotemporal relations. Nevertheless, both

(b)/(ii) and (b)/(iii) accord with 1.2. Consider immanent universals first. If all

immanent universal instances are greater than zero-dimensional, then there will be no

way to bundle whole immanent universals together so as to yield something that is

zero-dimensional. Nor will there be any way to bundle proper parts of immanent

universals, for (at least most) universals lack proper spatial parts. Trope theorists, by

contrast, need not deny that tropes have proper spatial parts. There is thus room in

logical space for a trope bundle theory according to which some greater-than-zero-

dimensional tropes get counted as members of some zero-dimensional bundle in virtue

23

of some relation that obtains among certain of their zero-dimensional proper parts.

However, even if the trope theorist allows that tropes might have proper spatial parts,

she will not allow that they might have proper parts that are not themselves tropes, on

pain of violating trope theory. After all, trope theorists typically motivate their view by

trumpeting its economical, one-category ontology (Campbell 1990). So even if the

trope theorist is willing to allow for tropes whose sizes outstrip that of their respective

bundles, which would allow at the limit for zero-dimensional bundles that have

greater-than-zero-dimensional tropes as elements, she necessarily will do so by

recourse to zero-dimensional tropes.

Premise 1.3 is that there are no concrete zero-dimensional property-instances

(whether understood as tropes or as instances of immanent universals) available to the

bundle theorist. The gist of the argument for 1.3 is that the bundle theorist’s

Ockhamist leanings—which are instrumental in arguing against bare particular

theory—ought to compel her to reject concrete zero-dimensional property instances.

The Ockhamism that I have in mind is captured in the following thesis: given a choice

between two otherwise equally meritorious theories, one of which posits n kinds of

entities and the other n-1 kinds of entities, we should endorse the latter. The bundle

theorist claims that, given the Ockhamist thesis, it is a mark in favor of bundle theory

that it accounts for exemplification with only one kind of item, properties, instead of

two, properties and bare particulars. (Notice that I have not said that the Ockhamist

thesis is sound, merely that the bundle theorist endorses it.) The present thought with

respect to 1.3, then, is that given a choice between two otherwise equally meritorious

theories of exemplification, one of which posits both extended and zero-dimensional

24

concrete property instances and one of which posits only extended concrete property

instances, the bundle theorist ought by parity of reasoning reject the former and accept

the latter. If this much is correct, then the bundle theorist has no way to construct

concrete zero-dimensional bundles, for the only property instances available to her

will either not be concrete or not be zero-dimensional.

What remains to be shown, then, is that there is a sufficiently meritorious

bundle theory that does away with concrete zero-dimensional property instances.

Toward that end, I suggest that zero-dimensional property instances be supplanted

with ordered pairs consisting of pairs of points and worlds in the first slot and

collections of extended property instances in the second. (The points in question may

either be understood as abstract idealizations from extended concrete regions or as

merely useful fictional entities, for if they are understood as concrete then they

become dangerously close to being bare particulars. See (Sider 2006).) To make this

proposal more precise, let ‘F’ name the property whose zero-dimensional instance is to

be replaced, let ‘p’ name its point location at world w, and let ‘the Gs’ pick out the

properties whose extended instances are contained in the second slot of the ordered

pair. Using alphabetical subscripts to distinguish instances of properties and numeric

subscripts to distinguish properties qualitatively, the ordered pair in question would be

of the form

Fa =supplanted by <<p, w>, {G1a,…,Gna}>

25

The Gs will be those properties that are held by the best fundamental theory in which

F figures at w as being law-like related to p in the relevant way, where relevance is

determined by the causal role attributed to F by the theory in question.

In short, all that is needed in order to describe the theoretical role of some

point-sized property instance is information about certain extended happenings in the

vicinity (which may be considerable) of the relevant point location. A property all of

whose instances are zero-dimensional at a given world could then be thought of as a

function from points to extended property instances at that world. The (would-be)

concrete zero-dimensional instances of such properties are supplanted by members of

the graphs of such functions, that is, ordered pairs consisting of point/world pairs and

the associated extended property instances. Any exclusive collections (plus a

“bundling relation”) of these ordered pairs would either be abstract or not zero-

dimensional, for ordered pairs will either be understood as abstract objects or as being

nothing over and above their collective members, which in this case include extended

items. If this ordered pairs suggestion holds water, then the bundle theorist ought to be

compelled by her antecedent commitment to Ockhamism to abandon concrete zero-

dimensional property instances in its favor.

I anticipate the following objections to the argument just run for 1.3.

Objection 1: What about the bundle theorist who antecedently is attracted to zero-

dimensional material objects and zero-dimensional concrete property instances? She

will deny that your suggested treatment is equally meritorious precisely because it

rejects such entities.

26

Response: Unless the theorist described presents some more direct reason for rejecting

the ordered pairs proposal, she is flouting the Ockhamist thesis and thus undermining

the chief argument for her view over bare particular theory. Her antecedent attraction

to extra entities does not suffice to undercut the proposal at hand; she must show that

her preferred entities are in some way indispensible or theoretically preferable.

Objection 2: Here is a reason to find concrete zero-dimensional property instances

theoretically preferable to the suggested ordered pairs. There are possible worlds that

contain just one point-sized, multi-propertied concrete object, say, object a at point p

with properties F and G. a’s F and G instances are zero-dimensional. But the ordered

pairs suggestion has no way to account for this since it requires recourse to extended

goings on, and yet the world in question, ex hypothesi, lacks anything extended. The

proposed account cannot distinguish the relevant F instance from the relevant G

instance. In both cases, the account yields <<p, w>, __ >, with nothing filling the

second slot.

Response: This objection begs the question by presupposing that the world described

is metaphysically possible. Recall that the proponent of the ordered pairs account

holds that points are mere idealized abstractions from extended items. She denies that

there is any possible world containing a concrete point-sized entity. So positing such a

possible world is not independent motivation for concrete property instances.

However, it is worth discussing the aspect of the objection according to which the

ordered pairs account cannot even distinguish between the idealized abstract

conceptions of F and G instances in a zero-dimensional world, for the world discussed

27

in the objection does seem to be conceptually possible even if not metaphysically

possible. This aspect of the objection is mistaken. Extended property instances are not

strictly required in order for the ordered pairs proposal to account for differences in

intrinsic character across point-sized property idealizations at a given world. Intrinsic

character is accounted for by membership in the graph of the function that the

proposal identifies with the relevant properties, in this case F and G, respectively. So

the account yields the very same ordered pair, <<p, w>, __ > for the F instance and the

G instance at issue. If w were a possible world, then this one ordered pair would be a

member of both the F graph and the G graph. Its membership in the two graphs

conceptually accounts for the idea that a single point location might involve multiple

intrinsic characters even in the absence of extended property instances; but to

presuppose that this idea corresponds to a metaphysical possibility is to beg the

question against the proponent of the ordered pairs account.

Objection 3: Why not favor a theory that does away with extended property instances

and keeps only zero-dimensional ones? This seems the most ontologically economical

view.

Response: No such theory will be meritorious enough for its ontological economy to

trump its deviation from pre-theoretic belief. We see extended property instances

around us all the time. Indeed, the only way we can make sense of zero-dimensional

property instances is in terms of extended ones; and even though it is consistent with

this epistemic fact to hold that the strict ontology allows only zero-dimensional goings

on, such a view is unduly strange. Moreover, it is dubious—quite independently of

28

any epistemic considerations—that properties like length or mass, which seem to have

extended instances, could be ontologically cashed solely in terms of zero-dimensional

instances. Typical instances of mass do not seem to presuppose point-instances of

mass.

Objection 4: The argument alluded to for premise 2 of the Main Argument—which will

be given in the next section—is itself evidence that the bundle theorist need not

endorse the Ockhamist maxim. Rather, she might simply argue against bare particular

theory directly. If the bundle theorist can argue that there are no bare particulars

without having to endorse Ockham’s razor, then she need not favor the ordered pairs

account over realism about concrete zero-dimensional property instances, and the

argument for 1.3 falls flat.

Response: Even if this objection were sound, it would not render the Main Argument

otiose. For it would still be a significant result if the bundle theorist were required to

relinquish her Ockhamist arguments against bare particular theory in order for it to be

plausible that zero-dimensional material objects exist, just as it would be significant if

one had to endorse nominalism in order to evade the Main Argument. Moreover, even

if it were true that the bundle theorist need not endorse Ockham’s razor, many

philosophers do endorse it; and those who do will still deny that the bundle theory can

furnish a plausible account of exemplification for zero-dimensional material objects,

so long as they are not moved by some further objection to the argument for premise

1.

29

If 1.3 holds up to these objections then, barring more forceful, unforeseen objections,

there is no way for an ontology of zero-dimensional material objects to be cashed as a

bundle theory. The bundle theorist holds that such objects are derivative on the tropes

(or universals) out of which they are bundled. But since the objects, qua material,

stand in spatiotemporal relations, they cannot themselves be derivative on mere

collections of entities that do not stand in such relations, namely, the proposed ordered

pairs understood as abstract objects. Yet understanding the relevant ordered pairs as

concrete renders them greater than zero-dimensional. This leaves no room for a bundle

theory of property exemplification for zero-dimensional objects, given 1.2. Given 1.1,

then, the default view is that such objects exist only if bare particulars exist.

4. Argument for Premise 2

The argument for premise 2 turns on the claim that in order for any particular to exist,

at least one sparse property must be exemplified. While some bare particular theorists

explicitly endorse this claim (Moreland and Pickavance 2003), I will argue that it is in

tension with bare particular theory. The argument for premise 2, like the main

argument, is a modus tollens.

Argument for Premise 2 of the Main Argument

2.1 Necessarily, if there are bare particulars then they might have existed without any

sparse properties having been exemplified.

2.2 Necessarily, there is no particular x such that x exists even though no sparse

property is exemplified.

30

2.3 Necessarily, there are no bare particulars.

The argument for 2.1 relies on the Humean doctrine that distinct existences are

independent. The bare particular theorist holds that bare particulars form a

fundamental ontological kind. If we accept a Humean combinatorialist approach to

modality then we will hold that instances of fundamental kinds may exist

independently of any instances of any (other) kinds. For the Humean who allows bare

particulars it thus follows that it is possible for, say, just one bare particular to exist

completely devoid of any exemplifications of any sparse properties.

One response to this argument is to reject the Humean combinatorial thesis in

the strong form in which I have presented it. The bare particular theorist may instead

accept the Humean view in only the following weaker form: while no instance Ki of

fundamental kind K requires the existence of any particular instance J1 of fundamental

kind J (≠K) in order to exist, each Ki will require the existence of some or other Ji.

Less abstractly, the bare particular theorist may hold that while bare particulars do not

need to be connected to any particular properties in order to exist, they do need to be

connected to some property or other. J.P. Moreland and Timothy Pickavance (2003)

advance this view in response to arguments like the one just given for 2.1. They

invoke the maxim that for a particular to exist is for it to exemplify a property, the

thought being that bare particular theory plus this maxim entails that bare particulars

exist only in the presence of exemplified properties. This argument is uncompelling,

however. The proponent of the maxim cannot accept both bare particulars and the

original “strong” Humean thesis; nor, accordingly, can the proponent of the strong

31

Humean thesis accept both bare particulars and the maxim; but the proponent of the

strong Humean thesis can accept the maxim. Moreland and Pickavance thus cannot

lean solely on the maxim to defend bare particular theory against the strong Humean

thesis. What they need in order to break the stalemate is independent support for

choosing bare particular theory plus the exemplification maxim over the strong

Humean thesis plus the exemplification maxim.

To appreciate the lack of independent support for endorsing bare particular

theory over the strong Humean thesis, let us look at a different but analogous case in

which an instance of some kind K cannot exist in the absence of some instance of a

distinct kind J. For example, an oblong rock cannot exist independently of being tied

to some or other size property, even if there is no particular size to which it must be

tied. But this is because of other properties the rock has, for example, being oblong

and being located in space. Nothing (let us assume) can be oblong or spatially located

if it is not also sized. But notice that this is a relationship between properties and is

thus very different from the bare particular case. The bare particular has no property F

that is responsible for underwriting Moreland and Pickavance’s claim that it must

exemplify some or other property G (≠F). So if, for all properties G, some bare

particular might exist even though it is not tied to G, then a bare particular might exist

without being tied to any properties. Interestingly, Moreland and Pickavance seem to

recognize this point in their response to D.W. Mertz (2001), who objects that a given

bare particular is ill equipped to preclude its being simultaneously tied in the relevant

sense to incompatible properties like roundness and squareness. Moreland and

Pickavance’s response is that there is no need to look to the bare particular to find

32

ground for precluding such cases of incompatible co-tying since the properties

themselves will preclude it. I think they are right. However, the same basic point sinks

their attempt to defend bare particular theory against the strong Humean thesis. There

is nothing about the bare particular available to ground the putative necessity of being

tied to some or other property; in analogous cases of necessary connection, what does

the grounding is a relationship between properties.

Let us move on, then, to consider the bare particular theorist who wishes to

deny the argument for 2.1 not by rejecting the (strong) Humean thesis but by rejecting

the claim that bare particulars form a fundamental ontological kind. Following David

Armstrong (1997), who gathers inspiration from early Wittgenstein and Brian Skyrms

(1981), she might contend that states of affairs (or facts) are fundamental and that bare

particulars and immanent universals are merely non-mereological constituents or

aspects of the instances of the fundamental kind, state of affairs. It would then be open

to the bare particular theorist to accept Humean combinatorialism about fundamental

items without being committed to the possible existence of any bare particulars devoid

of sparse exemplifications, for bare particulars would no longer be considered among

the fundamental items. States of affairs, the lone fundamental kind of item, can be

liberally recombined in Humean fashion without any worries about bareness.

Unfortunately, this response faces a dilemma. It is either a non sequitur or else

it sufficiently deflates the ontological role of bare particulars as to no longer be a

legitimate form of bare particular theory. The first horn of the dilemma stems from a

difference between the notion of fundamentality at work in the Humean combinatorial

claim and the notion of fundamentality at work in the states of affairs ontology. Let us

33

say that for contingently existing items x and y, x is ‘ontologically independent’ of y

just in case x might have existed even if y did not. The notion of fundamentality at

work in Humean combinatorialism is closely tied to the notion of ontological

independence in the following way. If K is a fundamental kind then, for all kinds J

(≠K), there is some K token that is ontologically independent of every J token. That is,

the Humean view entails that being an ontologically independent kind is a necessary

condition for being a fundamental kind. But on the states of affairs view, states of

affairs are not ontologically independent of bare particulars (or universals): one cannot

have a state of affairs without both a particular and a universal. States of affairs thus

fail the Humean criterion for being fundamental. So making states of affairs

‘fundamental’ in the sense of (Armstrong 1989, 1997) is not to the point of the

Humean modal claim about bare particulars. The states of affairs view

notwithstanding, the Humean will maintain that bare particulars are eligible for

ontological independence.

Indeed it is rather plausible that bare particulars are ontologically independent,

even given states of affairs, for there is something unsatisfying about saying that Ks

are fundamental and then going on to say that no K could possibly exist without

having an H and a J as components. Notice that sharing this sense of dissatisfaction

does not turn on whether one considers “constituents” or “unities” to be more

fundamental. Compare priority monism, the view that whole worlds are more

fundamental than their proper parts (Schaffer 2009, 2010). The priority monist does

not deny that there could be whole worlds that lack constituent parts (as the states of

affairs theorist, by contrast, does deny that there could be a state of affairs that (non-

34

mereologically) contains no bare particulars), so he does not hold that all instances of

the fundamental kind whole must co-exist with instances of the distinct kind proper

part.

Before explaining the second horn of the dilemma, it will be worthwhile to

discharge Armstrong’s (1989) two positive arguments for the claim that bare

particulars cannot exist independently, which are distinct from Moreland and

Pickavance’s argument discussed above. The first, the ‘abstraction’ argument (also

intimated in (Skryms 1981)), says that bare particulars cannot have independent

existence because they are nothing more than the products of our mentally abstracting

away from the collection of states of affairs in which they figure. I am dubious that the

states of affairs theorist can get away with holding that bare particulars are merely the

products of mental abstraction. Consider a mereologically simple world w with exactly

two co-located states of affairs: F is exemplified and G is exemplified. Are the two

states of affairs here such that there is some one particular x such that Fx and Gx or,

rather, are they such that there are two co-located particulars x and y such that Fx and

Gy (or Gx and Fy)? There is no fact of the matter unless it is first settled how many

individuals there are at w. When we are given a description of w that does not

presuppose fundamental ontological facts about bare particulars, we simply are not

equipped to do any “abstracting” of the sort Armstrong prescribes; yet the states of

affairs view entails that w contains a (or some) bare particular(s). So mere

“abstracting” cannot be all there is to the existence of bare particulars on the states of

affairs view. Some theorists who are skeptical of particulars (Dasgupta 2009) may

35

deny that there is any interesting fact as to whether w contains one “individual” or

two, but such a move rings at best ad hoc if endorsed by the friend of bare particulars.

The second argument from Armstrong (1989) contends that bare particulars

cannot be ontologically independent because, qua particulars, they must have some

unifying property such as being an object. The problem with this argument is that it

fails to take sparseness into account. Fundamental states of affairs are those and only

those that involve sparse property exemplification. So the fact that every particular

must exemplify non-sparse properties such as being an object brings nothing to bear

on the claim that bare particulars might well exist on their own, independently of

states of affairs.

Let us move on to the second horn of the proposed dilemma for the claim that

bare particulars evade the Humean combinatorial scheme because they are not

fundamental. Recall that the second horn is that the states of affairs view ends up

looking like it is no longer a species of bare particular theory. This second horn arises

from the suggestion—meant to block the first horn—that bare particulars are not

necessary for states of affairs. But this suggestion simply does not fit the view. States

of affairs are defined in terms of bare particulars (Armstrong 1989, 41). Taking the

present suggestion at face value makes the role of bare particular in the ontology of

states of affairs out to be little more than the role of a dent in the ontology of a dented

car; but this cannot be right since we can give a fully satisfying fundamental

ontological description of the dented car without any mention of dents. If bare

particulars are like dents in this respect, then they are not important to fundamental

ontology—states of affairs ought to be able, possibly, to get along without them. But

36

what might a state of affairs be that fails to involve any particular? Presumably, it

would just be a universal, or perhaps a first order universal plus the second order

universal being exemplified. Yet if universals—qua constituents or aspects of states of

affairs—can exist independently of any bare particular, then why not allow that bare

particulars might exist independently of any (sparse) universal being exemplified? I

see no independent reason for this disallowance. If there is no such reason, however,

then the present suggestion places the states of affairs theorist back in front of the

Humean crosshairs from which he seeks to escape.

The final pro-bare-particulars treatment of 2.1 that I would like to discharge,

found in (Sider 2006), is by my lights the most formidable since it denies neither the

Humean combinatorial scheme nor that bare particulars form a fundamental kind.

Indeed, Sider is happy to grant 2.1. He does, however, go on to argue that similar

considerations apply to bundle theory, thus rendering it vulnerable to 2.2 and 2.3. His

thought is that, given any plausible combinatorialism, the bundle theorist is committed

either to the possibility of unbundled universals or unbundled tropes. As Sider since

has acknowledged in personal correspondence, however, this tu quoque against the

argument for premise 2 is uncompelling. In the universal case, Sider’s thought is

beside the point: an unbundled universal is not a particular at all, so 2.2 simply does

not apply. In the trope case, Sider’s thought is false: though a trope that is unbundled

with any distinct trope is a particular, it is not a particular that fails to exemplify any

properties. After all, the trope bundle theorist is free to hold that every trope is trivially

bundled with—and thus exemplifies—itself. This concludes my argument for 2.1.

37

I will offer two arguments for 2.2. The first goes as follows. In order for

something to exist, it must be like something (and unlike other things); that is, it must

have a character;12 in order for something to have a character, it must exemplify at

least one sparse property; therefore (by the transitivity of being necessary for), in

order for something to exist it must exemplify at least one sparse property. Finally, if

an object itself must exemplify a sparse property, it follows that the object cannot exist

independently of at least one sparse property’s being exemplified.

Sider (2006) responds to this argument by drawing a distinction between a

strong and a weak sense of ‘having a character’. On the strong sense, it is true that

exemplifying at least one sparse property is necessary for having a character. On the

weak sense, however, no such condition holds. According to Sider, only the weak

sense is necessary for existence. What, then, is the weak sense? Sider tells us that x’s

having a character in the weak sense allows us to answer questions like ‘what is x like’

and ‘to what is x similar’? In the case of a bare particular that exemplifies no

properties, we will be able to answer these questions negatively: ‘x is not like __’;

‘Does x exemplify __? No!’, etc. So while x may not exemplify any properties, we

will still be able to furnish information about what x is like.

This response is unsatisfying. If all that we can say about some bare particular

x is that for each sparse property F, x does not exemplify F, then we lack the resources

for distinguishing a world in which only x exists, on the one hand, and, on the other

hand, a world in which there is nothing whatsoever. But surely if x has a character

then it will be possible to draw a clear distinction between x and nothing. Sider is right 12 Notice that the requirement that an item be like something does not entail that the item resembles any distinct item. It may be utterly unique and yet still be like something.

38

that some information can be expressed about objects like x, for negative information

is still information. Where he errs is in thinking that this sort of information is

sufficient for showing that x has a character.

The second argument in favor of 2.2 turns on the plausible claim that there is

some set of sparse properties for every world. If an object a at world w existed free of

any sparse exemplifications at w, then it would be a part of w that the sparse properties

failed to characterize. But, by definition, the sparse properties characterize the world

completely. So the existence of a violates the plausible claim that there are sparse

properties at arbitrary w. Assuming that this claim about sparse properties is true, there

can be no such object as a. Granted, there have been arguments over how to determine

which of the properties at a given world are the sparse ones (Schaffer 2004), but to my

knowledge there are no arguments in the literature to the effect that there are worlds

that contain no sparse properties.

I conclude that 2.2 holds and that the argument for premise 2 of the Main

Argument is sound. It follows from premises 1 and 2 of the Main Argument that zero-

dimensional material objects do not exist.

39

Chapter 2

T-Gunk and Exact Occupation ABSTRACT: An object is T-gunky just in case all of its parts (i) have proper parts and (ii) are of non-zero measure in every spatial dimension. I show that a recent argument due to Hud Hudson—though not intended as a threat to gunk—bears on the possibility of T-gunky material objects in non-gunky space. The friend of T-gunk can circumvent Hudson’s argument without abandoning pointy space or standard mereology, but only by taking on a novel understanding of the relation of exact occupation of pointy regions by T-gunky objects. I argue that this novel understanding is tenable from the perspective of the committed friend of material T-gunk. With this new conception of exact occupation in hand, the friend of material T-gunk need not look to theories of gunky spacetime in order to reconcile the possibility of material gunk with mathematical physics. 1. Introduction An item is gunky just in case all of its parts have proper parts.13 Many philosophers

take seriously the possibility that there are gunky material objects or spacetime

regions. Indeed, some philosophers refuse to rule out that the actual world contains

gunk.14 In this vein, some interesting work has been done on the issue of whether

actual physics can be carried out in a gunky spacetime.15 Other interesting work has

been done on the questions of whether gunky matter might be located at a non-gunky

13The dialectic with which this chapter is concerned presupposes spacetime substantivalism. I will assume throughout that all parts of material objects are material objects and that all parts of regions are regions. Following orthodoxy, I sometimes use ‘gunk’ as a mass noun to talk about objects or regions that are gunky. The term was introduced in Lewis, D., Parts of Classes, (Oxford: Basil Blackwell, 1991). 14 See J. Schaffer, ‘Is There a Fundamental Level?’, Nous, 37 (2003), pp. 498-517, and ‘Monism: The Priority of the Whole’, Philosophical Review (forthcoming); F. Arntzenius, ‘Is Quantum Mechanics Pointless?’, Philosophy of Science, 70 (2003), pp. 1447-1457. 15See B. Skyrms, ‘Logical Atoms and Combinatorial Possibility”, The Journal of Philosophy, 90 (1993) pp. 219-32; F. Arntzenius, ‘Gunk, Topology and Measure’ PhilSci Archive (2004); P. Roeper, ‘Region-Based Topology’, Journal of Philosophical Logic, 26 (1997), pp. 251-309; J. Russell, ‘The Structure of Gunk: Adventures in the Topology of Space’, Oxford Studies in Metaphysics, v.5 (Oxford: OUP, Forthcoming, D. Zimmerman, ed.). Important earlier work includes A.N. Whitehead, An Enquiry Concerning the Principles of Natural Knowledge, (Cambridge: Cambridge University Press, 1919) and The Concept of Nature, (Cambridge: Cambridge University, 1920); A. Tarski, ‘Foundations of the Geometry of Solids”, in J. H. Woodger, (ed.) Logic, Semantics, and Mathematics, (Oxford: Clarendon Press, 1956) pp. 24-9.

40

region and whether non-gunky matter might be located at a gunky region.16 To get

clear about how these considerations link up, some labels will be useful:

(Gunk): There are worlds with our physical laws that contain gunky material objects.

(Required): Gunky material objects can only exist in a gunky spacetime.

(Optional): Gunky material objects can exist in a non-gunky spacetime.

(Adequate): At least one theory of gunky spacetime that is adequate for doing physics

like ours is either presently available or forthcoming.

(Inadequate): No theory of gunky spacetime that is adequate for doing physics like

ours is either presently available or forthcoming.

There are four general camps into which philosophers who endorse (Gunk) and are

committed with respect to the other four theses might fall. Of course, the

(Required)/(Inadequate) camp is likely to be empty since those theses taken together

are in deep tension with (Gunk). The members of the (Required)/(Adequate) camp are

presumably at peace with themselves, though both (Required) and (Adequate) are

controversial. My interest will be in the two remaining camps, the members of which

endorse (Optional). As we have seen, any theorist who endorses both (Gunk) and

(Inadequate) in all likelihood endorses (Optional); and there is no prima facie barrier

to one’s endorsing (Gunk), (Optional), and (Adequate).

16 See F. Arntzenius and J. Hawthorne, ‘Gunk and Continuous Variation’, The Monist 88 (2005), pp. 441-465; K. McDaniel, ‘Gunky Objects in a Simple World’ Philo 9 (2006), pp. 47-54; R. Saucedo, ‘Parthood and Location’, Oxford Studies in Metaphysics, v.6 (Oxford: OUP, Forthcoming, D. Zimmerman, ed.). For present purposes, non-gunky spacetimes are to be understood exclusively as “pointy” spacetimes, that is, as continuous manifolds of infinitely many zero-dimensional spatiotemporal atoms.

41

I will show that a recent argument due to Hud Hudson threatens (Optional) in

an important way that previously has gone unnoticed.17 The argument, which will be

rehearsed in detail in section 3 below, deploys the mereological principle of weak

supplementation:18

Weak Supplementation (WS): Necessarily, for all material objects, x and y: if x has y

as a part and y does not have x as a part then there is some material object, z, such that

z is a part of x and z does not overlap y.19

In short, the argument is that since an arbitrary three dimensional object a in a pointy

region will have some proper part b that does not differ from a in positive volume, WS

tells us that there is some further part c of a that lacks positive volume. The threat the

WS argument poses to (Optional) has gone unnoticed because it only applies to

(Optional) under a certain intuitive conception of gunky material objects that I call ‘T-

gunk’. Unlike traditional gunky objects, the parts of T-gunky objects must be extended

in every spatial dimension. After motivating T-gunk as against alternative conceptions

of gunky material objects, I will defend (Optional) by showing that the WS argument

17 H. Hudson, ‘The Liberal View of Receptacles”, Australasian Journal of Philosophy 80 (2002), pp. 432-39 and The Metaphysics of Hyperspace, (Oxford: OUP, 2005). Very similar arguments are discussed (though not endorsed) in T. Bays, ‘Hudson on Receptacles’, Australasian Journal of Philosophy, 81 (2003), pp. 569-572 and G. Uzquiano, ‘Receptacles’, Philosophical Perspectives 20 (2006), pp. 427-52. An ancestor of this paper contained a detailed discussion of why these other instances of the argument do not fare any better than Hudson’s. I here suppress that discussion for the sake of space. 18 Hudson calls weak supplementation ‘the Remainder Principle’. 19 Notice that WS disallows exact co-location among wholes and proper parts. In the rest of the paper I will operate under an assumed rejection of such co-location. The present understanding of the mereological technical term ‘overlap’ and subsequent topological technical terms like ‘connected’ is the standard one of R. Cartwright, ‘Scattered Objects’, in Philosophical Essays, (Cambridge MA: MIT Press, 1987), pp. 171-86.

42

is uncompelling. In doing so, I will suggest and defend the controversial claim that T-

gunky objects exactly occupy more than one region at a single time. I will argue that

this surprising claim can be defended via independently plausible theses about

material objects that are neutral with respect to (Gunk) and (Optional).

If indeed the T-gunk readings of (Gunk) and (Optional) are tenable then my

defense of (Optional) as against the WS argument ought to be of fairly general

interest. Moreover, even those philosophers who do not endorse (Gunk) but rather the

weaker claim that material gunk only exists in very distant possible worlds might well

endorse (Optional). My arguments ought to be of interest to these theorists as well.

Finally, though my principal interest is in (Gunk) and (Optional), the WS argument is

aimed at the thesis that three-dimensional objects in general—not just gunky three-

dimensional objects—must have positive volume. My counter-arguments should be of

interest to those attracted to this more general thesis independently of their views

about gunk.

2. T-Theory and T-Gunk

Following Hudson, let us use ‘T-theory’ to name the thesis that, necessarily, all

material objects are of non-zero measure in every spatial dimension.20 Intuitively, the

thesis is that all material objects have positive volume. If T-theory is true then, in

three-dimensional space, zero-, one-, and two-dimensional regions are not eligible for

exact occupation by material objects; they cannot be ‘receptacles’. In this section, I

20 The ‘T’ in ‘T-theory’ abbreviates ‘three’, as in (at least) three dimensions.

43

explain the connections among T-theory, the WS argument, and the T-gunk reading of

(Optional). I then motivate the T-gunk reading against competing conceptions.

A material object is composed of T-gunk just in case each of its parts (i) has

proper parts and (ii) is of non-zero measure in each spatial dimension. To begin to see

how T-theory, the WS argument, and the T-gunk reading of (Optional) relate, notice

that the claim that T-gunky objects exist can be formulated as a restricted version of T-

theory. The idea is to relativize T-theory to all and only the parts of an arbitrary T-

gunky material object:

(T-Gunk): There is a class of material objects C whose members (i) consist of all and

only the parts of some gunky material object and (ii) are of non-zero measure in each

spatial dimension.

(T-Gunk) says that at least one T-gunky object exists. Clause (ii) makes it a version of

T-theory; clause (i) makes it a restricted version. As we have seen, the WS argument

against T-theory proper concerns an arbitrary three-dimensional material object and

attempts to show that, in pointy space, the object has at least one less-than-three-

dimensional part.21 So, since T-gunky objects are three-dimensional, if the argument

21 Hudson (op. cit.) explicitly registers the assumption that gunk is impossible, referring readers to his A Materialistic Metaphysics of the Human Person, (Ithaca: Cornell University Press, 2001) for arguments. So it is not quite correct to say that his argument against T-theory concerns a completely arbitrary three-dimensional object since he does not have gunky three-dimensional objects in mind in giving the argument. However, what matters for our purposes is that his argument does not explicitly presuppose that the three-dimensional object it concerns has any less-than-three-dimensional parts (lest it explicitly beg the question against T-theory). Since the T-gunk theorist will not share Hudson’s assumption that gunk is impossible, and since Hudson’s form of argument does not presuppose that the object it concerns has any less-than-three-dimensional proper parts, the T-gunk theorist who endorses (Optional)

44

goes through at all then it goes through against (T-Gunk), for it would show that

clause (ii) of (T-Gunk) is false. It would then follow that (Optional) is false for

material T-gunk. Since the WS argument presupposes a pointy spacetime, it threatens

only (Optional) and not (Required).

Before getting into the details of the WS argument in the following section, it

will be worthwhile to look at some alternative conceptions of gunk so that we can

better appreciate the import of T-gunk readings of (Gunk) and (Optional). One way

that a material object can be gunky without being T-gunky is for it to be point-sized.

At first blush this may sound strange since we typically think of gunky objects as

being extended in at least one dimension. If we allow for objects that are co-located

with their proper parts, however, then we can have pointy gunk. What is required is an

infinite collection of co-located point-sized objects, each of which has at least one of

the others as a proper part. Call this ‘strange gunk’. Even if we grant that strange

gunky objects are possible, however, they are surely too strange to be the kind of

objects that most theorists drawn to (Gunk) or (Optional) endorse. So if the T-gunk

reading of those theses is to be rejected, we will need to look elsewhere than strange

gunk for an alternative conception.

On a less strange alternative to T-gunk, gunky material objects are extended

only in one or two spatial dimensions. Call this ‘thin gunk’. I find thin gunk quite

implausible since there is no reason to restrict (Gunk) and (Optional) to less-than-

three-dimensional objects.

must face Hudson’s form of argument as a threat to the possibility of T-gunky material objects in non-gunky space.

45

A related alternative to T-gunk posits (spatially) three-dimensional gunky

material objects that have some less-than-three-dimensional parts. Call this conception

‘fit gunk’ (since it is neither as “fat” as T-gunk nor as “thin” as thin gunk). An

example of a fit gunky object is a cube, a, one proper part of which is a square, b. I am

happy to concede that an advocate of (Gunk) or (Optional) might well endorse fit

gunky objects since what is important for present purposes is that he need not endorse

them to the exclusion of T-gunky objects.

Moreover, I believe that a sensible case can be made as follows for the claim

that T-gunky material objects are more intuitive than fit gunky material objects.

Metaphorically speaking, if some square b is a proper part of some cube a then there is

a “chopping” mechanism that applies to a along the dimension in which b has no non-

zero extension, a mechanism capable of yielding proper parts of a that are not

extended in the relevant dimension.22 There is no principled reason to think that this

metaphorical mechanism must, of necessity, only be applicable with respect to one

particular dimension. In the most intuitive cases, it ought to be equally applicable to b

along one of the other dimensions, yielding a line-like object c that is a part of b (and

thus a part of a). Yet again there is no principled reason to deny that the mechanism

might apply to c along the remaining dimension, yielding a point-sized object, d. But

if the mechanism in question originally yielded a proper part (namely b) of the object

to which it was applied (namely a), then it ought to continue to be capable of yielding

proper parts when applied in other dimensions. If this much is granted, however, then

the result is that d is a proper part of a. So, given that a is gunky and d is point-sized, 22 The parts yielded by this metaphorical “chopping” mechanism may remain undetached from the relevant whole. Metaphorical “chopping” does not entail literal division.

46

we have either reached a contradiction or are required to hold that a has a strange

gunky part. Going the latter route, however, would render fit gunky objects far less

intuitive than T-gunky objects.

The final alternative to T-gunk that I will mention is what we might call

‘mixed gunk’. A mixed gunky material object has both (i) T-gunky parts and (ii) either

fit or (inclusive) strange gunky parts. For present purposes I am happy to allow that

the advocate of (Gunk) or (Optional) might endorse mixed gunk since the existence of

a mixed gunky object requires the existence of at least one T-gunky object.

In light of these considerations, T-gunk deserves to be taken seriously as a

competitive conception of material gunk, especially of three-dimensional gunky

material objects. Though the T-gunk conception has not been explicitly stated and

labeled in the literature until now (to my knowledge), I conjecture that it is what many

theorists have had in mind when conceiving of gunky three-dimensional material

objects. (This is not to deny, of course, that some mereologists would reject the T-

gunk conception.) At any rate, I will rest my case for the T-gunk reading of (Gunk)

and (Optional) here. In the next section, I will focus on Hudson’s actual argument

against T-theory proper, rather than saying at each step how it can be modified to

apply to a T-gunk reading of (Optional). Likewise, though I will respond to the

argument on behalf of the advocate of T-theory proper, my treatment applies

straightforwardly to the interests of the T-gunk theorist who endorses (Optional); one

needs only to restrict my treatment to T-gunky material objects. In the conclusion, I

return from examining the prospects of T-theory proper to discuss briefly how my

47

arguments bear on T-gunk readings of (Gunk) and (Optional). For the next several

sections, however, I will set concerns about gunk aside.

3. The Weak Supplementation Argument Against T-Theory

T-theory bars material objects that are extended in less than three dimensions, but its

adherents may well want to accept zero-, one-, and two-dimensional regions.

Consequently, the T-theorist needs a story about the relationships among material

objects, regions, proper parts of material objects, and proper subregions that allows for

less-than-three-dimensional regions while remaining consistent with T-theory. Hudson

indicates (correctly, I think) that T-theory is most difficult to refute when construed as

entailing the following mereological principle.23

T-theoretic Proper Parthood Principle (TPP): Necessarily, for any material object,

x, region which x exactly occupies, R, and proper subregion of R, r: r is exactly

occupied by a proper part of x just in case r is of non-zero measure in every spatial

dimension and either (i) r is connected or (ii) r is a disconnected fusion of regions

r0…rn such that each of r0…rn is itself both connected and of non-zero measure in

every spatial dimension.

23 Hudson (op. cit.), p. 54. (TPP) is not quite the principle that Hudson discusses, differing only in its inclusion of clause (ii). Clause (ii) is important, however, since without it (TPP) rules out the possibility that material object x might have a proper part y such that y is a scattered object. Thanks to ---------- for pointing out to me that (TPP) would be unattractive to many if it ruled out the possibility of proper parts like y. Similar remarks apply to (TPP*), clauses (ib) and (iib).

48

Hudson gives two versions of the WS argument. The first is directed at T-theory plus

TPP; the second is directed at T-theory plus the following stronger principle:

(TPP*): Necessarily, for any material object, x, region which x exactly occupies, R,

and proper subregion of R, r1: r1 is exactly occupied by a proper part of x just in case

(i) r1 is of non-zero measure in every spatial dimension and either (i.a) r1 is connected

or (i.b) r1 is a disconnected fusion of regions r10…r1

n such that each of r10…r1

n is itself

both connected and of non-zero measure in every spatial dimension; (ii) there is some

subregion of R, r2, which is of non-zero measure in every spatial dimension and either

(iia) r2 is connected or (iib) r2 is a disconnected fusion of regions r20…r2

n such that

each of r20…r2

n is itself both connected and of non-zero measure in every spatial

dimension; (iii) r1 and r2 do not overlap; and (iv) R is the union of r1 and r2.24

If the following principle is conjoined with WS then we can generate a

counterexample to each of T-theory, TPP and TPP*:

Contra-TPP (CTPP): Necessarily, for any material object x, region which x exactly

occupies, R, and subregion of R, r, where r is obtained by subtracting some zero-, one-

, or two-dimensional region from R: if r is exactly occupied by some part of x (call it

‘y’) then y is a proper part of x.

24 Notice that TPP* entails TPP.

49

Here is how to get the counterexample. Consider a material sphere a and one of its

parts b such that a exactly occupies a region that is one point “larger” than the region

exactly occupied by b. CTPP tells us that b is a proper part of a. But this entails (given

classical extensional mereology) that a is not a part of b. WS then tells us that there is

some material object z that is a proper part of a and does not overlap b. The only

eligible object(s) around to play the z role must exactly occupy the point that marks

the difference between a’s region and b’s. But any such object is zero-dimensional;

and since it is a proper part of a material object, it is a material object. So WS plus

CTPP generates a counterexample to T-theory, TPP, and TPP*. Notice, by contrast,

that WS by itself is quite consistent with each of T-theory, TPP, and TPP*. We are

now in position to look at Hudson’s versions of the argument, upon which the present

counterexample is based.

Version 1: Consider Sphere, a material sphere that is not exactly co-located with any

other material object. Consider the region exactly occupied by Sphere, Rs, and the

subregion of Rs, r, obtained by subtracting Rs’s center point. Consider some part of

Sphere that exactly occupies r; call it ‘SphereMinus’. SphereMinus is a part of Sphere;

Sphere is not a part of SphereMinus. So, by WS, there is some material object that is a

part of Sphere but that does not overlap SphereMinus. This object is clearly some

point-sized occupant of the center point of Rs. So there are proper parts of material

objects that are of zero measure in at least one spatial dimension.

50

Version 2: Consider Sphere and Rs once again. Consider the northern hemisphere of

Rs, minus the two-dimensional, disc-shaped region whose perimeter is Sphere’s

equator (call some exact occupant of this two-dimensional region ‘Disc’), and call

some exact occupant of the resultant region ‘Northerner’. Consider the southern

hemisphere of Rs minus the region exactly occupied by Disc and call some exact

occupant of the resultant region ‘Southerner’. Now suppose that Northerner and

Southerner are both parts of Sphere. Call the fusion of Northerner and Southerner

‘Almost-A-Sphere’. Almost-A-Sphere is a part of Sphere; Sphere is not a part of

Almost-A-Sphere. So, by WS, there is some material object that is a part of Sphere but

that does not overlap Almost-A-Sphere. Clearly Disc, which is two-dimensional, is

just such an object. So there are proper parts of material objects (which are themselves

material objects) that are of zero measure in at least one spatial dimension.

The idea behind replacing TPP with TPP* is that doing so blocks version 1. Here is

how. A crucial premise in version 1 is that Sphere is not a part of SphereMinus. This

claim follows (given classical mereology) from the claim (implicit in version 1) that

SphereMinus is a proper part of Sphere. And, indeed, SphereMinus meets the TPP

criteria for being a proper part of Sphere. But it does not meet the TPP* criteria. To

see this, consider Sphere as a test case for the universal principle TPP*. Plug in

‘Sphere’ for occurrences of the material object variable (‘x’) and plug in tokens of a

name of the region exactly occupied by SphereMinus for occurrences of the first

subregion variable (‘r1’). Notice that, in the test case at hand, the region most eligible

for playing the r2 role (that is, the region whose name ought to be plugged in for

51

occurrences of ‘r2’) is the point-sized region at the center of Rs. But this region—most

eligible though it may be—still fails to play the r2 role since, qua zero-dimensional, it

fails clause (ii) of TPP*. Since there is no further (non-question-begging) evidence for

the claim that Sphere is not a part of SphereMinus, version 1 relies on an unsupported

implicit premise when viewed under the lens of TPP*.

Version 2 is intended to go through even against the T-theorist who is

equipped with TPP*. Since the region occupied by Northerner (or Southerner,

respectively) is both (i) extended in all spatial dimensions and (ii) such that its union

with the distinct region occupied by the fusion of Southerner (Northerner) and Disc is

Rs, all TPP* criteria for Northerner’s (Southerner’s) being a proper part of Sphere are

met. I take it that this result is supposed to make version 2 a stronger threat to the T-

theorist than is version 1 since, while we have seen that the T-theorist (when equipped

with TPP*) has good reason to doubt that SphereMinus is a proper part of Sphere, she

cannot plausibly deny that Northerner and Southerner are legitimate proper parts of

Sphere.

However, the T-theorist’s inability to make the aforementioned denial is

irrelevant. Recall that what is both controversial and crucial in version 1 is the premise

that Sphere is not a part of SphereMinus. Similarly, the controversial, crucial premise

of version 2 is that Sphere is not a part of Almost-a-Sphere—not that Almost-a-Sphere

is a part of Sphere or that Northerner and Southerner are proper parts of Sphere. The

principal reason for accepting this premise seems to be that Almost-a-Sphere is a

proper part of Sphere. However, much like in version 1, the T-theorist who is

equipped with TPP* has ground to deny this claim. We can substitute ‘Sphere’ for

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occurrences of ‘x’ and substitute tokens of a name for the region exactly occupied by

Almost-a-Sphere for occurrences of ‘r1’. Since the only region eligible to play the r2

role in TPP* (given the preceding substitutions) is the two-dimensional region exactly

occupied by Disc, we again have a failure to satisfy clause (ii) of TPP*.

I conclude that neither version 1 nor version 2 of Hudson’s anti-T-theory line

is a decisive threat to T-theory plus TPP*. A general diagnosis can be made. In both

version 1 and version 2, the WS argument only goes through if we presuppose CTPP.

For without CTPP, one cannot conclude that SphereMinus or Almost-a-Sphere are

proper parts of Sphere. We have already seen that WS plus CTPP is inconsistent with

TPP, TPP* and T-theory. To rely exclusively on WS plus CTPP in order to run an

argument against T-theory plus TPP*, then, is to beg the question.

4. Parts and Exact Occupation

“Not so fast,” the anti-T-theorist might object at this point, “there is indeed further

non-question-begging evidence for the claim that Sphere is not a part of SphereMinus,

namely, the following mereological principle:

(Parts): x is a part of y just in case the region exactly occupied by x is included in the

region exactly occupied by y.

(Parts) entails that, since Sphere’s region is not included in SphereMinus’s, Sphere is

not a part of SphereMinus.”

53

Like WS, (Parts) is a widely accepted and intuitively attractive mereological

principle. Consequently, the T-theorist will do well to resist rejecting either of them, if

she can manage it. Fortunately, she can. I will argue that (Parts) only delivers the

conclusion that Sphere is not a part of SphereMinus if it is interpreted in a way that

presupposes the falsity of a thesis that any adherent to TPP* ought to endorse. Now, if

this TPP*-contravening interpretation of (Parts) were the only plausible interpretation

available, then the anti-T-theorist would be correct in claiming that the plausibility of

(Parts) constitutes non-question-begging evidence for the claim that Sphere is not a

part of SphereMinus. I will argue, however, that there is a plausible TPP*-friendly

alternative interpretation of (Parts) that leaves its intuitive mereological content intact.

If I am right, then to interpret (Parts) in the way required in order to infer from it that

Sphere is not a part of SphereMinus is to beg the question against the T-theorist.

Toward seeing that (Parts) is open to a TPP*-friendly interpretation, notice that

it is consistent with the following very similar principle:

(Parts*): x is a part of y just in case there is some region rx which x exactly occupies

and some region ry which y exactly occupies, and ry includes rx.

(Parts*) plays the same mereological role for the anti-T-theorist as does (Parts), for the

anti-T-theorist assumes that every material object exactly occupies only one region at

any given time, and the only significant difference between (Parts) and (Parts*) is that

the former is formulated using definite descriptions to denote exactly occupied regions

while the latter is formulated using existential quantification. However, (Parts) and

54

(Parts*)—though expressions of (basically) the same mereological principle—differ

greatly in the extent to which the language of each is amenable to the T-theorist. I

submit that the T-theorist ought to understand the exact occupation relation, as it

applies among material objects and pointy regions, differently than does the anti-T-

theorist. Specifically, the T-theorist may well hold that a given material object exactly

occupies more than one region, so long as none of the regions it exactly occupies have

a difference of positive volume.25 Call this thesis ‘multi-regional exact occupation’.

To more precisely understand multi-regional exact occupation for some

material object x, begin by considering some region R such that the traditional

understanding of exact occupation tells us that x exactly occupies R. Now consider the

class C of regions R0...Rn such that (i) each Ri is less-than-three-dimensional and (ii)

every subregion of each Ri is at zero distance from some subregion of R. According to

multi-regional exact occupation, x does not only “exactly occupy” R; it also “exactly

occupies” every region that results from taking R and adding or subtracting any

member of C.

The TPP*-friendly interpretation of (Parts) that I want to suggest, then,

employs this multi-regional understanding of exact occupation. On this interpretation,

it does not follow from (Parts) that Sphere is not a part of SphereMinus. To see this,

notice that by multi-regional exact occupation there is at least one region (indeed there

are infinitely many regions) that Sphere and SphereMinus both exactly occupy. Since

25 There is one small modification to TPP* that is required in order for it to accord with multi-regional exact occupation, but it is not a modification that affects TPP*’s role in furnishing a way around the WS argument. The modification is to clause (iii). We will see below that the friend of multi-regional exact occupation must allow for a certain kind of possible overlap between r1 and r2; what she cannot allow is for the intersection of r1 and r2 to have positive volume. So the modified version of clause (iii) would read: ‘r1 and r2 do not have an intersection of positive volume’.

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any region is included in itself, (Parts), on the interpretation I am suggesting on behalf

of the T-theorist, tells us that Sphere is indeed a part of SphereMinus.

5. Prospects for Defending Multi-Regional Exact Occupation

Multi-regional exact occupation is a radical suggestion—indeed prima facie it may

ring oxymoronic. I will argue, however, that it is neither ad hoc nor implausible, given

an antecedent commitment to T-theory plus TPP*. Moreover, multi-regional exact

occupation does not throw T-theoretic mereology out of whack, for (Parts) as

interpreted via multi-regional exact occupation (let us call (Parts) so interpreted

‘(Parts)T’) plays the same general mereological role as does (Parts) on its traditional

interpretation. The need for a second interpretation merely reflects what is at issue

between the T-theorist and her opponent, namely, whether certain regions are ever

exactly occupied by material objects. Since the notion of exact occupation is central to

this issue, it should come as little surprise that the T-theorist might understand it

differently than does her opponent.

Multi-regional exact occupation is a sufficiently interesting and useful

suggestion for the advocate of T-theory that she might just accept it and bite the

dialectical bullet in the face of those who would demand independent motivation. It

will be worthwhile, however, to see whether she can do more than bite the bullet. One

promising strategy is to appeal to a more general thesis that she and her opponent both

endorse and then explain that it is her dual commitment to this more general thesis and

to T-theory that leads her to multi-regional exact occupation. While this motivating

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line is thus not entirely independent of T-theory, it does indicate that multi-regional

exact occupation is crucial to the general plausibility of T-theory.

One bipartisan general thesis that serves this purpose concerns the

supervenience of facts about exact occupation on facts about parthood. Let us

understand the exact occupation relation as a class EO of ordered triples, each

consisting of a material object, a region, and a time. The T-theorist who endorses

multi-regional exact occupation will affirm, and her opponent will deny, that for some

material object x and time t, there are multiple ordered triples in EO that contain both x

and t. The supervenience thesis that both the T-theorist and her opponent endorse can

now be stated as follows.

(Supervenience): For any material object x, facts about which regions and times x is

tripled with in EO supervene on facts about the gain, loss, and movement of x’s

(spatial) parts over time.26

Put differently, if x exists at t and t' (≠t) and is tripled with r and t but not with r and t',

then x must have gained, lost, or moved a (spatial) part between t and t'. If the T-

theorist were to tell a story about exact occupation that violated (Supervenience), then

she would be venturing out on a dangerous limb indeed, for how could there be a

26 Recall that I have set aside cases of proper part/whole co-location. I also ignore concerns about the possibility of extended simples. (Hudson explicitly rejects extended simples). Finally, I ignore cases of “internal” movement, that is, cases where a proper part of an object moves without thereby disrupting the whole object’s exact occupation relation with the region(s) it originally exactly occupied. For example, consider an inner three inch cubic undetached proper part of a five inch cube of matter. The inner cube might shift within the one-inch thick outer “box” of matter without the five inch cube failing to exactly occupy the region it occupied before the shift. At any rate, I mention this sort of case only to set it aside.

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change in which region an object exactly occupies unless there were a change in the

inventory or location of its parts? If she wants to uphold (Supervenience), however,

then she had better endorse multi-regional exact occupation for material objects.

To see this, suppose that the T-theorist accepts traditional, mono-regional exact

occupation. She then seems to have no principled reason to deny that a given sphere

might exactly occupy a certain open region at t and then “grow” to exactly occupy its

closure at the next instant t′.27 But the T-theorist seems unable to allow, on pain of

contradicting T-theory, that the parthood facts in which the sphere is involved can

subvene this intuitively possible case of “growth,” given that we have agreed to set

aside the possibility that the sphere is an extended simple. Notice that (Supervenience)

mentions three kinds of parthood facts: movement, loss, and gain. Let us take each in

turn. The T-theorist cannot plausibly maintain that the sphere has “grown” merely by

having its parts move unless she maintains that some of the parts in question are

extended simples. Nor can she plausibly maintain that the sphere has “grown” by

losing proper parts since it seems clear that no material object can “grow” merely by

losing proper parts. This leaves only the possibility that the sphere has “grown” in

virtue of gaining some proper parts between t and t′. But the T-theorist cannot go this

route either since the would-be newly gained part of the sphere, namely, the part that

now exactly occupies (in the traditional sense) the boundary of the original open

27 Instead of this talk of ‘growth’ across time, one might consider ‘growth’ across worlds. That is, if we accept mono-regional exact occupation then we would seem to have no principled reason to deny that some actual open sphere might have been closed. (Supervenience) applies in both the temporal and modal cases.

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region, is less-than-three dimensional.28 The upshot is that the T-theorist must either

give up T-theory and maintain that the sphere has “grown” by gaining a two-

dimensional proper part or reject (Supervenience). Notice, in contrast, that if she

accepts multi-regional exact occupation then she will have a principled reason to deny

straightaway that any material object might “grow” from exactly occupying a given

open region to exactly occupying its closure, for on multi-regional exact occupation,

any material object that exactly occupies an open region already exactly occupies its

closure.

Recourse to multi-regional exact occupation in response to the WS argument is

thus not ad hoc, for it is not merely a way to accommodate (Parts). It is also the most

plausible means by which the T-theorist can accept (Supervenience). Nor is multi-

regional exact occupation implausible, given antecedent commitment to T-theory,

since (Supervenience) is plausible across party lines.29

None of this is to say, however, that (Parts)T is uncontroversial. Before

concluding, I want to address two more worries about the sort of story I have been

urging the T-theorist to tell. First, one might suspect that—(Parts)T notwithstanding—

the defense of T-theory that I have described is inconsistent with an important

principle of extensional mereology. According to this principle (which is known

commonly as ‘anti-symmetry’), if x and y are distinct and x is a part of y then y is not a 28 I explicitly cancel the following re-description of the case at hand: The sphere “loses” a three-dimensional part a near its surface at t and “gains” in its place at t′ a new three-dimensional part b that is only boundary-size larger than a, thus resulting in boundary-sized “growth” from gain in parts that does not require the gaining of any part that fails to be three-dimensional. If this attempted re-description were accurate, then the case at hand would not raise a problem for the T-theorist. 29 For the sake of the dialectic strategy in play, it is worth pointing out that (Parts) does not entail (Supervenience). The “growth” example shows this. I can accept that an open sphere might “grow” to be a closed sphere without having to accept that boundary-sized regions contain proper parts of spheres. In such a case, I would be free to accept (Parts) while rejecting (Supervenience).

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part of x. Consider again Sphere and SphereMinus. Now, the point of my introducing

(Parts)T is that—unlike (Parts) as typically interpreted—it tells us that Sphere is a part

of SphereMinus (and, of course, that SphereMinus is a part of Sphere). But, the worry

goes, SphereMinus was introduced as exactly occupying a region which is included in,

but does not include, the region which Sphere was introduced as exactly occupying.

Presumably, then, Sphere and SphereMinus were introduced as being two entities. Yet

if they are indeed two then they cannot both be parts of each other without violating

anti-symmetry.

The answer to this worry is straightforward. The fact that the denotations of

‘Sphere’ and ‘SphereMinus’ were fixed via conceptually distinct descriptions does not

entail that they name numerically distinct objects. The theorist who endorses multi-

regional exact occupation is free to hold instead that the two names pick out a single

object under different descriptions. One of the descriptions highlights one of the

regions that the object exactly occupies while the other description highlights a

distinct region that the object exactly occupies. Anti-symmetry is upheld.

The second worry about multi-regional exact occupation is that it entails some

counterintuitive views about boundaries and contact for material objects. For example,

according to the T-theorist who endorses multi-regional exact occupation, no two

material objects x and y can be at zero distance from one another without it being the

case that x exactly occupies some region that overlaps some region that y exactly

occupies. To see this, suppose that x exactly occupies some region r1 and y exactly

occupies some region r2 (r1≠r2). Now either r1 and r2 are connected or not. If they are,

this can only be because at the places where they are connected one is open and the

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other closed. Suppose then that, where they are connected, r1 is open and r2 is closed.

Ex hypothesi, x exactly occupies r1; but by multi-regional exact occupation and the

fact that r1 exactly touches some boundary points of r2 (call the region composed of

these boundary points of r2 ‘r3’), it follows that x exactly occupies some region r4

which is the union of r1 and r3. But, ex hypothesi, y also exactly occupies a region that

has r3 as a subregion, namely, r2. So, if r1 and r2 are connected then x and y must

“overlap” in the sense described, that is, they must exactly occupy overlapping

regions. Suppose then that r1 and r2 are not connected. Since x and y are at zero

distance from one another, it must be the case that at least one of the regions

responsible for r1 and r2 being disconnected, call it ‘r5’, is itself no greater than two-

dimensional. Suppose for simplicity that r5 is such that all of its parts are at zero

distance from both r1 and r2. From multi-regional exact occupation and the fact that r5

is (i) less than three-dimensional and (ii) such that each of its parts is at zero distance

from both r1 and r2, we can infer that x exactly occupies some region r6 that is the

union of r1 and r5 and that y exactly occupies some region r7 that is the union of r2 and

r5. Since r6 and r7 overlap by having r5 in common, it follows that x and y exactly

occupy overlapping regions. Our T-theorist, it seems, is forced to hold that the

smallest distance relation that any two non-overlapping material objects can bear to

one another is non-zero. Notice, however, that the “overlapping” required here is not

material part-sharing, for the overlapping required involves only part-sharing for

regions: r4 and r2 have r3 in common and r6 and r7 have r5 in common, but no part of

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either x or y exactly occupies (even in the traditional sense) either of r3 or r5. Still,

there seems to be something counterintuitive going on.30

I am optimistic, however, that the counterintuitive element can be mitigated by

recourse to the above methodology, that is, by an argument for this non-traditional

view about contact from T-theory plus some attractive bipartisan theses. The idea is

that from the T-theoretic perspective of endorsing the bipartisan theses, non-

overlapping material objects—in the sense described—should not ever be at zero

distances from one another. The theses:

(Touch1) For any two material objects x and y and region of empty space R, if R is

‘between’ x and y in the sense that no direct path can be traced from any part of x to

any part of y without traversing R, then x and y are not touching.

(Touch2) For any two material objects x and y that are capable of touching one

another, if x and y are not touching then the (smallest) region precluding them from

touching (by being ‘between’ them) must have at least one subregion that is a

receptacle for at least one part of x or at least one part of y.31

30 Also, as mentioned in an earlier note, this form of overlap requires a minor modification to clause (iii) of TPP*. 31 Depending on how we parse (Touch2), it may be vulnerable to counterexample. However, since I have little doubt that (Touch2) could be reformulated with an exception clause that would circumvent the putative counterexample, and since doing so would render (Touch2) rather cumbersome, I have included the present footnote rather then the reformulation. Here is the counterexample. There is a possible world in which x and y are as close as each can possibly be to opposing sides of some cube, C, which, in the world in question, is nomologically immovable and impenetrable. In this case, x and y are precluded from touching one another by an object that occupies a region that lacks any subregions that are receptacles for any parts of x or y. Why does the counterexample depend on how we parse (Touch2)? Because if we understand ‘capable’ with sufficient modal conservatism, then x and y in the C world do not fit the bill. I have suppressed explicit modal interpretation of (Touch2) to preserve its generality.

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Call the conjunction of (Touch1) and (Touch2) ‘(Touch)’. (Touch) is consistent with T-

theory (still maintaining the assumption that spacetime is pointy) only if we allow that

touching material objects must overlap in the sense described above. To see why,

recall that the claim that touching material objects must “overlap” is entailed by T-

theory plus multi-regional exact occupation. Consider then whether (Touch) is

consistent with T-theory modulo multi-regional exact occupation. The following

reductio shows that it is not. Consider two arbitrary three-dimensional material objects

a and b such that (i) they are capable of touching and (ii) there is nothing but an empty

two-dimensional region R between them. According to (Touch1), these objects are not

touching. (Touch2) thereby requires, while T-theory forbids, that R has some

subregion that is a receptacle for some part of one of the objects. That is, (Touch) and

T-theory entail contradictory treatments of this case. Notice, however, that if we

supplement T-theory with multi-regional exact occupation, then the T-theorist will not

allow the reductio to get off the ground, for she will not allow that it is possible for

there to be nothing but an empty two-dimensional region between any two three-

dimensional objects. Rather, she will hold that any putative such region is occupied by

both of the material objects in question.32

32 As a useful further litmus test for the view that I have been urging on behalf of the T-theorist, one would do well to look at the four ‘constraints on any adequate metaphysics of extended objects’ offered in D. Zimmerman, ‘Could Extended Objects Be Made Out of Simple Parts? An Argument for “Atomless Gunk”’, Philosophy and Phenomenological Research 56 (1996), pp. 1-29. I will only mention one of these here, but I mention it because it is the only one of the four that is not met straightforwardly by the view that I have been developing. Zimmerman labels this constraint ‘(A)’: Every extended object has a left and right half which are discrete and are themselves extended objects. The reason this constraint is not obviously met by the view that I have been pushing is that it is not obvious on that view whether right and left halves of material objects ought to count as being discrete. I have two comments about this. First, while Zimmerman takes care to argue for this constraint against

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6. Taking Stock

It is not difficult to imagine the anti-T-theorist sounding the following complaint right

about now. “Look, you have done nothing to convince me that T-theory is any more

plausible than I thought it was at the outset. All you have done is to reshape the

discussion so that, instead of the T-theorist having a view that does not sit well with

one widely accepted principle, namely WS, she now has a view that does not sit well

with some other widely accepted principle, namely mono-regional exact occupation.”

But this complaint is overstated insofar as the reshaping that I have done is not

superficial. The reshaping falls out of a more specific and sophisticated

characterization of T-theory than Hudson considers, namely, a characterization

whereby T-theory does sit well with WS, as well as with (Parts). The original problem

for T-theory raised in Hudson’s version 1 was that T-theory leads to trouble given a

principle, WS, which the T-theorist herself ought to endorse. My reshaping shows that

the putative trouble does not follow so long as the T-theorist rejects mono-regional

exact occupation. However, and this is the crux of why the reshaping is not superficial,

I have argued from attractive bipartisan premises that the T-theorist ought not to

accept mono-regional exact occupation.

7. Conclusion

If one maintains that CTPP can be justified independently, then one owes the T-

theorist an argument. I have tried to show that the most charitably reconstructed

those who would deny that extended objects have right and left halves, he does not give an argument for requiring that the halves be discrete. Second, and more importantly, it is not clear that the contact-requires-overlap aspect of the view that I have been urging requires the right kind of overlap to be in conflict with discreteness.

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implicit argument (the argument from (Parts)) still begs the question against T-theory.

Moreover, the T-theorist can tell a perfectly coherent mereological story that does not

violate WS, anti-symmetry, (Supervenience), or (Touch). The T-theorist, as I have

defended her, does have to give up contact without a certain constrained form of

“overlap” for material objects, but it is far from obvious that this result should decide

the debate over T-theory.

I will close by saying something about how the above defense of T-theory

proper applies to the advocate of (Gunk) and (Optional) who endorses T-gunk. The

move to multi-regional exact occupation asks us—if I may invoke the going metaphor

in the literature33—to put on blurry goggles with respect to the relation of exact

occupation of regions of pointy spacetime by T-gunky objects. However, unlike other

ways of attempting to reconcile (Gunk) with some conception of spacetime that is

adequate for doing physics (e.g. Arntzenius’s measure theoretic approach), the present

suggestion does not ask us to blur any intrinsic properties of spacetime itself, whether

topological, mereological, or metrical. Rather, it asks us to “blur” the relation of exact

occupation. While there are certain drawbacks to this approach, it has the advantage of

leaving intact the traditional pointy spacetime presupposed in the standard formalisms

of contemporary physics.

33 Arntzenius (op. cit.)

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Chapter 3 Problems From Whole-Self-Distance Whole-self-distance is the phenomenon whereby an object is located at a distance from its whole self. Call the claim that whole-self-distance is possible ‘WSDP’. After arguing that WSDP is implausible, I discuss its relationship to (i) realism about immanent universals, (ii) realism about extended simples, and (iii) three-dimensionalism about persistence through time. The upshot of the discussion is that each of these views is tied sufficiently closely to WSDP for it to be the case that each is itself implausible. The chapter concludes with a discussion of recent attempts to defend each of (i)-(iii), some of which circumvent commitment to WSDP. I argue that each such attempt either fails to circumvent commitment to WSDP or is committed to something at least as problematic. The chapter thus serves to motivate away from universals, extended simples, and three-dimensionalism. 1. Introduction Let us take the relations of distance, exact location (of objects at regions) and

parthood as primitive, with proper parthood analyzed via parthood: for objects x and y,

x is a proper part of y just in case x is a part of y and y is not a part of x.34 We can now

characterize what I will call ‘strict partial location’. Object x is strictly partially

located at region r just in case some proper part of x is exactly located at r. Intuitively,

you are strictly partially located at the region where your nose is exactly located. We

can also characterize what I will call ‘loose partial location’. Object x is loosely

partially located at region r just in case (i) x is strictly partially located at some region

r1, (ii) r1 is an element in r (i.e. r1 is a subregion of r), (iii) there is some region r2

(≠r1) that is not an element in r; and (iv) x is strictly partially located at r2. Consider

the region that corresponds roughly to being “inside” my car at a given time; call it ‘r’.

34 See (Parsons 2007) for an insightful discussion of location relations. According to Parsons, exact location is not necessarily the most helpful location relation to take as primitive and it can be defined in terms of other location relations. By my lights, however, exact location is the most intuitive primitive. Indeed, in Parsons’s gloss of other location relations—specifically his ‘pervasive’ location relation—he slips in talk of exact occupation (2007, 203). This indicates that perhaps exact location is the most helpful primitive. At any rate, I do not think that anything I will say below turns on the issue of which location relation to take as primitive.

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There was a time t this morning, as I was stepping into my car, at which I was loosely

partially located at r: at t, my foot (say) was exactly located at some subregion of r

while all-of-me-minus-that-foot was exactly located at some region disjoint from r.

Let us return for a moment to your nose, which is located at a region at which

you are strictly partially located. The same holds for your chin. Moreover, your nose

and your chin stand in a non-zero distance relation to one another; correspondingly,

you are strictly partially located at regions that stand in a non-zero distance relation to

one another. Notice, however, that you do not stand in a non-zero distance relation to

yourself. You merely have proper parts that are exactly located at distant regions. To

claim that you stand at a non-zero distance to yourself would be absurd. The absurdity

is no fault of yours, of course. It is the fault of whole-self-distance (WSD), the

phenomenon whereby an object is located at a non-zero distance from its whole self.

Call the claim that whole-self-distance is possible ‘WSDP’. By my lights, the

absurdity of WSDP does not decrease when we conceive of objects more rarefied than

you, for example, immanent universals or extended simples, nor does it decrease when

we consider common sense objects like you as they persist across time.

Yet many philosophers defend theses that implicitly commit them to WSDP. I

have in mind realists about extended simples, realists about immanent universals, and

three-dimensionalists about persistence through time. In this chapter, I show that the

theorists just mentioned—given a standard first pass interpretation of their views—are

committed to the possibility of WSD. In section 2, I argue that WSDP is implausible,

leaving those whose views commit them to WSDP in dialectical trouble. Of course,

not all defenders of immanent universals, extended simples, and three-dimensional

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objects fit the first pass interpretations. In each case, however, the modified

interpretations either fail to circumvent WSDP or are independently problematic. The

extended simples case is dealt with in sections 3 and 4, the immanent universals case

in section 5, and the three-dimensionalist case in section 6. The chapter is thus a

sustained argument against extended simples, immanent universals, and three-

dimensional objects. To the extent that the argument is compelling, it gives us good

reason to look to tropes (if we are naturalists) in order to solve the problems of

property exemplification and to four-dimensionalism in order to give an account of

persistence. Moreover, if the argument against zero-dimensional material objects

given in Chapter 1 is sound, then we can conjoin the present argument against

extended simples to yield a powerful by-default motivation for material atomless

gunk.

2. Argument against WSDP

Let us assume that worlds are just spacetimes-plus-their-occupants and that some

worlds are capable of expansion and contraction.35 One perfectly kosher way for these

worlds to contract is by having some of their regions spontaneously pop out of

existence. Call this phenomenon the ‘annihilation’ of regions. There is a similar case

of annihilation for objects whereby occupants of regions can spontaneously pop out of

existence. Consider now one of these worlds, w. w contains an object x at time t such

35 Nothing I will say here turns on substantivalism (the thesis that spacetime is a substance) or supersubstantivalism (the thesis that every material object is numerically identical to some spacetime region). That is, my assumption about expansion and contraction is neutral as to whether spacetime is reducible to relations among material objects and as to whether material objects just are spacetime regions.

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that all of x is located at region r1, which stands in a non-zero distance relation to

some disjoint region, r2. Suppose r2 is annihilated, leaving the rest of w in tact. I

submit that whatever may have been the occupant(s) of r2, it is possible for x to

survive r2’s annihilation; after all, all of x is located at r1, and there is no reason to

believe that r2’s annihilation would necessarily have any affect on some object all of

which is located at r1. This neo-Humean claim that x might survive the annihilation of

r2—whatever the occupant(s) of r2 may be—can be generalized as follows.

Humean thesis: for any object x, time t, and disjoint regions r1 and r2, it is

compossible (i) for all of x to be located at r1 at t; (ii) for r2 to be annihilated at t; and

(iii) for x to survive through t.

Now consider a second thesis.

Annihilation thesis: for any object x, region r, and time t: if some part y of x is located

at r at t, and r is annihilated at t, then y is annihilated at t.

Suppose now that some object x stands in a WSD relation at time t by being

wholly at r1 and wholly at r2. Suppose r2 is annihilated at t. What happens to x? If x

must be annihilated, then the Humean thesis is false. However, if x were to survive

then the annihilation thesis is false. This leaves the defender of WSDP with a

dilemma. She must reject either the Humean thesis or the annihilation thesis. Neither

option strikes me as at all plausible. Since the defender of WSDP must accept one of

these options, she defends an implausible view.

I suspect that the defender of WSDP will not be immediately troubled by this

first dilemma. She will reject the annihilation thesis without flinching, citing x itself as

a counterexample. Since x still exists at r1 at t, it is not the case that x’s having had all

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of its parts located at r2 at t led to x’s annihilation when r2 was annihilated at t. Indeed

the defender of WSDP might even enlist the Humean thesis in support of the claim

that x constitutes a counterexample to the annihilation thesis; after all, all of x was

located elsewhere than at r2 at t. This is all fair enough. But the opponent of WSDP

will rightly press the proponent at this point as to what happened to x when r2 was

annihilated. After all, all of x was located at r2 at t, and r2 was annihilated at t, so

something must have happened to all of x at t. By my lights, the best available answer

on behalf of WSDP is that x simply changed its location from being at r1 and r2 to

being exclusively at r1. This is not only consistent with the survival of x; it entails the

survival of x.

We have seen that it was fair for the proponent of WSDP to reject the

annihilation thesis; likewise, it was fair for the opponent to ask her what happened to x

if not annihilation. What remains to be seen is whether it was fair for the proponent to

respond that x merely changed its location. I am dubious. In order for x to have merely

changed location, as the proponent of WSDP maintains, x must have changed location

at some time or other. Either the time at which x changed location was t or not. If it

was, then x was not located at r2 at t; but this is just to deny the stipulated set-up of the

thought experiment. However, if x did not change location until after t, then x was

located at r2 at t—the time that r2 was annihilated, so whatever it was that happened to

x at t, it was not a change in location. Yet something must have happened to x-at-r2 at

t. I submit that far and away the best explanation of what happened to x-at-r2 at t was

that it simply went out of existence. If this is right then either (i) x was never wholly

located at r1 or r2, or (ii) x was both entirely annihilated at t and entirely survived

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through t (which, of course, entails that x was not entirely annihilated at t—a

contradiction). On option (i), we allow that x-at-r2 was annihilated even though x-at-r1

survived because we allow that x-at-r2 and x-at-r1 are each a proper part of x. If this is

the case, then x cannot be wholly located at either of r1 or r2 at t. On option (ii), we

adhere with sheer faith to WSDP and are burdened with the resulting commitment that

all of x was annihilated and yet all of x also survived. Being forced to take one of these

options would be bad news for WSDP since (i) amounts to a rejection of WSD and (ii)

entails a contradiction. The attempt to escape the first dilemma has led the proponent

of WSDP to a second one that is even more clearly damning.

3. Being Committed to WSDP

In this section, I show that the realist about immanent universals, the realist about

extended simples, and the three-dimensionalist about persistence across time are all

committed to WSDP, given certain first-pass interpretations of their respective

theories.

The realist about immanent universals holds that one “object”, namely the

immanent universal greenness, both (i) lacks proper parts and (ii) is exactly located at

a certain scattered region rG, where rG is the fusion of all regions at which green

objects (or, more carefully, entirely green objects) are exactly located. Consider the

unique oldest dollar bill in my wallet, Bill, and the unique longest blade of grass in my

yard, Blade. Bill and Blade are both green. Suppose that Bill is exactly located at some

region r1 and that Blade is exactly located at some disjoint region r2, which stands in a

non-zero distance relation from r1. Since Bill and Blade are green, r1 and r2 are

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elements in rG. Since greenness is exactly located at rG, and since r1 and r2 are

elements in rG, it must be the case that greenness is located at each of r1 and r2. Now,

since greenness lacks proper parts, it cannot be the case that greenness is loosely

partially located at either r1 or r2. But if an object is located at a region and is not

loosely partially located at that region, then it must be the case that the whole object is

located at that region. It follows that all of greenness is located at r1 and that all of

greenness is located at r2. Since r1 and r2 stand in a non-zero distance relation, it

follows that greenness stands in a non-zero distance relation from its whole self. The

realist about immanent universals is committed to WSDP.

The realist about extended simples holds that there are certain objects that both

(i) lack proper parts and (ii) are exactly located at regions of non-zero measure.

Consider one of these extended simples, Sphere. Sphere is exactly located at a certain

spherical region, rS, which is the fusion of two hemispheric regions, r1 and r2.

Consider now the third of r1 that is closest to the r1-pole; call it ‘r1a’. Call the third of

r2 that is closest to the r2-pole ‘r2a’. Sphere is located at both r1a and r2a. r1a and r2a

stand in a non-zero distance relation. Now, since Sphere lacks proper parts, it cannot

be the case that Sphere is loosely partially located at r1a or r2a. But, as we saw above,

if an object is located at a region and is not loosely partially located at that region, then

it must be the case that the whole object is located at that region. If this is correct, then

it follows that all of Sphere is located at r1 and that all of Sphere is located at r2. Since

r1 and r2 stand in a non-zero distance relation, it follows that Sphere stands in a non-

zero distance relation from its whole self. The realist about extended simples is

committed to WSDP.

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The three-dimensionalist about persistence holds (i) that objects lack proper

temporal parts and (ii) that objects persist by being exactly located at non-zero

intervals. You are exactly located at the interval that exactly runs the course of your

lifespan. Call the first year of your life ‘y1’ and your most recent year ‘yn’. You are

located at both y1 and yn. y1 and yn stand in a non-zero (temporal) distance relation.

According to the three-dimensionalist, you lack proper temporal parts, so it cannot be

the case on his view that you are loosely partially located at y1 and loosely partially

located at yn. But once again, if an object is located at a (temporal) region and is not

loosely partially located at that region, then it must be the case that the whole object is

located at that region. If this is correct, then it follows that all of you is located at y1

and that all of you is located at yn. Since y1 and yn stand in a non-zero distance

relation, it follows that you stand in a non-zero distance relation to yourself. The three-

dimensionalist is committed to WSDP.

In each of the three above cases, instances of the following general premises

were used to argue for commitment to WSDP:

(P1) If regions r1 and r2 are elements in region R, and object o is exactly located at R,

then o is located at each of r1 and r2.

(P2) If object o lacks proper parts then o cannot be loosely partially located anywhere

(by definition of loose partial location).

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(P3) If object o is located at region r but is not loosely partially located at r, then it

must be the case that all of o is located at r.

As noted, P2 follows straightaway from the definition of loose partial location, which

requires proper parthood. P3 can be proved by reductio. Begin by assuming that o is

located at r but not loosely partially located at r, and that not all of o is located at r.

Since o is not loosely partially located at r, it cannot have any proper part that is

exactly located at some region r1 that is disjoint from r. But this is inconsistent with

the claim that not all of o is located at r, for if it were true that not all of o is located at

r then it would be true that there is some proper part of o that is exactly located at

some region disjoint from r. Contradiction.

This leaves us with P1. Some philosophers (e.g. Lewis (1991), Gilmore (2004),

Hudson (2006) and McDaniel (2007)) have suggested that, in the case of extended

simples at least, P1 can simply be rejected. According to these philosophers, there are

possible objects—called ‘spanners’—that (i) are extended, (ii) lack proper parts, and

(iii) are not located at any of the subregions of the region at which they are exactly

located.36

This move is particularly unattractive in the immanent universal and

persistence cases because in those cases partial location is needed in order to make

sense of how immanent universals or three-dimensional objects could possibly play

36 The term ‘spanner’ originated, I believe, with McDaniel. The earliest explicit mention of the possibility of this kind of object occurs in (Lewis 1991), though Lewis’s discussion is about sets, not material objects. Moreover, Lewis does not seem to care much about defending the possibility against objections. Indeed, the philosophers mentioned in the text treat the possibility of spanners with varying degrees of respect.

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the theoretical roles that they are invoked to play. For example, in order to make sense

of universal realist theories of property exemplification, immanent universals have to

spatiotemporally overlap—in some way or other—the objects that exemplify them. If

greenness is located somewhere in space at time t, but not where Bill is located, then

how can Bill exemplify greenness at t? Similar reasoning applies in the persistence

case. If Jones runs a marathon in the summer of 1980 and is located somewhere in

time, then how can she fail to be located at the summer of 1980?

Perhaps this is too quick and some sense can be made of the roles that

immanent universals and three-dimensional objects are invoked to play even if such

things never enter into partial location relations. Perhaps whatever the relation is that

obtains between spanners and subregions also obtains between greenness and Bill on

the one hand and between Jones and the summer of 1980 on the other. The realist

about immanent universals and the three-dimensionalist can simply import whatever

defense the realist about spanners employs. Perhaps. But even if going this route has

some appeal, I cannot see that it will save the day. The problem is that spanners only

superficially avoid commitment to WSDP.

For the sake of argument, then, let us grant the possibility of spanners. Even

once we do, the fact remains that some significant relation obtains between spanners

and the subregions of the regions that they exactly occupy, independently of whether

this relation is to be called ‘location’, ‘occupation’ or something else entirely. Let us

call the relation ‘schmocation’. So every spanner x is such that x is exactly located at a

region r but merely schmocated at every subregion of r, r1…rn. Now, since r is

extended (by the definition of spanner), it follows that some of the subregions of r will

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stand in non-zero distance relations to one another. Consider two such subregions, r1

and r2. Our spanner x has no proper parts around to be possible candidates for being

schmocated at r1 or r2, respectively, so it must be the case that all of x is schmocated

at r1 and that all of x is schmocated at r2. Consider now the relationship between the

location relation and the distance relation, as well as the relationship between location

and schmocation. With these abstract considerations in mind to guide us, we can

introduce a “schmistance” relation, where schmistance is to distance as schmocation is

to location. Here is the rub. To whatever extent we can make sense of schmistance

(which turns on the extent to which we can make sense of schmocation, which turns

on the extent to which we can make sense of spanners), whole-self-schmistance is just

as troubling a notion to be committed to as is whole-self-distance. The move to

spanners merely pushes the absurdity that haunts WSDP back a step; it does not

dissolve it.

I am thus inclined to conclude that spanners save neither the extended simples

theorist, the realist about immanent universals, nor the three-dimensionalist from the

problems of whole-self-distance. If these views are to be cogently defended, it will

need to be by recourse to some formulation of them that is not committed to WSDP.

That is, it will require recourse to versions of these views that stray in one way or

another from the first pass characterizations that I provide of each above. Later I will

look at some recent attempts to formulate realism about immanent universals and

three-dimensionalism that evade commitment to WSDP. We will see that each of them

falls short. In the meantime, I want to address a recent attempt to defend whole multi-

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location (which is only a short step from WSD) for extended simples, a defense that

does not take recourse to spanners.

4. Entension

Josh Parsons (MS) defends the claim that an object might be wholly located at more

than one location, from which claim WSDP is only a small step away.37 Parsons (MS

and 2007) draws a distinction between being ‘wholly’ located at a region and being

‘entirely’ located at a region. An object is wholly located at a region just in case every

part of it is located there. An object is entirely located at a region just in case there is

no disjoint region such that some part of the object is located at this latter region.

Parsons’s distinction forms the basis of his defense of entension against the objection

that no object could wholly exist at some one region if it also exists at some disjoint

region. As Parsons notes, this sort of objection is only compelling if ‘wholly’ is

understood as Parsons understands ‘entirely’. After all, there is no incoherence in the

conjunctive entensionist claim that all of x is located at both r1 and (disjoint) r2. The

incoherence only arises when one claims that all of x is located at both r1 and r2 and

no part of x is located at r2; and to hold that the entensionist is committed to this last

conjunct is to beg the question against him.

I mention Parsons’s distinction only to dissuade those who might have thought

that it could help the defender of WSDP to navigate the Humean/Annihilation

dilemma given in section 2 above. In generating the dilemma, we did not presuppose

37 Parsons dubs the notion of an object’s being wholly multiply located ‘entension’ to accord with the terminology of the literature on persistence. Entension is to endurance (or ‘three-dimensionalism’) as pertension is to perdurance (‘four-dimensionalism’) and extension is to persistence.

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that the friend of WSD is committed to entire multi-location, in Parsons’s sense; we

presupposed merely that she is committed to whole multi-location, in Parsons’s sense.

Let us turn now to an attempt to vindicate realism about immanent universals in the

wake of WSD-based objections.

5. Running without Hiding, Part 1: Immanent Universals

Arguably the foremost contemporary defender of realism about immanent universals

is David Armstrong. Armstrong (1988, 1989, 1997) holds that the most fundamental

items at the ontological ground floor are states of affairs, which he understands in a

technical sense as being ‘thick’ particulars: bi-categorical items consisting of bare

particulars and immanent universals.38 (The bare particular considered in complete

isolation from its universals is called a ‘thin’ particular.) Armstrong’s answer to the

WSD worry involves a sort of reneging on the quintessential immanent universal

realist claim that immanent universals are ‘located in’ spacetime. His thought is that

spacetime itself is just a certain highly complex state of affairs and that universals are

thus not really located in spacetime, but rather serve to help constitute spacetime

(1988, 99). Since universals are not strictly speaking located in spacetime, WSD

simply does not arise.

I think Armstrong’s response to the objection from WSDP is uncompelling.

Even if certain universals in fact help to constitute spacetime, and even if we are

moved by the inference from this constitutive role on the part of universals to the

38 This sort of ‘fact’ based ontology is found in the early work of Wittgenstein. See also (Skyrms 1981) and (Barwise and Perry 1983). In contradistinction to Armstrong, neither Wittgenstein, Skyrms nor Barwise/Perry uses the fact-based ontology to promote realism about universals, and none of those philosophers is overtly committed to realism about bare particulars.

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suggestion that the expression ‘located in spacetime’ is no longer strictly applicable to

them, it simply does not follow that all universals fail to stand in non-zero distance

relations to themselves. Armstrong’s claim that universals play a constitutive role in

the fabric of spacetime is thus quite beside the point of the WSDP worry.

The nail in the coffin of Armstrong’s response is that if the response works

with respect to WSDP-as-geared-toward-immanent-universals, then it ought to work

equally well with respect to WSDP-as-geared-toward-particulars. Consider some

instance of this latter form of WSD: a particular x stands in a non-zero distance

relation to its whole self. Not even Armstrong holds that the preceding describes a

legitimately possible state of affairs (though certain friends of extended simples and

three-dimensionalism do). Now consider some extremely implausible theory T, which

on its face is committed to the claim that our blade of grass, Blade, is capable of

standing in a (spatial) WSD relation. The problem for Armstrong’s mode of response

is that it can be applied in this case just as effectively as it can be applied in the

immanent universals case. Blade is a thick particular; it is thus a state of affairs, in

Armstrong’s technical sense; it is thus constitutively involved in the grand, complex

state of affairs that just is the whole spacetime continuum; it is thus not ‘located in’

spacetime; it thus cannot enter into a WSD relation. However much the implausible

theory T seems to be committed to WSDP, it in fact is not so committed. If

Armstrong’s form of response to the WSDP worry holds water, then T is just as easy

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to defend (with respect to WSDP) as is realism about immanent universals. I conclude

that Armstrong’s response does not succeed.39

6. Running without Hiding, Part 2: Three-Dimensionalism

Finally, let us turn to some recent attempts to formulate the central three-

dimensionalist notion of being ‘wholly present at distinct times’ that are not

committed to WSD. The three attempts with which I will be concerned are put forward

by Trenton Merricks (1999), Kit Fine (2006), and Thomas Hofweber/David Velleman

(MS).

Merricks argues that the best way to spell out three-dimensionalism is via

commitment to presentism, the view that only present objects exist. I will not spend

much time on Merricks’s view since it will suffice for my purposes to emphasize the

controversy surrounding presentism. If WSD shows that Merricks’s version of three-

dimensionalism is indeed the most plausible, then by my lights my task in this paper

with respect to persistence—namely, to motivate four-dimensionalism—has been met.

I will not argue directly against presentism here (now!). Let us turn then to Fine’s

version of three-dimensionalism.

As I understand him, Fine proposes a version of three-dimensionalism that

need not deny that persisting objects have temporal parts. If this is correct, then the

problems from WSD will not arise for Fine’s three-dimensionalist.

39 In his (2003), Cody Gilmore attempts to defend realism about immanent universals against the worry that it is committed to contradictions of the form: universal U both is and is not such and such distance from object x. I will not discuss Gilmore’s proposal here except to note that it presupposes WSDP.

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Fine presupposes a distinction between “events” and “things” in our “ordinary

ways of thinking about part-whole,” namely, that events have mereologically

disjunctive (i.e. union of parts) conditions of presence in both space and time, while

things have disjunctive conditions of “presence” (i.e. location) only for space, with

conjunctive (i.e. intersection of parts) conditions for time. For example, a quart of milk

q, composed of two pints, p1 and p2, is such that it is only present at time t if both p1

and p2 are present at t, but it is located (note: not exactly located, just located) at

spatial region r so long as at least one of p1 or p2 is located at r. However, an

occurrence of lightning, L, composed of an earlier streak s1 and a later streak s2, is

present at t so long as either s1 or s2 is present at t, and is located at r so long as either

s1 or s2 is located at r.

Fine then poses two questions: why should there be this disparity between the

presence conditions for things and events; and why should there be disparity between

the spatial “presence” conditions for things and the temporal presence conditions for

things? (It seems to me that answering the second will answer the first

simultaneously.)

His answer to the first question is that things endure (Fine uses ‘exist’ in an

idiosyncratic technical sense in place of ‘endure’) through time while events do not;

rather than enduring, events perdure (Fine uses ‘extend’ in place of ‘perdure’) through

time. His answer to the latter is that neither a thing nor an event can endure (‘exist’) at

a time unless all of its parts exist (non-technical sense of ‘exist’!) at that time.

Presumably, since this is the case, no useful endurance-based account of event

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persistence will be forthcoming, for events very often fail to have all of their (spatial)

parts at every time at which they occur.

Fine notes that his answers rest on two assumptions. First, that there is a

difference between the way things and events are present in time; second, that there is

a difference in the conditions of presence/location for a sum (whether a thing or an

event) depending on whether the sum is thought to persist by enduring or by

perduring.

It seems to me that Fine is stacking the deck with respect to what “our ordinary

ways of thinking” might be. I would have thought it quite ordinary to suppose that

apples, say, count as things, and that they are the type of things that might survive the

annihilation of certain of their proper parts. But, according to Fine, as I understand

him: (i) things persist by enduring across time and (ii) an item endures from t0 to tn

just in case it has (at least) all of the parts it had at t0 at every time between and

including t0 and tn (it seems to me that Fine is neutral as to whether the item is

allowed to gain parts). So, if apples are indeed items that can survive loss of parts

then, given (ii), they cannot endure; and then, given (i), they cannot be things. So

apples, on the view that they can survive loss of parts, do not count as things for Fine.

This is an unattractive result. I do not find it unattractive out of hand to hold

that in order for an item to endure, it must not lose any parts. What I find unattractive

is the distinction between things and events to which Fine is committed, especially

given his concern for “ordinary” ways of thinking and speaking.

I am far more inclined to take the traditional 4D (and 3D, for that matter) tack

and hold that there is just one way to be “present” or “located”, namely, to be located

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at a spacetime region. This is all quite consistent with Fine’s remarks about

conjunctive and disjunctive presence conditions. What I deny is that the

conjunctive/disjunctive differences reflect some deep distinction between “things” and

“events.” I am inclined to hold that both things and events are, with respect to

persistence, the way Fine holds that only events are. The putative thing this quart of

milk, on my view, just is the putative event this milk spatiotemporally occupying a

fusion of temporally connected momentary spacetime regions each of which is the

correct size for exactly containing a quart of milk. Compare this “milk event” M with

the occurrence of lightning that Fine describes, L. L is present at certain times even if

some of its parts are not. But, I submit, this is also true of M. For there is a time t at

which M is present—the milk is in the process of occupying the relevant fusion of

regions—but at which some of M’s parts, namely, the parts that exactly occupy some

of the other momentary sub-regions of the fusion, are not present. So I am inclined to

allow for the disjunctive/conjunctive distinction while denying that it marks items

relevantly like quarts of milk from items relevantly like occurrences of lightning.

What sort of item is such that I am happy to allow that it has conjunctive conditions of

presence? The only sort that comes to mind is momentary items. But I’m also inclined

to deny that there are any momentary items.40 So, while I allow that the

disjunctive/conjunctive distinction is quite intelligible, I am doubtful that the

conjunctive criterion for presence has a non-empty extension.

Fine anticipates the objection that items like quarts of milk are actually event-

like in the way I suggest. In his discussion of one attempt to make the quart of milk 40 See Chapter 6 for a version of four-dimensionalism that does not require the existence of (zero measure) momentary items.

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event-like, Fine (i) denies that the fusion of the two pints is the quart and also (ii)

denies that the fusion of the two pints is anything that we would “normally” take to

“exist.” But one is left to wonder whether this is Fine’s technical term ‘exist’, which is

supposed to be synonymous with ‘endure’? If so, then his claim that we would not

take the fusion of the two pints to exist is obvious from what he tells us about how to

understand endurance, and his appeal to our “normal” leanings with respect to his

technical term seems dialectically unmoving. If, however, ‘exists’ is used in an

innocent, pre-theoretical sense here, then it is simply not true that those who do not

share his antecedent view with respect to persistence would shy away from saying that

the fusion in question exists. And with respect to (i)—his claim that the fusion is not

the quart—one is left to wonder just what the referent of ‘the quart’ might then be.

This example is not the same as his original example in which the quart of milk is

introduced, for in that example it was stipulated that a quart of milk is given and that it

exists just when all of its parts exist. But then, it follows from the way this later

example is given (i.e. from the fact that there are times at which the fusion exists but

one of the pints does not) that the fusion cannot be the quart from the first example. So

it is no mark against the 4D-ist that she cannot identify the fusion with the quart from

the first example. Notice that she can still tell a perfectly acceptable story about there

being a quart of milk and about when such a thing exists: there is a quart of milk from

t1 until t2 since that is the interval at which the relevant two-pint quantity of milk is

such that it is spatiotemporally occupying a fusion of temporally connected momentary

spacetime regions each of which is the correct size for exactly containing a quart of

milk.

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I conclude that Fine’s attempt to undermine the parity between events and

things fails. His attempted clarification of three-dimensionalism follows suit.

Like Fine, Hofweber and Velleman seek to illuminate what is really going on

in disagreements between three-dimensionalists and four-dimensionalists. They begin

with the notion of a property’s being intrinsic to a time: property F is intrinsic to time t

just in case something x has F and would still have F as long as everything at t retained

its (non-temporally) intrinsic properties, even if something not at t had different (non-

temporally) intrinsic properties. With this notion in hand, they characterize three-

dimensionalism as the view that an object persists over some interval just in case its

“identity”—that is, the property of being that object—supervenes on properties that

are intrinsic to each moment in the interval. An object that meets this condition is said

to endure. They then characterize four-dimensionalism as the view that an object

persists over some interval just in case it exists at each moment in the interval but does

not do so by enduring.

Notice that the Hofweber/Velleman explication of three-dimensionalism is not

committed to WSDP. Their story commits the three-dimensionalist to the claim that a

given object might be the very object that it is at multiple times, but it does not

commit her to the claim that a given object might have all of its temporal parts at time

t1 and all of its temporal parts at time t2. In fact, Hofweber and Velleman reject

whole-multi-location— and thus would presumably reject WSD—as incoherent. This

in itself might lead us to question whether they have successfully captured three-

dimensionalism. After all, there are actual three-dimensionalists who take on the task

of defending whole-multi-location; presumably, these three-dimensionalists accept

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this difficult task because they take their commitment to three-dimensionalism to

entail commitment to the possibility of whole-multi-location.

However, since there may be philosophers with just the sort of intuitions about

persistence that Hofweber and Velleman attribute to the three-dimensionalist, it will

be worthwhile to further explore the extent to which the Hofweber/Velleman version

of three-dimensionalism is problematic, independently of WSD. I have two worries for

their approach. First (Hofweber and Velleman MS) commits the three-dimensionalist

to the existence of unextended times. This is a problem because time might be gunky,

and if time is gunky then any theory that turns on there being unextended quantities of

time will be false.41 In personal correspondence, Hofweber has expressed optimism

that the Hofweber/Velleman account can be reformulated in such a way that the three-

dimensionalist can allow for gunky time. Perhaps his optimism is well founded, but

for now the worry remains.

My second worry stems from the fact that Hofweber and Velleman’s three-

dimensionalist cannot allow for spatiotemporal-continuity as a possible criterion of

identity over time. (Indeed they make this point as an example of how to distinguish

the three-dimensionalist from the four-dimensionalist.) The worry is that there is room

in three-dimensionalist logical space—the boundaries of which I take to be determined

by the writings of self-proclaimed three-dimensionalists—for spatiotemporal

continuity to play a role in diachronic identity. The denial that Hofweber and

Velleman attribute to the three-dimensionalist ought not to be built into three-

dimensionalism.

41 See (Stuchlik 2003).

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For an example of the kind of theorist who does not fit the Hofweber/Velleman

picture, consider an austere metaphysician who holds that only fundamental particles

exist and that all “ordinary objects” are merely derivative constructions from

spatiotemporal relations among fundamental objects. This metaphysician holds that

these particles persist across time, even though they do not undergo intrinsic changes

in doing so. Now, this metaphysician might hold that each fundamental particle

persists just in case it has appropriately related proper temporal parts at distinct times.

But, she might just as well hold that the fundamental particles have no proper temporal

parts and that they persist by having all of their parts at every time that they exist, so

long as there is a continuous spatiotemporal path connecting each of these times. But

if our metaphysician goes this second route then her view of persistence for the objects

in question is much more in line with the writings of actual three-dimensionalists than

it is in line with that of actual four-dimensionalists. In correspondence, Hofweber has

responded to this worry by claiming that objects that persist without undergoing

intrinsic change do not really persist in the metaphysically interesting way that

concerns actual three- and four-dimensionalists. But the difference between the two

ways that our hypothetical metaphysician might have gone shows that there is an

interesting debate to be had even for objects that cannot survive intrinsic change. So

Hofweber and Velleman cannot have captured what is really at the heart of the debate

between three- and four-dimensionalists.42

42 I also have a third worry for the Hofweber/Velleman notion of a property’s being intrinsic to a time. The worry is that the analysis that they give of a certain notion (that of a property’s being local to a region), which is required in order to give their explication of intrinsicness to a time, is susceptible to fatal counterexample. I will not get into the details of this worry here since (i) doing so would require a

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7. Conclusion

I have shown that WSDP is deeply problematic. I have also shown that the first pass

versions of realism about immanent universals, extended simples, and three

dimensional objects are committed to WSDP. Finally, I have argued that recent

attempts to formulate versions of these theories that circumvent commitment to WSDP

are independently implausible. If I am right then we have good reason to turn away

from realism about immanent universals, extended simples, and three-dimensionalism.

good deal of space and (ii) I take the two more easily-expressed worries already given in the text to suffice for the purposes of this chapter.

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Part Two

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Chapter 4

Gloppy Trope Bundles

The issue of property exemplification in contemporary metaphysics concerns the underlying ontology of material objects and their properties. This chapter defends the view that objects just are appropriately related collections or ‘bundles’ of particular properties or ‘tropes’. After highlighting some ways in which trope bundle theory is more attractive than other realist theories of property exemplification (specifically, universal and bare particular theories), I clarify the details behind its deepest problem, that of saying what metaphysically determines or makes it the case that some given tropes form a bundle. I then suggest and defend a novel account of trope bundling. At the core of this suggestion is the claim that there is a (previously unfamiliar) fundamental property—markedness—the chief role of which is to mark certain locations from others, thereby furnishing a sort of binary code for fundamental ontology. On a first approximation of the corresponding account of bundling, some ordinary qualitative tropes are bundled just in case they are appropriately located with respect to a markedness trope. Along the way, I discuss how best to understand tropes as existing in space and time, de-motivate alternative trope bundle theories, and sketch a prospective theoretical benefit of markedness that goes beyond the bundling problem.

1. Introduction

Let us say that a property F is ‘monadic’ just in case there is some individual x and

world w such that (i) x is F at w and (ii) no contingent individuals other than x and its

parts exist at w.43 Material objects exemplify many monadic properties. For example,

the apple on my kitchen counter is red, round, and juicy. Metaphysicians vary widely

in their treatments of such examples. Indeed, some go so far as to deny that there are

any individuals or monadic properties, at least at the fundamental level (Ladyman

2007, Maudlin 2007). I will not engage these extreme revisionary ontologies here,

though I believe that they can be resisted. Instead, I will confine my scope to those

ontologies that maintain a place for individuals and/or monadic properties. Even with

43 This definition of ‘monadic’ is basically the definition of ‘intrinsic’ typically attributed to Kim (1982). Lewis (1983a) characterized clause (ii) of the definition as the property being lonely and famously showed that this property is itself (negatively) extrinsic. However, as Lewis notes, Kim’s definition seems unproblematic insofar as it characterizes a good many properties that are not positively extrinsic. For my purposes, Kim’s definition will suffice. I use ‘monadic’ instead of ‘intrinsic’ or ‘not positively extrinsic’ to minimize confusion and clutter. Some confusion may linger since my usage allows that certain relational properties are monadic, but let us set that subtlety aside.

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the scope so constrained, however, there remains substantial disagreement about

examples like the one above. There is live debate as to whether the properties are

anything over and above the particular apple (or classes of like objects), whether the

apple is anything over and above the properties (or their instances), and whether the

properties are universals. I will be concerned with a view according to which

properties are indeed distinct from material objects or classes thereof (realism), objects

are nothing over and above their collective properties/instances (bundle theory), and

properties are not universals but particulars (trope theory).44 Trope bundle theory is

motivated by its ontological economy, for it purports to explain monadic property

exemplification with only one fundamental ontological category and yet without the

obscurity of universals. That said, I will not argue here that it is superior to competing

ontologies of monadic property exemplification.45 Rather, my principal aims will be,

first, to explain and defend the details of my favored characterization of trope theory,

which differs in important respects from traditional trope theory; second, to clarify the

central problem for trope bundle theory, namely, the problem of what it is for some

tropes to be bundled; third, to argue that prior strategies for solving the problem come

up short; and finally, to suggest and defend a novel solution.

The solution to be suggested ties the central ontological role of tropes to their

status as spatiotemporal. The idea is that bundling is determined by tropes of a

heretofore unknown primitive property—markedness—the chief role of which is to 44 Present usage of ‘trope’ to talk about features understood as particular instead of universal originated with D.C. Williams (1953). Present usage of ‘realism’ whereby trope theories are realist is not to be confused with certain traditional uses whereby only universal theories are realist. 45 For example, I will set the issue of how trope theory ought to treat the exemplification of non-monadic relations (e.g. being the brother of) entirely to one side. I am optimistic about the prospects for a reductive treatment of such relations (at least those that are not perfectly natural), though I will not attempt to give one here.

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mark certain locations in spacetime from others. At first blush this strategy may seem

puzzling. How can adding one more monadic trope to some collection of tropes make

the collection a bundle? The puzzle can be solved by utilizing the spatiotemporal

status of tropes: some tropes form a bundle just in case they are appropriately located

with respect to a markedness trope. Moreover, if we take tropes to be world-bound

entities, which move comports with the overarching particularity of tropes, then we

can understand the appropriate kind of relative location as an ontologically innocuous

relation, for it will be fixed at a given world and time by the very existence of the

tropes in question. The advantage of this approach is that it solves the bundling

problem without invoking any of the controversial items to which competing trope

bundle theories are committed, for example, a primitive ‘compresence’ relation

(Campbell 1990), substantival spacetime regions, or essential connections among

distinct qualitative properties (Simons 1994).46 Of course, the view to be defended

does carry commitment to a new primitive property, markedness. I will argue,

however, that markedness is a richer theoretical asset than its competitors for the role

of trope bundler. Specifically, it furnishes new replies to important recent objections to

trope resemblance class theories of properties and priority pluralist theories of

fundamentality (Manley 2002, Schaffer 2004, 2007).

The plan for the rest of the chapter is as follows. The next section provides

some terminological clarification and a brief motivation for general trope bundle

theory. Section 3 is a discussion of how best to understand monadic tropes as being

46 Spacetime substantivalism is the view that spacetime is itself a substance, irreducible to distance relations among material objects. That the view is controversial is clear from the literature it has inspired, both for and against. For a helpful overview of much recent debate, see section 6 of (Nerlich 2003).

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spatiotemporally located. In section 4, I present the bundling problem, consider

whether it might be a pseudo-problem, and argue that it is not. Section 5 raises some

worries for earlier attempted solutions to the problem. Section 6 introduces a novel

trope ontology based on the idea of markedness—glop theory—and uses it to give an

account of bundling that circumvents the worries raised in section 5. Finally, in section

7, I address the objection that markedness is an unduly expensive theoretical primitive.

2. Terminology and Motivation

A trope is a non-repeatable, particular feature.47 For example, the color of the ‘e’ in

the last word token of this sentence is a trope. The ‘e’ in the last word token of this

sentence has a numerically distinct but exactly resembling color trope. According to

trope bundle theories, properties are resemblance classes of tropes;48 propertied

individuals are certain appropriately related collections or ‘bundles’ of tropes;49 and

what it is for arbitrary propertied individual a to exemplify arbitrary monadic property

F is for there to be some trope that is a member of both the F class and the a bundle.50

47 ‘Particular’ is ambiguous between (i) not universal and (ii) not generic. My characterization of tropes is to be read in accord with (i) since some theorists accept non-repeatable universals, e.g., being Socrates. It is not to be read in accord with (ii) since some theorists accept generic tropes, e.g. color tropes. 48 For the trope theorist, resemblance is a primitive, ontologically innocuous relation (Campbell 1990). The ontological innocuity stems from the fact that resemblance is an internal relation between tropes: the intrinsic nature of any two tropes fixes whether or not they resemble. 49 In rather loose accord with (Hawthorne and Sider 2003), I wish to remain officially neutral as to whether bundles are anything over and above certain distinguished pluralities of tropes. For example, everything I say—including the claim that material objects just are trope bundles—will be consistent with eliminativism about material objects, assuming that ‘material objects just are certain pluralities of tropes’ expresses a form of eliminativism about material objects. 50 If some trope T is a member of both the F class and the a bundle, then in addition to a’s exemplifying F, we can say as a shorthand that a ‘exemplifies’ T. Notice that trope bundle theory is a general theory of property exemplification, not a specific theory of predication. Trope bundle theorists may well vary among themselves as to which tropes and which bundles must exist in order for an utterance of a given instance of the schema ‘a is Φ’ to be true or correctly assertable.

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In order for trope bundle theory to be an informative theory of property

exemplification, then, it must say what it is for some tropes to be bundled. How best to

do so will be the subject of sections 4 through 6. For now, it will be helpful to clear up

some potentially confusing terminology from the traditional literature on tropes.

Tropes are sometimes called ‘abstract particulars’. This is potentially

confusing because ‘abstract’ commonly is understood as exclusively applying to non-

spatial (and perhaps non-temporal) items. However, on the most plausible

understanding of tropes, tropes are located in space and time.51 It is thus more accurate

to describe tropes as concrete than as abstract. Those theorists who have called tropes

‘abstract particulars’ have had different notions of abstraction in mind than the notion

that conflicts with spatiotemporality, namely, the notion of mentally abstracting away

certain features from a given object (Williams 1953, Campbell 1990) or the notion of

being predicable (Stout 1923).

Though I will not attempt a full-fledged argument for trope bundle theory as

against alternative realist theories of monadic property exemplification for material

objects, some initial motivation is in order.52 To see the appeal of bundle theories over

theories that take recourse to so-called ‘bare’ particulars in order to individuate or

unify material objects (Sider 2006), notice that bundle theories require one less

primitive ontological category. Instead of requiring tropes (or universals) and bare

51 For arguments to this effect see (Schaffer 2001). Schaffer argues that characterizing tropes as spatiotemporal accords far better with the core motivations for trope theory than do alternative characterizations. 52 Nor will I offer an argument for realism about properties, though I believe that rather powerful arguments are available (Goodman 1966 and Armstrong 1978, 1989, 2004). For an interesting recent defense of (one version of) property nominalism, see Rodriguez-Pereyra (2002).

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particulars, bundle theories require only tropes (or universals). The promise of a more

economical ontology motivates bundle theory.

To see the appeal of tropes over universals, notice that the former are neither

mysteriously located outside of space and time, as is required of transcendent

universals, nor such that they might be simultaneously wholly located at scattered

places, as is required of (multiply exemplified) immanent universals. For example,

suppose that the property of being 3kg mass is exemplified by a stone near the top of

Mont Blanc and also by a lava rock near the top of the Olympus Mons. If this property

is an immanent universal then it is both wholly located in France and wholly located on

Mars. By contrast, tropes—like the stones themselves—are singly located in

spacetime. Tropes may well have a part here and a part there, but they will never be

mysteriously wholly here and wholly there. The relative lack of mystery motivates

trope theory.53

3. Tropes in Spacetime

Generic trope bundle theory as I have characterized it has four principal theses:

(i) Sparse monadic tropes are ontologically fundamental;54

(ii) Primitive resemblance relations obtain among some monadic tropes;

53 This line of motivation goes back at least as far as Plato’s Parmenides. See the introduction to (Landesman 1971) for a useful discussion. See (Parsons MS) for an attempt to explicate ‘wholly’ in a way that foils the motivation, and (Armstrong 1988, 1989) for an attempt to explicate the role that immanent universals play in constituting spacetime in a way that foils the motivation. I offer a more thorough defense of the motivation, including an argument that Parsons’s and Armstrong’s remarks do not successfully undercut it, in the preceding chapter. 54 The notion of sparseness that I have in mind is the standard Armstrong (1978)/Lewis (1983b, 1986) notion, according to which the sparse properties are just those properties needed to describe the world completely and without redundancy.

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(iii) To exemplify monadic property F just is to be a bundle that contains a trope from

the F resemblance class; and

(iv) All tropes are located in spacetime.

Given thesis (iv), it is intuitive to hold that each trope has a particular, non-repeatable

spatial volume, spatial shape, and temporal duration.55 Since sizes, shapes, and

durations are features of the items that have them, and since tropes just are particular,

non-repeatable features, it follows that particular, non-repeatable sizes, shapes, and

durations of arbitrary tropes are themselves tropes. It then follows from thesis (iii)

(and the fact that size, shape, and duration are monadic) that in order for some trope T

to have a size, shape, and duration, T must be a bundle that has size, shape, and

duration tropes as members. Prima facie, this attempt to stretch first order trope bundle

theory into a second order theory gives rise to a puzzle: what is it for a trope to be a

bundle that contains other tropes? One way to discharge the puzzle is to require that

thesis (iii) be read as an exclusively first order claim. However, this move has the

serious drawback of leaving one with no account of second order exemplification with

respect to size, shape and duration—the second order properties whose

exemplification thesis (iv) requires.

As a first step toward explaining away the puzzle without giving up a unified

account of the requisite instances of second order exemplification, I submit that every

trope is trivially bundled with itself. That is, every trope T is a bundle, one of the

members of which is T itself; thus, every trope exemplifies itself. The redness trope of

55 Hereafter I will speak simply of ‘size’, ‘shape’, and ‘duration’, leaving implicit the respective ‘spatial/temporal’ modifiers. Sizes and durations may be of arbitrarily small measure and shapes may be arbitrarily simple.

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the apple is itself red. Let us call this suggested phenomenon ‘reflexive

exemplification’.56 The second step invokes the following notion:

(Inter-Exemplification): for any collection of tropes, the Ts, the Ts inter-exemplify

one another just in case each T exemplifies every T.

I will argue that inter-exemplification helps to furnish a solution to the puzzle about

second order bundling because (a) inter-exemplification only obtains for single tropes,

that is, if the Ts inter-exemplify one “another” then there is only one T;57 and (b) the

cases of second order trope exemplification that give rise to the puzzle—namely,

exemplifications of second order size, shape, and duration tropes—are cases of inter-

exemplification.

To see that inter-exemplification only obtains for single tropes, consider

without loss of generality some trope M1, say, the mass trope of the head of a

sledgehammer. M1 has a mass trope, namely itself, a duration trope, DM1, a size trope,

VM1, and a shape trope, SM1. If we stipulate the inter-exemplification of M1, DM1, VM1,

56 Reflexive exemplification is not unique to trope theory. For example, Cody Gilmore, a defender of immanent universals, considers reflexive occupation by regions plausible (Gilmore 2003, 423). Moreover, reflexive exemplification is not, as it might seem at first blush, threatened by Russell-style paradox. The key to circumventing the threat is to disallow that the sort of predicates that seem to give rise to paradox (e.g. ‘not being self-exemplifying’) express genuine properties. Finally, there are certain putative properties that would seem to give rise to counterexamples to reflexive exemplification for tropes, e.g. sortal properties like being an oak tree, which seems to work as a counterexample because, intuitively, no tropes are oak trees. But it is not obvious that such complex tropes ought not to be identified with the relevant material objects, if they are indeed genuine tropes. Nor is it obvious, even if we reject identification, that being an oak tree tropes, if they exist, are only exemplified by oak trees and not also by themselves, for exemplification facts need not link up in obvious ways with our intuitions about predication. For more on this last point, see section 6.2. 57 If this is correct then inter-exemplification and reflexive exemplification turn out to be extensionally equivalent. Every case of reflexive exemplification is a trivial case of inter-exemplification, and if every case in which inter-exemplification obtains for some class of tropes is a case in which that class has only one member, then every case of inter-exemplification is a case of reflexive exemplification as well.

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and SM1, it follows from (iii) that M1 just is a bundle that includes itself, DM1, VM1, and

SM1 as elements. Given the inter-exemplification stipulation, this same form of

argument applies for DM1, VM1, and SM1, respectively. Since there are no other

monadic features of any of M1, DM1, VM1, and SM1, it follows that each of these tropes

just is numerically identical to the same bundle, that is,

M1=DM1=VM1=SM1=M1DM1VM1SM1.58, 59 If this much is correct, then requiring that

tropes be spatiotemporal does not lead to a bundle-confounding explosion of size,

shape, and duration tropes. Rather, the size, shape, and duration of arbitrary trope T

are nothing over and above T itself.

What remains to be shown is that it is plausible that inter-exemplification ever

in fact obtains. Fortunately, this can be done. Given that DM1 has a size and a shape,

why shouldn’t its size trope be VM1 and its shape trope SM1? After all, whatever size

(or shape) trope DM1 exemplifies will exactly resemble VM1 (SM1) with respect to size

(shape), so for the theorist motivated by ontological economy, inter-exemplification is

the natural choice. That we can “abstract” apart the size, shape, and duration of a

given trope in our ideology does not entail that these features are in fact numerically

58 I assume here that bundles cannot be individuated intensionally. 59 One might worry that M1 cannot be identical to SM1 since M1 is intrinsic and yet, following (Skow 2007), there is reason to doubt that shapes are intrinsic. One attractive way of understanding shapes is as being equivalent to a series of distance relations of the form the distance from x to y is n, where x and y range over parts of the shaped object in question and n ranges over numbers. Skow’s reason for rejecting this as an account of intrinsic shapes is that it quantifies over numbers: the shape of an object turns on relations its parts bear to numbers, not on how the object is in itself. I have two comments in response to this worry. First, it is not at all clear that if exemplification of property F by object a requires that some of a’s parts be related to abstracta then F is extrinsic. Second, even if we stipulate that the preceding is a good guide for our use of the predicate ‘extrinsic’, it is not clear that mass itself should not then count as extrinsic, for we might well understand mass as a quantity of the form the mass of x is n, which understanding involves quantification over numbers. On this understanding of mass, the proposed disparity between M1 and SM1 does not hold up. Finally, notice that quantification over numbers in order to characterize a property F does not, on my use of ‘monadic’, preclude F from being monadic. I here assume that numbers exist necessarily.

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distinct spatiotemporal items. What is perhaps more difficult to make intuitive is the

claim that DM1 (=VM1=SM1) is itself massive. The key is to notice that without their

being inter-exemplified, there would be no account of the exemplification of DM1 by

M1. After all, the head of the sledgehammer has other tropes besides M1, each of

which might well persist for just the career of M1 and thus exemplify a duration trope

that exactly resembles DM1 with respect to duration. So why should DM1 and not one of

these other duration tropes be exemplified by M1? The inter-exemplification

hypothesis provides a direct answer to this question.

Importantly, however, I am not suggesting that our intuitions about how to

numerically carve up features are wildly inaccurate and that seemingly distinct tropes

can just be numerically identified willy-nilly, resulting in some form of property

nominalism. To better articulate what I am suggesting, let us say that a trope is

‘merely spatiotemporal’ just in case it is a size, shape, or duration trope that does not

exemplify any intrinsic character beyond size, shape, or duration. For example, M1 is

not merely spatiotemporal, for it exemplifies an intrinsic character—being massive—

that is qualitatively additional (though not, in the case of M1, ontologically additional)

to its status as sized, shaped, and persistent. I wish to remain neutral as to whether

there are any merely spatiotemporal tropes. Let us say further that merely

spatiotemporal properties are ‘non-qualitative’ and that all other familiar monadic

properties such as mass, color and the like are ‘qualitative’.60 Far from advocating

widespread trope identifications, my view is that inter-exemplification does not

60 Notice that this is a technical use of ‘qualitative’ that does not jibe with certain more traditional uses. For example, some philosophers use ‘qualitative’ to mean something like descriptive, as in the distinction between descriptive expressions and (most) names. On this more traditional usage, mere sizes and shapes would count as qualitative. Not so on my usage.

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generalize beyond a given qualitative trope and its size, shape, and duration tropes.

Specifically, it does not generalize to the identification of tropes of two qualitative

properties such as mass and color. Toward seeing this, notice that inter-

exemplification would seem to be an all or nothing phenomenon. That is, given thesis

(iv), any qualitative trope F must have the following four features: it must be F, sized,

shaped, and persistent. By contrast, there is no qualitative trope F that must have some

distinct qualitative feature G. For example, there could be colorless mass tropes

(indeed, there could be colorless, massive objects). While this asymmetry between

mass, size, shape, and duration on the one hand, and mass and color on the other, does

not prove that inter-exemplification cannot possibly obtain between certain mass and

color tropes, it does strongly suggest it. Otherwise, what would explain why some red,

massive material objects are such that their redness and mass tropes inter-exemplify

one another while other red, massive material objects (for example, ones that will soon

change color while retaining their mass) are such that their redness and mass tropes

are merely co-bundled but do not inter-exemplify one another? I maintain that—while

any qualitative trope is numerically identical to its size, shape, and duration tropes—

any two sparse qualitative tropes are primitively numerically distinct.

Though logically independent of (i)-(iv), a further thesis entailed by the sort of

spatiotemporal trope theory that I favor is that tropes are world-bound. To see the

importance of this thesis, notice that if M1 were not world-bound then there would be

no apparent reason why it is not possible for it to have a different size than the size it

actually has, which (putative) possibility is highly problematic for the claim that M1 is

identical to its own actual size trope VM1. Similar reasoning suggests that it is best for

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the trope theory that I have been describing to deny that tropes can survive changes in

size and shape. Notice that these theses dovetail with the overarching particularity of

tropes: just as no trope is both wholly here and wholly there, so no trope is both in this

world and in that one, or of this size (shape, duration) and of that one.

Since tropes may be either qualitative or non-qualitative and either scattered or

connected, we can carve out four species of tropes: Scattered Qualitative (SQ) tropes,

Scattered Non-Qualitative (SN) tropes; Connected Qualitative (CQ) tropes; and

Connected Non-Qualitative (CN) tropes.61 Examples: ordinary, macro instances of

color such as the redness of the skin of a whole apple are SQ tropes, for redness is

qualitative and yet there are non-red regions of space between the apple skin’s

smallest (red) proper parts. Merely spatiotemporal size tropes can be examples of

either SN or CN tropes since merely spatiotemporal size is a non-qualitative property

that has both scattered and connected tropes. A mass trope will work as an example of

a CQ trope so long as we stipulate that the mass trope in question is connected

(otherwise mass is SQ). In the actual world, this kind of trope will not be found in

(whole) macro objects but may well be exemplified by, say, fundamental particles.

Before proceeding to the bundling problem, it will be helpful to make explicit

some additional ways in which my proposed understanding of tropes as spatiotemporal

is new, especially to those who are not antecedently sympathetic or familiar with trope

theory. Notice first that, for all I have said, tropes do not have to be bundled with

distinct tropes in order to exist. Tropes are not metaphysically posterior to material

objects. For example, the qualitative tropes of a hammer do not have their characters

61 Here I have in mind the explication of these topological notions found in (Cartwright 1975).

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in virtue of being tropes of the hammer. Rather, the hammer has its character in virtue

of the existence of those tropes. Notice, further, that tropes as I have characterized

them are not metaphysically posterior to spacetime or to distance relations. (Whether

tropes are metaphysically prior to distance relations is a different matter, about which

I say more in section 6.3) On the present view, spacetime does not confer sizes and

shapes upon tropes. Rather, the spatiotemporal features of tropes, including their

relative locations, are ontologically built in from the get go. Though radical in some

ways, this view is consistent with versions of both substantivalism and relationalism

about spacetime.62 With the stage now set for a better understanding of tropes as

spatiotemporal items, let us turn to the bundling problem.

4. The Bundling Problem

The bundling problem is captured in the following question. What metaphysically

determines or makes it the case that some given tropes form a bundle? Proponents and

opponents of trope theory alike have recognized the depth of the problem (Simons

1994, Daly 1997). To better see it, let us return to the quadripartite trope taxonomy

from the preceding section, according to which tropes can be either qualitative or non-

qualitative and either connected or scattered. The bundling problem is deeper for

qualitative than for non-qualitative tropes since the latter can plausibly be bundled by

co-location. For example, it is plausible that the size trope and shape trope of an apple

62 There are some modifications to the standard versions of these theories that the present view of tropes suggests. Specifically, substantival regions would be identified with bundles of merely spatiotemporal tropes; and fundamental distance relations for the relationalist would obtain not among material objects but among tropes. The view I have presented is free to be substantivalist in the sense described since it is neutral as to the existence of merely spatiotemporal tropes.

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(if there are any such ‘merely spatiotemporal’ tropes) are bundled in virtue of being

co-located. By contrast, qualitative tropes like juiciness tropes and redness tropes

cannot—at least prima facie—be bundled by co-location since the juicy parts of the

apple are not co-located with the red parts.

Among qualitative tropes, the problem is deepest for CQ tropes. This is

because (i) for many non-fundamental qualitative tropes (i.e. tropes that are members

of non-fundamental qualitative properties), it is plausible that supervenience on the

arrangements of fundamental (i.e. sparse) qualitative properties determines bundling

and (ii) many fundamental qualitative properties (e.g. mass) include CQ tropes. These

claims require some elaboration. In stating (i), I am assuming that ordinary sortal

terms like ‘apple’ possess some necessary and jointly sufficient application conditions

with respect to spatiotemporal arrangements of fundamental qualitative tropes and that

certain arrangements of fundamental qualitative tropes can and do meet these

conditions. (Notice that I am not assuming that we can state these conditions or that

we explicitly attend to them when employing sortal terms.) Given this assumption,

there will be a fact of the matter as to which arrangements of (tropes of) fundamental

qualitative properties count as being apples, at least in the most straightforward cases

(perhaps there will be some vagueness). We can then say that a given apple’s non-

fundamental monadic qualitative properties are just those that supervene on the

relevant arrangements of (tropes of) fundamental properties.63 If we then assume, as is

plausible, that the tropes of non-fundamental monadic qualitative properties are just

63 The notion of supervenience at work here is familiar: if two apples are duplicates with respect to the fundamental properties exemplified by their most basic parts, then they will be duplicates with respect to their non-fundamental properties. As such, I will sometimes say as a shorthand that the latter properties (or tropes) are ‘fixed by’ the former.

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the SQ tropes, we arrive at the thesis that some SQ tropes are co-bundled just in case

they supervene on one of the relevant arrangements of (tropes of) fundamental

properties.64

The force of (ii) is that this supervenience approach to bundling—to whatever

extent it works with respect to SQ tropes (more on this in section 6.2)—will not work

in the case of fundamental CQ tropes. To see this, notice that even if we assume that

there are some necessary and jointly sufficient conditions for applying the sortal term

‘fundamental particle’ to collections of CQ tropes, and even if we assume that some

collections of CQ tropes meet these conditions, we will not have made any progress on

the bundling problem for CQ tropes. All that we will have said is that there are some

conditions for fundamental-particle-bundling that some CQ tropes do in fact meet; we

will not have specified those conditions. In the next section, I will consider some more

promising attempts to specify bundling conditions and will argue that each is

unsatisfying. Before getting there, however, I want to consider the charge that the

bundling problem, even in its most difficult, CQ-specific form, is merely a pseudo-

problem.

64 It might be contended that emergent properties (understood here as properties of wholes that do not supervene on any properties of the relevant parts)—which are usually taken to be fundamental—can have SQ tropes. If this contention were true it would undermine the claim that all SQ properties are non-fundamental. This is an interesting issue and I do not have space to treat it in detail here. My quick response is that it is not so clear that emergent properties can ever have SQ tropes, for the connected proper parts of any SQ tropes are themselves tropes (lest tropes be composed of non-tropes, which would threaten the one-category trope ontology), and it is hard to see how some arrangement of the connected tropes that compose a given SQ trope (call them ‘the xs’) could fail to subvene that SQ trope (call it ‘T’). To see this, note first that T is, of course, a trope of some putatively emergent property F. Now, either all of the xs are F tropes or not. If so, then T is fixed by their arrangement. If not, then F would seem not to be fundamental. I am thus inclined to suggest that if there are any (fundamental) emergent tropes then they are CQ.

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One might object to the depth of the bundling problem as follows. “Look, there

is no need to posit some deep metaphysical such-and-such in order to explain which

tropes count as being ‘bundled’ as an apple, or even which tropes count as being

‘bundled’ as an electron. We know which properties apples and electrons have—that

is, we already know which among the many collections of tropes out there get to count

as being apple bundles or electron bundles—because we know what apples and

electrons are like. Our concepts of objects determine bundling. There is no big

mystery. The bundling problem is a pseudo-problem.”

This formulation of the pseudo-problem objection is ambiguous with respect to

whether its advocate is a strong or weak deflationist about bundling. According to the

strong deflationist, there just is no ontological fact of the matter as to whether some

tropes form a bundle until some agent or sufficiently unified community of agents

forms a concept of those-tropes-as-bundled. According to the weak deflationist, there

is an ontological fact of the matter as to whether some tropes form a bundle even

before some agent forms the corresponding concept, it’s just that there is nothing

puzzling about the ontology—nothing further needs to be said about what “makes it

the case” that the tropes form the bundle. By my lights, the strong disambiguation is

the less charitable since it entails that ontological facts about material objects are

mind-dependent. Let us shift focus, then, to the weak reading of the objection.

There are two ways one might be a weak deflationist, depending on whether

one accepts the following thesis:

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(Unrestricted Bundling): For any collection of tropes, the Ts, there is some object

that is a bundle of just the Ts.

If one rejects Unrestricted Bundling then one forfeits the freedom from explanatory

burden that weak deflationism would otherwise afford, for one would then need to

explain why some collections of tropes form bundles while other collections do not.

That is, the bundling problem would resurface for this weak deflationist, leaving it

difficult to see how he could remain a genuine deflationist about bundling.

This leaves the weak deflationist who endorses Unrestricted Bundling as the

most formidable of the possible sources of the ambiguous deflationist objection.

However, this species of deflationism faces its own serious problem, namely, that it

allows for the existence of material objects that exemplify free-floating qualitative

tropes. To see this, consider the collection of tropes that consists of (i) all of the tropes

that are members of the apple bundle on my kitchen counter and (ii) the present mass

trope of the Eiffel Tower. According to Unrestricted Bundling, there is some material

object a that exemplifies all and only these tropes. Assuming a fairly uncontroversial

form of Humean modal recombination, the deflationist will then allow for there to be a

world in which the only material object is a (or a counterpart thereof). But in such a

world there is a material object that exemplifies free-floating mass, for it exemplifies a

mass trope that is not grounded in or co-exemplified with any other tropes in its

location. (Recall that the mass trope of the Eiffel tower is not some abstract quantity

but rather a specifically sized and shaped spatiotemporal entity; as such, its

counterpart in the world being considered will wildly outstrip the apple counterpart’s

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location). Now, I will not argue that there could not be such a free-floating qualitative

trope. The point that I wish to emphasize is that Unrestricted Bundling (plus the

plausible modal principle) requires the widespread possibility of material objects that

exemplify such tropes. It is surely better to avoid this sort of controversial commitment

when giving a general theory of exemplification (in this case, a deflationary theory),

even if one holds that certain free-floating qualitative tropes are not obviously

impossible. In light of this worry for the best form of the deflationist objection, I

submit that the bundling problem is not a pseudo-problem.65

5. Worries for Prior Approaches

Before offering a new answer to the bundling problem, some motivation for rejecting

prior strategies is in order. I will consider four: the distributional co-location approach,

the connectedness approach, Peter Simons’s (1994) ‘nuclear’ approach, and the

primitive compresence approach. We have already seen that a fifth strategy, the naïve

co-location approach, is not viable since not all co-bundled tropes are co-located (the

redness trope of the apple is in a different location than the juiciness trope).66 The

distributional co-location approach is a more sophisticated version of the general co-

65 A second problem for Unrestricted Bundling is that it carries commitment to widespread material co-location without any intuitive material part-sharing. Since I take the first problem to suffice for showing that the bundling problem is not a pseudo-problem, however, I will not pursue this second problem here. 66 One might suggest that naïve co-location is actually more promising in the case of fundamental CQ tropes than it is in the case of SQ tropes like (macro) redness tropes or juiciness tropes. After all, the tropes of paradigmatic fundamental particles, e.g. the mass and charge tropes of an electron, plausibly are co-located. Non-fundamental tropes could then be bundled via supervenience on these fundamental CQ tropes, which are bundled via co-location. However, while this suggestion may have some appeal with respect to certain actual world examples, it is not clear that all actual fundamental particles exemplify co-located tropes. More importantly, the present suggestion rules out the very possibility of bundled, non-co-located CQ tropes. It would be preferable to have a theory of bundling that was not modally constrained in this way.

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location strategy. On this view, tropes are to be understood distributionally as global,

field-like entities that take values at locations.67 The approach is monistic in the sense

that each field-like trope is exemplified by the whole world. Co-location is thought to

be vindicated by the distributional/monist approach because the global property color

and the global property juiciness (to stick, without loss of generality, to the apple

example) are indeed such that there is a location—namely, the region at which the

whole apple is exactly located—at which both take values. The subregion of the

region exactly containing the apple at which the skin of the apple is exactly located

takes the value “red” of the property color and the value “not juicy” (or “zero”) of the

property juiciness. The subregion at which the flesh of the apple is exactly located

takes the values “yellow” and “juicy” (or “one”), respectively. So even though the red

part of the apple is not co-located with the positively juicy part, there is still a co-

locational bundling story available with respect to the apple’s color and juiciness.

I have two worries for the distributional/monistic approach. First, it is an

unappealing way to understand tropes. Intuitively, qualitative tropes are individuated

in part by the simple characters that they embody, and red and yellow (or juicy and not

juicy) are certainly distinct qualitative characters. Second, and more importantly for

present purposes, the distributionalist/monist cannot use mere co-location as his means

of outputting apple bundles, for no (actual) apple is co-located with the entire

qualitative world. Instead, as we have seen, the distributionalist/monist is required to

67 For more on distributional properties, see (Parsons 2004). For a trope-theoretic approach of this sort, see (Campbell 1990). Campbell is concerned with fundamental properties like mass, but his treatment generalizes.

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take recourse to fundamental spacetime regions in order to bundle tropes.68 To do so,

however, is to abandon austere trope bundle theory for a theory according to which

something in addition to tropes is needed at the ground floor of ontology, namely,

non-derivative spacetime regions.69 In addition to harboring an extra ontological

commitment, this move bears the expense of committing the disributionalist/monist to

the independently controversial thesis of spacetime substantivalism. It would be better

for trope bundle theory to be neutral with respect to the substantivalist/relationalist

debate.

A second prior strategy for bundling is based on the topological notion of

connectedness. On this view, some tropes, the Ts, are bundled just in case a

continuous path can be traced through all of the Ts without failing to overlap a T. The

problem with this strategy is that it is implausible that connectedness is either

sufficient or necessary for bundling. Making connectedness sufficient would rule out

the possibility of distinct but touching material objects. As in the case of the

widespread possibility of material objects that exemplify free-floating tropes, the

prohibition of distinct but touching material objects is too controversial to be required

by the best theory of property exemplification. And even if we wish not to brook free-

68 See (Schaffer 2009) for a version of monism that is not committed to tropes and (Campbell 1990) for a version of monism that is committed to tropes but invokes primitive compresence in response to the bundling problem. 69 It might be suggested that the distributionalist/monist can get around this worry by adopting the account of substantival regions that I suggested in a prior footnote, according to which regions just are bundles of ‘merely spatiotemporal’ size, shape, and duration tropes. I am dubious, however, about the prospects for such a move. The distributionalist/monist holds that tropes are global. So on his view there is only one size trope, for example. It is thus difficult to see how the distributionalist/monist could individuate regions using size trope “values” ala my above account if regions are supposed to be composed in part of such items, for regions are what he uses to individuate the “value-instances” of his tropes. What is most important for present dialectical purposes, however, is that even if the distributionalist/monist were to succeed in going this route, he would remain saddled with commitment to substantivalism.

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floating tropes, we might well want to allow, in the interest of modal liberality, for

some possible jointly disconnected CQ tropes to be co-bundled. Making

connectedness necessary for bundling, however, blocks this allowance.

A third prior approach is Peter Simons’s very interesting ‘nuclear’ theory of

trope bundling. Some tropes, the Ts, form what Simons calls a ‘nucleus’ just in case

each T symmetrically ontologically depends on every other. Simons’s story about

bundling then proceeds in two parts. First, all of the tropes in a given nucleus are

bundled. Second, further contingent tropes are bundled in virtue of being the

determinate instances of some determinable tropes contained in the nucleus. Though

Simons’s view is ingenious in a number of ways, it comes at a great expense. By

invoking ontological dependence, the view requires deep necessary connections

among distinct qualitative properties. As in the cases of substantivalism, widespread

possibility of material objects with free-floating tropes, and criterial connectedness, it

is better not to build a controversial commitment of this sort into one’s theory of

bundling. Moreover, Simons’s view requires that bundles contain both determinate

tropes and their corresponding determinables. While this move is not untenable, it is

once again costly. A more economical trope theory will not reify determinable tropes

but will rather explain our use of determinable predicates via facts about the relevant

determinate tropes.

A final prior strategy for solving the bundling problem is to posit a primitive

compresence relation that obtains among all and only those tropes that are members of

some one bundle. There are three related reasons for finding primitive compresence

unappealing. First, much of the attraction of trope bundle theory for monadic

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properties of material objects stems from its ontological economy, which is hampered

by the introduction of a primitive, external, non-monadic trope.70 Second, even if one

were to take recourse to some external or non-monadic relation in order to give an

account of bundling, doing so would not be as ontologically expensive if the relation

in question were already well understood (e.g. spatiotemporal relations), for the

resulting theory would serve a more informative explanatory role than one according

to which bundling is simply primitive; but primitive compresence is not antecedently

well understood. Third, even if some new primitive property—whether it be monadic,

non-monadic, internal, or external—is needed in order to solve the bundling problem,

the resulting damage to the ontological economy of trope theory will be mitigated if

the primitive property in question does important philosophical work beyond the

bundling problem; but compresence does no such work.71

Unlike primitive compresence, markedness (i) is internal and monadic; (ii)

works in conjunction with antecedently well-understood location relations to give an

informative analysis of bundling;72 and (iii) is equipped to do more philosophical work

70 Given the nature of compresence, it may well seem obvious that it is an external and non-monadic relation. Still, it might be suggested that if compresence is a reflexively exemplified trope then it is not so obvious that it fails my criteria for being monadic, for there could be a world in which only one individual (and its parts) exist, namely a compresence trope. But this suggestion will not work, given that tropes are spatiotemporal items. Any compresence trope—qua trope—will have a size, shape, and duration. Now, either these second order features of the compresence trope will be identical to the trope itself (as I suggest above for other tropes) or not. If so, then no single compresence trope can be exemplified by tropes of distinct sizes, shapes, or durations, which is clearly an inadequate result for any theory of compresence tropes. If not, however, then it will not be possible for a compresence trope and its parts to be the only individuals at any world, for the existence of any compresence trope will require the existence of numerically distinct individuals, namely, some size, shape, and duration tropes. 71 A fourth concern is that compresence seems to lead to an uncomfortable regress, for what ties an nth-order compresence relation to the relevant tropes if not some (n+1)th-order compresence relation? However, I am inclined to agree with Campbell (1990, 37) and Armstrong (1989, 55-6) that this fourth concern is not deep since the entities involved at each new step supervene on those involved at the first step. 72 I will attempt (in 6.3 below) to cash location as an ontologically innocuous relation.

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than merely to bundle tropes. This last feature of markedness is the topic of section 7.

Let us now turn to the details of (i) and (ii).

6. Markedness and Glop Theory

This section introduces a new trope bundle theory that eschews primitive compresence

and essential connections between distinct qualitative properties, respects intuitions

about location and connectedness, and is neutral on substantivalism. The core idea is

that there is a special monadic property the tropes of which function as the

metaphorical adhesive for bundling qualitative tropes. I call this property

‘markedness’ and its tropes ‘mark tropes’ since its definitive nature is to mark certain

regions from others.73 There is nothing more to the intrinsic character of markedness.

Mark tropes are not red or fuzzy or loud, and they will not be directly detected in the

laboratory. Consequently, markedness is a ‘non-qualitative’ property in the technical

sense described in section 3, though it is not a ‘merely spatiotemporal’ property since

its tropes embody an intrinsic character beyond size, shape, and duration, namely,

being marked. Even though it is not posited by the empirical sciences and has been

until now unfamiliar in philosophy, markedness is a perfectly natural, fundamental

property.74 Its distribution at a given world is primitive in the sense that it is not fixed

73 Here and elsewhere I presuppose talk of regions entirely for ease of exposition. What I say is consistent with the reduction of regions to spatiotemporal relations among primitive tropes. 74 Some philosophers maintain that in order for a property to be perfectly natural it must be posited by our best physical theories. However, even these philosophers would agree that if, say, bare particular theory is the correct theory of property exemplification then physics is tacitly committed to bare particulars, notwithstanding the fact that our best physical theories do not explicitly posit bare particulars. The same line of reasoning applies in the case of markedness. In short, philosophers who turn to physicists for the catalog of fundamental items ought to be neutral—at least at the outset—as to which fundamental items a given theory of exemplification posits, for physicists are not explicitly concerned with theories of exemplification. (Since I am working from a realist stance about properties, I

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by the distribution of any other property. As a heuristic, we might think of the

difference between regions that contain mark tropes and those that do not as furnishing

a sort of ontologically fundamental binary code.

Markedness might seem mysterious at first, but it is in fact simple and

straightforward. Just as mass tropes distinguish the regions at which they are present

from those at which they are not in virtue of being massive, so mark tropes distinguish

the regions at which they are present from all others in virtue of being marked. One

important disanalogy, of course, is that mass is characterized (or at least recognized at

the actual world) in large degree by its causal and functional roles with respect to other

qualitative properties, where the fulfillment of these roles can be observed empirically.

But this disanalogy need not render markedness any more mysterious than its

metaphysical competitors like the ‘thin’ bare particular, the compresence relation, or

the Aristotelian kind, for none of these will ever be empirically detected either. Rather,

to whatever degree these strictly metaphysical posits are intelligible, they are so

entirely in virtue of the philosophical roles assigned to them by their proponents. In

the same way, markedness is intelligible via its philosophical role. Mark tropes are just

those primitively sized, shaped, and persisting fundamental items that play the role

attributed to them throughout this paper.

Though the key technical concept here is the concept of markedness, the

greater ontological system that underwrites the present version of bundle theory is

called ‘glop theory’. The funny sounding term ‘glop’ plays two grammatical roles: it

here assume that physics concerns objects and properties and thus must be tacitly committed to some underlying theory of exemplification. As noted at the outset, I will not enter into debate here with those philosophers who reject this assumption and contend that all physics commits one to is, say, some sort of relational structure.)

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functions as an acronym for Grounding Local Ontological Primitive and it functions

as a mass noun that picks out instances of this primitive. Despite the complicated

terminology, instances of the grounding primitive just are mark tropes; so something

is a mark trope or a collection of mark tropes just in case it is a quantity of glop.75

Again, competing ontologies of exemplification feature their own grounding

primitives like bare particulars (Sider 2006), substantival spacetime (Schaffer 2009),

compresence (Campbell 1990), and Aristotelian kinds (Loux 1998a, 1998b). The key

insight of glop theory with respect to the metaphysics of property exemplification is

that the work that these competing primitives are invoked to do can be done from

within the single ontological category of monadic trope. It is time to see how.

Let us say that a trope T1 is ‘local to’ a distinct trope T2 just in case T1 is

entirely located at a region contained by a region at which T2 located.76 For example,

M1, the mass trope of the head of the sledgehammer, is local to the mass trope of the

whole hammer. This relation—locality—is irreflexive, non-symmetric, and transitive.

Call the converse of being local to ‘being local for’ (so the mass trope of the whole

75 Why introduce two new pieces of terminology? The central idea behind markedness, namely, that it serves as ground for every qualitative property without itself having any non-spatiotemporal features other than marking regions at which it exists from those at which it does not, might well appeal to philosophers who dislike certain details of glop theory that I develop elsewhere. Such philosophers may then want to adopt the term ‘markedness’ without being committed to the idiosyncrasies of glop theory. 76 In the interest of avoiding commitment to substantivalism, here is an analysis of locality that does not mention regions: trope T1 is local to distinct trope T2 just in case for each spatiotemporal part of T1, t1, there is some spatiotemporal part of T2, t2, such that t1 is at zero distance from t2. Intuitively, this means that if T1 is local to T2, then every part of T1 touches some part of T2. Now, one might worry that this non-substantivalist gloss does not capture the analysis of locality given in the text. After all, to revert back to talk of regions, why couldn’t T2 exactly occupy an open region and T1 exactly occupy the boundary of that region? If this were the case then T1 would not intuitively “occupy a region” that is contained by a “region” that T2 “occupies”, notwithstanding the fact that every part of T1 touches some part of T2. However, the glop theorist is free to deny that mark tropes can be as small as the size of the boundary of a region. With this denial in place, the worry does not arise.

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hammer is local for M1). Here is a first pass at how markedness and locality furnish a

solution to the bundling problem:

(Mark BundleFirst Pass) If any two qualitative tropes are local to the same mark trope

then they are members of the same bundle.

This first pass provides a sufficient condition for bundling certain kinds of monadic

tropes. We will see presently that the development of necessary and further sufficient

conditions will depend upon which kind of monadic tropes we have in mind.

There is a privileged subclass of mark tropes, called ‘bits’, which function as

the glue for bundling CQ tropes.77 To get to the class of bits, we begin with the class

of all possible mark tropes, M. For a number of reasons that are largely tangential to

the bundling problem, glop theory allows that there are mark tropes that are not local

for any qualitative tropes.78 Consider then the subclass of M—MQ—that contains all

and only the mark tropes that are local for at least one qualitative trope. The class of

bits—MB—is a subclass of MQ. The key criterion that figures in whittling MQ down to

MB is maximal connectedness, where a mark trope is maximally connected just in case

it is not scattered and not in contact with any other mark trope. This topological

criterion blocks certain absurd commitments involving counting.79 (It also equips glop

77 ‘Bit’ is a technical term. My usage is not to be confused with other uses of ‘bit’ in the trope literature. See, for example, (Bacon 1997). 78 For example, with this allowance the glop theorist protects the role of markedness as a robustly grounding ontological primitive, for it entails that markedness does not supervene on the qualitative. 79 Consider a mark trope T1 that is local for exactly two qualitative tropes, TF and TG. Suppose that TF and TG are both significantly smaller than T1. If we consider proper parts of mark tropes to themselves be mark tropes (which seems plausible), then there will be many mark tropes that are local for TF and TG. But we do not want there to be many different ways for the same two qualitative tropes to be

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theory with some additional philosophical benefits to be discussed in section 7.) For

CQ tropes, then, bundling is determined by locality to some one bit:

(Mark Bundle) For any two CQ tropes T1 and T2, T1 and T2 are members of the same

bundle just in case there is some one bit b that is local for both T1 and T2.

Bundling for SQ tropes is then determined by supervenience upon

arrangements of fundamental tropes (which typically will be CQ), as mentioned in

section 4. (SQ bundling is discussed in more detail in subsection 6.2 below.) As for

SN and CN tropes, bundling is determined by co-location. Gloppy trope bundling is

thus pluralist in a way that corresponds to the quadripartite trope taxonomy. This

result should not be surprising. CQ tropes were the principal source of the depth of the

bundling problem, so it makes sense that their bundling requires a wholly novel

treatment even if bundling for the other species of tropes does not.

6.1 Some Clarifications about Markedness

Since markedness is an entirely new philosophical posit, it is important to make its

principal features explicit. In this subsection, I anticipate and attempt to forestall ten

potential confusions.

bundled at a given time. The topological criterion for being a bit blocks this kind of case because it singles out the maximally connected T1 as being the unique mark trope relevant to the bundling of TF and TG.

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(1) Mark tropes are not identical to spacetime regions. If mark tropes were identical to

regions, then they would not be able to distinguish marked locations from un-marked

locations, which would make markedness a conceptual non-starter.

(2) Mark tropes can exist unexemplified by spacetime regions. Mark tropes exist even

if relationalism is true and there are no non-derivative spacetime regions.80

(3) Mark tropes can exist unexemplified by material objects. Material objects are

bundles of qualitative tropes. Mark tropes are metaphysically prior to, and many are

wholly distinct from, such bundles.

(4) (2) and (3) do not entail that markedness can exist unexemplified. Mark tropes are

reflexively exemplified, so they are never unexemplified.

(5) Markedness is monadic. Its status as reflexively exemplified and ontologically

independent allow markedness to meet the criteria for being monadic.

(6) Mark tropes are qualitatively simple and homogeneous. Markedness is not a

conjunctive or structural property; it does not “qualitatively decompose.”

(7) Markedness is a categorical property. Markedness is not a higher order role for

some qualitative property to play, but a first-order property itself. The role it plays

cannot be played by any qualitative property on pain of forfeiting sufficient modal

generality. For example, if mass were responsible for bundling, then there would be no

available account of bundling for CQ tropes that are not local to mass tropes.

(8) Markedness is not bare particularity. Mark tropes necessarily have sizes, shapes,

and durations. Bare particulars, by contrast, have sizes, shapes and durations only

80 My talk of ‘marked regions’ and ‘unmarked regions’ is shorthand for ‘regions at which mark tropes are present’ and ‘regions at which no mark trope is present’. It does not carry commitment to the claim that regions exemplify markedness.

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contingently, if at all.81 Moreover, markedness is a sparse property; being a bare

particular is not a sparse property, for it is essentially exemplified by bare particulars

and yet it is plausible that bare particulars are such that they might not have

exemplified any sparse properties (Sider 2006). Finally, notice that that which

exemplifies being a bare particular (namely, a bare particular) may well also

exemplify a cluster of contingent qualitative properties. This is not the case for that

which exemplifies markedness (namely, a mark trope), for mark tropes are not

massive or green or…etc.

There is a sense in which markedness is ‘bare’, but then again there is a sense

in which all properties are ‘bare’. For example, once we consider mass apart from its

relations to other properties, it becomes very difficult to intuit any intrinsic character

for mass. But for all this, mass considered apart from its relations to other properties is

not as ‘bare’ as is, say, the Chrysler Building when considered apart from all of its

property exemplifications. Mass tropes cannot ever be as ‘bare’ as a ‘thin’ or ‘truly’

bare particular (Armstrong 1989, Sider 2006). Markedness parallels mass in this

respect.

(9) Markedness is neither quiddity for properties nor haecceity for objects. It will very

often be the case that distinct qualitative tropes, whether merely numerically distinct

or also qualitatively distinct, are local to one mark trope; so the mark trope cannot be

what gives each of the qualitative tropes its individual particularity. What does that job

81 David Armstrong, a proponent of bare particulars, has recently (2004) modified his view by dropping contingent exemplification. Armstrong now contends (i) that if arbitrary bare particular a exemplifies arbitrary universal F, then a is necessarily F in the sense that a is F in every world in which a exists and (ii) that all particulars are world-bound. By my lights, Armstrong’s modification yields a version of bare particular theory that is not deeply different from bundle theory, though I will not pursue this point here.

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will be difference in location, qualitative character, or both. Nor does markedness

grant individual particularity for objects. Complex material objects, for example, will

(via their many scattered parts) involve many distinct mark tropes. So markedness is

not what gives these objects their individual particularity.

(10) Markedness is not compresence. Consider an arbitrary unmarked substantival

spacetime region, r. I have suggested (footnote 61) that the trope theorist should

understand r as being a bundle of certain ‘merely spatiotemporal’ size, shape, and

duration tropes. However we cash compresence ontologically, it will be the case that

these tropes form the relevant bundle by being compresent. But, ex hypothesi, r is

unmarked. So markedness is not identical to compresence.

6.2 More on SQ Bundling

We have seen that gloppy bundling is pluralist and that markedness is needed chiefly

to bundle CQ tropes. In this subsection, I want to address a worry for the proposed

account of SQ bundling. Recall that on this account, an object has an SQ trope T of

property F in its bundle just in case (i) F supervenes on the fundamental properties of

the collective fundamental particles that compose the object in question and (ii) T is

the SQ F trope that is fixed by the relevant particles. The worry is that this account is

too liberal since it allows for the bundling of many tropes that do not seem intuitively

to cohere with predications we typically endorse. For example, the account entails that

the red apple on my counter has a brown trope since the apple has a seed with a brown

trope, which trope supervenes on the apple’s fundamental particles; yet we would not

typically say that the apple is brown. So, on the present view, there is a rift between

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predication and exemplification. Other examples in this vein are easy to generate: part

of the apple is cubical, so there is a cubical trope fixed by the fundamental parts of the

apple, so the apple has a cubical trope in its bundle, so the apple is cubical; part of the

apple has a one cm diameter, so there is a one cm diameter trope fixed by the

fundamental parts, so the apple is one cm in diameter; part of the apple is a stem, so

there is a being a stem trope fixed by the fundamental parts, so the apple is a stem. But

intuitively ‘the apple is cubical’, ‘the apple is one cm in diameter’, and ‘the apple is a

stem’ are all false.

As a point of entry for defending the present account of SQ bundling against

this kind of worry, I submit as a datum that felicitous predication and strict

exemplification come apart ubiquitously. For example, consider a standard baseball

with white rawhide and red stitching. Few contexts license felicitous utterances of ‘the

baseball is red’. Yet if a white trope is a member of the baseball bundle, then a red

trope is as well, for while the difference between how much of the surface of the

baseball is covered by rawhide and how much is covered by stitching is relevant to our

color talk about baseballs, it is not sufficient for excluding redness from a complete

ontological account of the baseball in question. So, if the complete ontological account

of the baseball is to be entailed by its being a certain trope bundle, then a red trope had

better be in the relevant bundle. Examples like this one are easy to generate.

Still, there is admittedly something uncomfortable about the severity of the rift

between exemplification and predication that the proposed account of SQ bundling

seems to require, so it will be worthwhile to say more about how to mend it. I will

mention three strategies for doing so. The first is to endorse what we might call

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‘macro-property nihilism’. On this view, the only tropes that exist are tropes of

fundamental particles. All other putative tropes are really just tropes of fundamental

particles arranged thus and so. So on this view, ‘the apple is red’ is either false or

shorthand for something like ‘the apple is composed in part of fundamental particles

with tropes arranged “red-wise”’. Similar treatments apply for claims like ‘the apple is

brown’, ‘the apple is cubical’, ‘the apple is a stem’, etc. I prefer to resist this strategy

since I would like to remain neutral as to how many (and how many kinds of) tropes

exist. This is not to say that the strategy is untenable.

The second strategy is analogous to the stage theory of persistence championed

in (Sider 1996, 2001) and (Hawley 2001). On this analogous view, the apple is

composed of spatial parts, which are in some sense “spatial-counterpart” related. Some

of these parts are seeds, one is the stem, some have a one cm diameter, etc. Now, on

stage theory, we are urged to adopt an atypical semantics according to which singular

terms like ‘that apple’ pick out different instantaneous objects on different occasions

of utterance, even if the speaker demonstrates the only piece of fruit in the kitchen on

each of several occasions within a short temporal span. On the spatial analog theory,

then, singular terms like ‘that apple’ pick out different spatial parts of the apple on

different occasions of utterance. I will not attempt to flesh out just what semantic

mechanism is responsible for determining which parts are picked out on which

occasions. It may have to do with speaker intention or some other aspect of the context

of utterance. At any rate, ‘the apple is brown’ will come out false when ‘the apple’

picks out the whole apple (or, as is more common for utterances that involve color

predicates, when ‘the apple’ picks out the surface of the apple), but will be true when

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‘the apple’ picks out a seed or all of the seeds. Since we do not typically operate in

contexts where we would use ‘the apple’ to pick out the seeds, we do not think of ‘the

apple is brown’ as plausibly being true. Nevertheless, there are such contexts, for

example, ones in which the interlocutors are, say, chiefly interested in the seed colors

of various fruits.

The third way to smooth the rift is to posit an ambiguity in the predicative

copulae (‘is’, ‘has’, ‘possesses’, etc.) that corresponds to an ambiguity in

‘exemplifies’. As a first step in sketching this third strategy, let us say that arbitrary

material object x ‘entirety exemplifies’ arbitrary property F at time t just in case x’s

being F at t counterfactually depends on every part of x at t. Let us say that x ‘segment

exemplifies’ F just in case (i) some proper part y of x entirety exemplifies F and (ii) x

does not entirety exemplify F. For example, an apple entirety exemplifies its total

mass but only segment exemplifies the mass of its westernmost half. On this view,

many correct disambiguations for monadic predications of material objects are going

to favor segment exemplification. For example, ‘the apple is red’ is true on the

segment disambiguation but false on the entirety disambiguation, for only the surface

of the apple is red. Similarly, ‘the apple is brown’ comes out true on the (appropriate)

segment disambiguation, for the seeds are a segment of the apple and they are brown.

Our intuition that ‘the apple is brown’ is false comes from our convention of using

color predicates to apply only to segments of material objects which are surfaces of

the objects in question. But there is no reason to believe that our communicative

conventions must correspond one-to-one with what is strictly true, as shown in the

baseball example. Again, I will not give a semantic theory for determining the correct

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disambiguation, but I imagine that speaker intention and other contextual factors

would be relevant. Nor will I choose here between “spatial counterpart” theory and the

present copulae ambiguity suggestion. My concern has been to show that each is

tenable. If I have succeeded then the aforementioned worry for SQ bundling is

tractable.

6.3 Austerity Preserved

It is time to take stock of the ontological commitments of gloppy trope bundling in

order to ensure that they do not outstrip the desired austere monadic trope ontology.

The pluralist bundling strategy in play invokes co-location (for SN and CN bundling),

supervenience (for SQ bundling), and locality (for CQ bundling). Do any of these

relations offend against the monadic trope ontology?

Let us begin with co-location. Since on the present view tropes are world-

bound, any two co-located tropes at arbitrary world w and time t are essentially co-

located in the sense that they would not be the very tropes that they are if they were to

fail to be co-located at t. For in order for them to fail to be co-located at t, they would

have to exist in some other world wherein they are not co-located at t; but this is illicit

on the world-bound understanding of tropes. The upshot is that co-location is an

ontologically innocuous relation among tropes insofar as the existence of some tropes

at a given time fixes whether they are co-located. Consequently, co-location is free to

determine bundling for SN and CN tropes without adding anything to the monadic

trope ontology.

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Supervenience is also an ontologically innocuous relation, so the account of

bundling for SQ tropes does not outstrip the monadic trope ontology either, so long as

CQ bundling does not.

The task remains, then, of showing that locality—the key relation for gloppy

bundling of CQ tropes—does not carry any ontological commitments beyond monadic

tropes. We have seen (footnote 75) that locality can be cashed in terms of distance and

parthood. This way of cashing locality is not ideal, however, since it relies on distance

relations, for which it is challenging (though I think not impossible) to offer an

account without committing oneself to primitive external relations. Fortunately, the

task of showing that locality does not outstrip the austere monadic trope ontology can

be met by cashing it in terms of co-location and parthood. Trope T1 is local to trope T2

just in case T1 is co-located with some trope T3 that is part of T2. Since neither the co-

location nor the parthood relation carries any fundamental ontological commitment

that offends against austere monadic trope theory, neither does locality.

An important worry remains, however. Even if locality does not have to be

cashed in terms of distance, it is not clear that we can understand primitive tropes as

ever being sized, shaped, or located in the first place unless we take external distance

relations to be primitive. One response to this worry is to bite the bullet and simply

allow for primitive external distance relations in addition to primitive internal,

monadic tropes (a move which, of course, remains consistent with relationalism about

spacetime). Doing so does not keep glop theory from maintaining a leaner ontology

than competing theories of bundling; it just makes it less lean than it would have been

otherwise. However, there is an alternative to biting the bullet. The idea would be to

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allow the lengths of fundamental tropes to be primitive and to cash all distance

relations derivatively as factors of the primitive lengths. Since lengths are internal for

tropes (the existence of a trope fixes its length), this suggestion affords an account of

distance relations in terms of internal properties. Admittedly, this suggestion is

inchoate, but I do not see any immediate reason to reject it as untenable.

7. Costs and Benefits

Even if locality is not problematic for the ontological economy of gloppy trope

bundling, there remains an elephant in the glop theorist’s ontology room, namely,

markedness. Why not reject gloppy bundling precisely because it—unlike other

theories of bundling—commits us to this new primitive property?

The answer is that the theoretical benefits of markedness outweigh its cost.

There are three important ways in which commitment to markedness is preferable to

the commitments of competing approaches to bundling. First, mark tropes are not a

new kind of entity for the monadic trope theorist, but a new instance of the lone

antecedently accepted kind: monadic trope. This is not the case for primitive

compresence relations, which are not monadic (see footnote 69), or substantival

regions, which—when understood in the way required by distributionalism/monism—

are not trope bundles (see footnote 68). Second, the competing commitments face

independently formidable problems that glop theory circumvents. Here I have

substantivalism and Simons’s notion of ontological dependence among distinct

qualitative properties chiefly in mind. The former has inspired an entire literature of

objections (and, in fairness, replies), while the latter runs counter to powerful Humean

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modal intuitions. Third, markedness reaps unique philosophical benefits beyond trope

bundling. I lack the room to do more than gesture toward these benefits here, but I will

try in the space remaining to make the case for one of them plausible.

The benefit I have in mind is a new response to an important recent objection

to trope resemblance classes raised by David Manley (2002), an objection that (to my

knowledge) has not yet been addressed adequately by trope theorists. Manley holds

that general predicates like ‘colored’, ‘pale’, and ‘reddish’ pick out highly natural

properties and that consequently the trope resemblance class theorist ought to be able

to class color or pale tropes together using her lone available resource for constructing

classes, degree of primitive resemblance. The problem is that Manley has devised

ingenious versions of the co-extension and imperfect community problems that seem

to show that the trope resemblance class theorist cannot with any plausibility form the

desired classes. I suppress the details of Manley’s examples, but the upshot is that

general tropes of the sort mentioned allow for cross-cutting resemblances of equal

degree that preclude intuitive class constructions. The trope theorist wants to class

pink and pale blue tropes into the ‘pale’ camp to the exclusion of non-pale purple

tropes, but she cannot do so because the purple tropes, qua ‘reddish’, resemble the

pink tropes, and qua ‘bluish’, resemble the pale blue tropes, just as much as they

resemble one another qua pale.

Manley anticipates the response that such general predicates as ‘colored’,

‘bluish’, ‘pale’ and the like do not express properties that are natural enough for the

classing problems to be metaphysically worrisome. He attempts to forestall this

response by urging that it leaves one with no way to account for the resemblance facts

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that guide our usage of such predicates, the best such account being that the predicates

do indeed express highly natural properties.

However, glop theory has resources that Manley does not anticipate.

Specifically, the glop theorist can maintain that every mark trope that is local for an

intuitively ‘pale’ qualitative trope at a given world has, say, a certain kind of shape.

The deep ontological structure that underwrites the resemblance-tracking use of ‘pale’

at that world, on this suggestion, does not involve some universal property being pale,

but rather the relevant class of appropriately shaped mark tropes. Notice, further, that

the suggestion does not require each of the relevant whole mark tropes to be of some

particular shape. All that is needed is for each of the relevant mark tropes to have

some arbitrarily small “tag” somewhere in its spatial arrangement such that this “tag”

is of a particular shape. This way, single mark tropes can ontologically underwrite

felicitous utterances of multiple general predicates of the sort in question. For

example, a mark trope that is local for a pink trope will have both a “tag” that

corresponds to ‘reddish’ and a distinct “tag” that corresponds to ‘pale’, without there

having to be any tropes other than the relevant mark trope and the relevant pink trope.

Finally, notice that the present suggestion does not saddle the glop theorist with undue

modal restrictions on distributions of qualitative properties. This is because glop

theory allows that mark tropes may be larger than the qualitative tropes for which they

are local. As such, the “tags” invoked might well turn out to be extremely small parts

of mark tropes that do not overlap any qualitative tropes. This would allow for a

sufficiently liberal take on the possible arrangement of qualitative tropes without

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giving up the requisite correlation between “tags” and sufficiently natural qualitative

predicates at a given world.

In light of this prospective benefit, I submit that gloppy trope bundling is part

of a potentially rich philosophical package deal that mitigates the cost of committing

to markedness as a primitive. Mark tropes may be as recherché as primitive

compresence relations, bare particulars, Aristotelian kinds, Simons’s nuclei, or

substantival regions, but they seem to be more useful across the gauntlet of

philosophical puzzles than all but perhaps the last of these. At the least, they deserve

to be taken seriously as a new competitor in the ontology of monadic property

exemplification.

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Chapter 5 How to Reconcile “Sparse” Fundamentality with Infinite Complexity and Emergence ABSTRACT: I introduce and defend a new version of the view according to which ontological fundamentality works by “building” from the bottom up via asymmetric supervenience relations upon a base of locally exemplified properties. This general view has been criticized recently for failing to allow for the possibilities of (i) infinite mereological complexity (gunk), (ii) infinite qualitative complexity and (iii) emergent properties. I attempt to discharge the problem from (i) by denying that being sufficiently ‘local’ requires being mereologically simple; to discharge the problem from (ii) by positing a single, novel property as the ultimate subvener across worlds; and to discharge the problem from (iii) by rejecting the chief arguments in favor of emergence. Each of these moves is controversial—the second especially so. The move is defended via theoretical cost-benefit analysis.

1. The Builder and the Heir

The ontologically fundamental items at arbitrary world w are all and only those needed

in order to fix all of the facts at w.82 As a first approximation of what is meant here by

‘fix’, consider the familiar metaphor of creation.83 Once God settles which

fundamental objects exist and which fundamental properties and relations they

exemplify, He thereby settles which non-fundamental (or perhaps less fundamental)

objects exist and which properties and relations they exemplify, as well as which non-

fundamental properties and relations the fundamental objects exemplify. There is of

82 ‘Facts’ is here intended in a pre-theoretical sense. Similarly, ‘fundamental items’ is purposefully ambiguous. It may designate properties, bearers of first-order properties, states of affairs, structures, or more exotic entities still. One’s view on how to disambiguate will depend largely on one’s views about the metaphysics of property exemplification. The bundle theorist (e.g. Campbell 1990), the bare particular theorist (e.g. Sider 2006), the states of affairs theorist (e.g. Armstrong 1997), the nominalist (e.g. Quine 1980), the object nihilist (e.g. Hawthorne and Cortens 1995), and the structural realist (e.g. Ladyman 2007) will all have different stories to tell. The issues about fundamentality to be discussed in the text are live for each of these approaches. My expository focus in what follows will be on first-order properties, though I will have more to say about bearers as well. 83 The converse of the fixing relation is expressed ‘is fixed by’, which, as I will use it, may be taken as loosely synonymous with ‘is grounded by’ and ‘exists (or obtains) in virtue of’. It should be noted, however, that I understand the fixing relation to be non-reflexive (neither reflexive nor irreflexive), which runs counter to the traditional way of understanding grounding.

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course room to make the metaphorical first approximation more precise, but it will

suffice to get us started.

According to one popular way of thinking about ontological fundamentality,

the world is qualitatively fixed from the bottom up: the exemplification of a select few

basic properties completely fix all of the facts of the world. The properties that play

this role are the sparse properties, those that characterize the world completely and

without redundancy (Armstrong 1978, Lewis 1983, 1986). Accordingly, let us call this

the ‘sparse’ conception of ontological fundamentality.

According to another popular way of thinking about ontological

fundamentality, the world is locally fixed from the bottom up: localized facts

collectively fix global facts. Locality may here be thought of as spatiotemporal, as

mereological (so that the local is more mereologically basic than the global), or as

some combination of the two.84 Let us call this general approach the ‘local’ conception

of ontological fundamentality.

Continuing the taxonomy, let us call the even more general view whereby

ontological fundamentality proceeds from the bottom up—whether qualitatively or

locally—the ‘builder’ conception of fundamentality. Both the sparse and the local

conceptions, then, are builder conceptions. The builder conception is highly intuitive.

We typically think that the qualitatively abundant/macro/mereologically complex facts

at our world are best explained by recourse to certain sparse/micro/mereologically

simpler goings on.

84 In the serious ontology room, where extended simples and zero-volume complexes crop up, spatiotemporal relations and mereological relations may come apart. For present purposes, however, I will concentrate on cases where size and parthood intuitively accord.

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Yet the various builder approaches do not by any means exhaust the logical

space. It may be that ontological fundamentality proceeds from the top down. Perhaps

it is illusory to hold that God only needed to settle the facts about some privileged

items in order to fix the facts about all items. Why not hold instead that He could only

fix the ontology at a given world by specifying all of the facts? The world is not built

according to this way of thinking about ontological fundamentality; it is given.

Accordingly, let us call this general top-down approach the ‘heir’ conception of

ontological fundamentality. Rounding out the terminology, let us call the heir

approach with respect to qualitative facts the ‘abundant’ conception, and the heir

approach with respect to spatiotemporal/mereological facts the ‘global’ conception.

So, on the abundant conception, there is no special, non-redundant subset of properties

whose distribution at a given world fixes the qualitative facts at that world. Rather, all

of the exemplified properties are ontologically on a par. And on the global conception,

facts about spatiotemporally local and mereologically simpler items do not fix the

facts about spatiotemporally global and mereologically more complex items. Rather,

facts about the whole cosmos fix all of the spatiotemporally local and mereologically

simpler goings on.

Three points of clarification are in order. First, notice that the heir views are

quite consistent with how ontological fundamentality was described at the outset. On

the abundant conception, it is still the case that the fundamental property facts

(vacuously) fix the non-fundamental property facts, for there are no non-fundamental

property facts. Similarly, on the global conception, it is still the case that the

fundamental object facts fix the non-fundamental object facts, it’s just that there is

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only one fundamental object, namely, the whole world, and the non-fundamental

objects are its proper parts. The second point of clarification is terminological. The

approach that I have dubbed ‘global’ is very similar to the view that has been called

‘priority monism’ in the recent literature, and the approach that I call ‘local’ is very

similar to what has been called ‘priority pluralism’ (Schaffer 2009, 2010,

Forthcoming).85 My preference for the neologism has to do with the third point of

clarification, which is that there are moderate views available that complicate the

taxonomy. For example, what we might call the ‘regional’ theory of fundamentality

says that the fundamental items are neither mereological simples nor whole worlds,

but rather mereological complexes that are nonetheless proper parts of worlds.86

Another example is what we might call the ‘minimally natural’ theory of

fundamentality, according to which the set of fundamental properties is neither so

conservative that it includes only sparse properties nor so liberal that it includes all

abundant ones. Rather, on this view, the fundamental properties are all and only those

above some threshold of naturalness.87 I think there is more to be said in favor of the

regional view than the minimally natural view. Indeed, we will see that the builder

theory that I will develop is regional. For now, however, I will set aside discussion of

these moderate positions.

85 Priority monism is the view that the whole concrete world is more fundamental than (‘is metaphysically prior to’) its concrete proper parts. Understood in this way, priority monism is strictly neutral as to whether the sparse or abundant conception is correct. Schaffer, in his recent writing, endorses the sparse conception. 86 Clarification: ‘regional’ is intended here in a metaphorically geographical sense (as between ‘local’ and ‘global’), not as a conjugate of ‘region’ in the spatiotemporal sense. Schaffer (2003) calls the regional view ‘molecular’. 87 The relevant notion of naturalness for properties is that of (Lewis 1983, 1986).

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At first blush it may seem that—moderate alternatives aside—the builder and

heir camps are mutually exclusive. But they are not. Hybrid theories are available. It is

consistent to endorse the builder conception for the qualitative and the heir conception

for the spatiotemporal/mereological. Likewise, it is consistent to endorse the heir

conception for the qualitative and the builder conception for the

spatiotemporal/mereological. The first sort of theorist will hold that the ontologically

fundamental facts are those involving the exemplification of sparse properties by the

world taken as a whole, unified item. The second sort of theorist will hold that the

ontologically fundamental facts are those involving the exemplification of

properties—which need not be sparse in the Armstrong/Lewis sense—by maximally

local items, perhaps spacetime points or point-sized objects.

Using broad brushstrokes, we can thus portray four competing views about

ontological fundamentality: sparse local, sparse global, abundant local, and abundant

global. (If we wanted to use a finer brush, we could include sparse regional, minimally

natural regional, abundant regional, minimally natural local, etc., for a total of nine

available positions. But we have agreed for now to set the moderate views aside.) To

my knowledge, there is only one of these four that has not been taken up at one time or

another in the relatively recent literature: the abundant local theory. This is probably

for good reason. The abundant local theorist is committed to holding that all facts are

fixed by local facts and yet that abundant exemplification facts are not fixed by

Armstrong/Lewis sparse exemplification facts. In worlds that feature abundant

properties, however, this commitment entails that the fundamental level of property

bearers is very small (say, involving quarks) and yet the fundamental level of

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properties involves abundant properties (say, being a low-income senior). The

problem here is not that the view entails that quarks are low-income seniors, for the

abundant local theorist could perhaps maintain that the semantics for ‘low-income

senior’ involves a complex field-like entity that takes certain fundamental values at

various quarks. The problem, rather, is that the being a low-income senior value at a

given quark is, on the view in question, a fundamental qualitative property; yet,

presumably, it is not a property that could ever be empirically detected in quarks. This

seems like trouble. Fundamental qualitative properties ought to be empirically

detectable. Perhaps this is not a knockdown worry, but since the view is not defended

in print and since its prospects are uncertain at best, I will set it aside for the remainder

of the paper.

From among our immoderate quartet, then, we are left with the sparse local

theorist, the sparse global theorist, and the abundant global theorist. An example of the

first is David Lewis (1983, 1986a, 1986b), whose Humean Supervenience is perhaps

the most well known of all sparse local theories. An example of the second is Jonathan

Schaffer (2009, 2010, forthcoming), whose recent defense of priority monism takes a

sparse global turn. Interestingly, the only example of the third (to my knowledge) is

also Schaffer, though a past version of Schaffer (2004), who defends what is actually a

minimally natural approach to properties while gesturing toward a global approach to

mereology and spacetime. Accordingly, Schaffer’s work is the locus of argumentation

against the sparse local theory, emphasizing a deep puzzle that arises for its adherents

(though some aspects of the puzzle are anticipated by Leibniz (1998), Armstrong

(1978), Lewis (1983), Maudlin (1998), and Oppy (2000), probably among others). The

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puzzle concerns how the sparse local conception can apply in worlds that contain

infinite mereological complexity, infinite qualitative complexity, or emergent

properties—all phenomena that seem to be metaphysically possible. After all, how can

a given world be built up from local fundamental parts if it contains no simple parts?

How can it be built up from basic property exemplifications if it contains no basic

properties? And even if a given world contains simple parts and basic properties, how

can it be built up completely from them if it contains higher level properties that elude

supervenience? Let us call the problem brought out by the possibilities of infinite

complexity and emergence ‘the puzzle of modal generality’ for theories of ontological

fundamentality.

It will be worthwhile to clarify the details of each element of the puzzle, which

are sometimes blurred in the literature. Mereologically infinitely complex (M) worlds

come in two mutually inclusive varieties. The first (MG) are gunky worlds: worlds that

contain items all of whose parts have proper parts.88 The second (MKJ) are

knuggy/junky worlds: worlds that lack any maximal complex. Every part in such

worlds is a proper part.89

Qualitatively infinitely complex (Q) worlds also come in two mutually

inclusive varieties, supervenience (QS) and conjunction (QC). QS worlds lack a

88 The term ‘gunk’ is first used in this way in (Lewis 1991). The mereology with which I am concerned here is such that (i) only items located in spacetime—whether regions, material objects, tropes, etc.—can be parts; and (ii) all parts of regions are regions; all parts of material objects are material objects; all parts of tropes are tropes, etc. 89 The term ‘knug’ is coined in (Parsons 2007). ‘Junky’ is first used in this way in (Bohn Forthcoming) and (Schaffer 2010). Schaffer (2010) argues against the metaphysical possibility of knug/junk, claiming that junky objects must be “worldless” since “a world that contained junk would be an entity (that is) not a part of another entity at that world” (65). This seems to me to beg the question. Schaffer also argues against knug/junk that it contravenes classical mereology. But as an argument against the possibility of knug/junk, this point is at best inconclusive. See (Bohn Forthcoming) for a defense of the possibility of knug/junk.

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minimal supervenience base: for every collection C of properties exemplified at such

worlds, every member F of C is such that there is some exemplified property G where

G is not in C and F supervenes on G. To characterize QC worlds, let me introduce the

terms ‘sub-conjunct’ and ‘congunktion’.90 For some conjunction p and item x: x is a

sub-conjunct of p just in case there is some conjunction q such that (i) q is a

constituent element of p and (ii) x is one of q’s conjuncts.91 A congunktion, then, is a

conjunction every sub-conjunct of which is a conjunction. QC worlds contain at least

one instance of at least one congunktive property—that is, a property picked out by a

(non-trivially) congunktive predicate.92

Emergence (E) worlds contain an instance of at least one strongly emergent

property, where a property F of a complex object a is ‘strongly’ emergent just in case

F is perfectly natural and does not locally supervene on the properties of and relations

among a’s most mereologically basic proper parts (McDaniel 2008, Sider MS).

M worlds and Q worlds are mutually inclusive yet independent; the same is

true of M objects and Q objects. M-and-Q objects are easy to imagine: for a given

object a and all of the properties (the Fs) that it and its parts exemplify down to

arbitrary mereological level L, just think of a further property G that subvenes the Fs

and is exemplified by some proper parts of a below L. It is also easy to imagine non-Q

M objects: just think of proper parts below L without thinking of them as

90 The notion that I call ‘congunktion’ does not, of course, originate with me. 91 I am thinking of constituent elements as being like parts of conjunctions, though not exclusively proper parts. So one of the constituent elements of p is p itself. 92 To get infinite conjunctive complexity, strict congunktion is not in fact required. All that is required is a string of conjunctions “all the way down,” which is consistent with some of the conjuncts not being conjunctions. I will set this subtlety aside and focus on congunktion in what follows. There is also a structural version of Q worlds, in which properties have infinite structure. However, the differences between the conjunctive and structural versions are not important for present purposes.

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exemplifying any further subvening or sub-conjunctive properties. To see that there

are possible non-M Q objects, consider a mereologically simple object b in a non-MKJ

world that exemplifies some class of properties, the Fs, where either (i) no F is in or

supervenes upon a minimal base or (ii) at least one F is congunktive. Granted, b is a

curious sort of object; but I see no reason to deny that it is possible. Though I will

suppress detailed discussion, I see no immediate barrier to supposing that E objects

might be M, Q, or both, without needing to be either.

We are now in position to get more precise about the different prongs of the

modal puzzle as it arises for the sparse local theorist.

MG Puzzle: Possibly, there are MG objects; at worlds that contain such objects, the

maximally local property distribution facts cannot be fundamental, for there will be

property instances that cannot be traced to any atomic locations/bearers. (This puzzle

is discussed in (Leibniz 1998) and (Schaffer 2003, 2010).)

QS Puzzle: Possibly, there are QS objects; at worlds that contain such objects, the

sparse property distribution facts cannot be fundamental, for there will be no sparse

properties: the properties exemplified by the objects in question will fail to supervene

on a minimal base, which violates sparsity. (This puzzle is presented in (Schaffer

2004).)93

93 One aspect of Schaffer’s (2004) critical line against the builder that I will not discuss in the text is his claim that the sparseness-criterial roles of resemblance-grounding and causal-joint-carving cannot be adequately filled by the Armstrong/Lewis conception of sparse properties. In short: I have doubts that his criticism goes through on a suitably charitable understanding of what these roles amount to.

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QC Puzzle: Possibly, there are QC objects, for which no non-conjunctive properties are

available to preclude redundancy. At worlds that contain such objects, sparsity must

either fail or else harbor redundancy to the point of not being able to maintain its

putative ontological economy. (Elements of this version are recognized in (Armstrong

1978, Lewis 1983, and Schaffer 2004).)

E Puzzle: Possibly, strongly emergent properties are exemplified; at such worlds, the

maximally local facts do not fix all of the facts at higher mereological levels. (Schaffer

2010, McDaniel 2008)

To appreciate the depth of the puzzle of modal generality, notice that it is an open

question whether the actual world is any of MG, Q, or E. Since we do not at present

know whether the actual world has any of these features, there is pressure to ensure

that one’s theory of ontological fundamentality applies in worlds that do. Moreover,

we will see in the next section that there is pressure from considerations of

explanatory power to think that any true theory of ontological fundamentality must

In more detail: to argue that the Armstrong/Lewis conception fails the resemblance-grounding role, Schaffer points out that many cases of higher level resemblance are such that the resembling items are multiply realizable with respect to fundamental items. I think that this betrays a biased conception of grounding. That more than one type of arrangement of fundamental items can underwrite a high level item of kind K does not hamper the fact that every K item will be fixed by appropriately arranged fundamental items. And this latter fact, by my lights, is sufficient for the ontological role of grounding resemblances. A similar point holds for Schaffer’s objection from causal-joint-carving. Schaffer argues that since, for example, being a synapse involves possession of a causal power to transmit a pulse between neurons, and since no individual fundamental property has this power, fundamental properties are ill suited to play the causal-joint-carving role. But no argument is given for the presupposition that all joints must be carved only by individual fundamental properties as opposed to being carved by appropriate arrangements of instances of multiple fundamental properties.

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hold at worlds that contain infinite complexity. If this is correct then the mere

metaphysical possibility of MG, Q, and E constitutes a prima facie threat to the sparse

local conception quite independently of how the actual world happens to be.94

The goal of this chapter is to discharge each of the above puzzles without

abandoning the explanatory and Ockhamist intuitions that favor the builder conception

of ontological fundamentality. This is not to say, however, that the solution to be

proposed will embody the most familiar builder form, according to which there is a

plurality of sparse properties exemplified at maximally local, point-sized instances.

Rather, on the view to be suggested, there is a single ultimately fundamental property

whose distribution into localized but not mereologically simple instances fixes all facts

across modal space. That is, the view to be defended is a moderate builder theory: the

sparse regional theory. It will be explained how the localized property instances can

play their subvening role without having to be mereologically simple or decompose

into mereological simples. The MG puzzle will thus be discharged. Moreover, it will

be explained how the proposed single fundamental property can subvene every

qualitative property F1…Fn at a given world without it having to be the case that any

of the Fis ultimately subvene all of the others. The key will be to explain how, in a

certain idiosyncratic sense, the novel fundamental property is in fact non-qualitative.

So, insofar as the intuitions in favor of the possibility of QS are intuitions about

qualitative properties, the QS puzzle will be discharged. A similar treatment will be

94 Ross Cameron (2008) is dubious that the explanatory power of theories of ontological fundamentality is enough to render them any more than contingently true at the actual world, though he does think it works for that more modest purpose. Cameron does not consider the specific argument to be given in section 2.

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proposed for the QC puzzle. As for the E puzzle, some reasons for doubting the most

prominent arguments for strong emergence will be discussed at the end of the paper.

Here is the itinerary. In the next section, I say more about what a theory of

ontological fundamentality ought to look like and argue that if any such theory is true

at the actual world then it ought to be true at worlds that contain infinite complexity.

In section 3, I contrast the sparse local/regional (“non-global”) conception with both

the sparse global and abundant global conceptions. I argue that it is better equipped to

address the possibility of MKJ than the former, and more ontologically economical

than the latter. In section 4, I outline my proposed species of the sparse regional

conception and use it to solve the MG and Q puzzles. Section 5 then sketches what I

take to be the most promising response to the E puzzle.

2. Fundamentality* and Modal Generality

Naturally, we want to make the notion of ontological fundamentality as clear as

possible. Yet any attempt to capture fully our pre-theoretic intuitions with a precise

analysis of the predicate ‘fundamental’ is likely to fail. Consequently, I will opt for

introducing an entirely new predicate, with the hope that its extension sufficiently

overlaps that which we pre-theoretically attribute to ‘fundamental’ for the ensuing

discussion of the sparse/abundant and local/global debates to be compelling to a wide

audience. Let us say, then, that an item is ‘fundamental*’ if it is not dependent upon,

(asymmetrically) supervenient upon, reducible to, posterior to, or derivable from other

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entities.95 Continuing with terminology, let us say that an analysis of some

phenomenon X is a ‘complete ontological’ analysis of X with respect to arbitrary

world w only if the terms in its analysans designate entities that are fundamental* at w;

and that a theory T of phenomenon X is ‘Metaphysical’ only if:

(i) T applies to X at all (or all suitably proximate) worlds in which X (or at

least one of its counterparts) exists;96 and

(ii) for all worlds w at which T applies, T offers a complete ontological

analysis of X with respect to w,

where a theory T applies to some phenomenon X at world w just in case it is intended

to be true of X at w. So, intuitively, a theory of X is Metaphysical only if it gives a

fundamental level account of X at every world. The notion of a complete ontological

analysis is relativized to worlds because there are many individuals and properties that

are fundamental* at only some of the worlds in which they exist or are exemplified.

Quarks, for example, might have been reducible to some other kind of item even if

they are not reducible in actuality. Similarly, quarks might be involved in a complete

ontological analysis of some phenomenon X with respect to the actual world and yet

fail to be involved in a complete ontological analysis of X with respect to some other

world in which X exists. In this case, the correct Metaphysical theory of X would be

disjunctive: it would invoke quarks in giving its analysis of X with respect to some

worlds and not with respect to others. As a contrasting example—one in which the

95 No doubt the notions driving this characterization of fundamentality* are themselves controversial. I am optimistic, however, that the various ways of understanding them among the readers of this paper overlap to an extent sufficient for excusing further analyses. 96 The first parenthetical qualification reflects the fact that some robust metaphysical theories are contingent; the second reflects the fact that some metaphysicians reject transworld identity. I will hereafter be less careful to make these qualifications.

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complete ontological analysis of some X is constant across all worlds—consider a

theory of laws of nature ala (Armstrong 1983, Dretske 1977, Tooley 1977) (hereafter

‘ADT’), according to which laws just are the obtainings of second-order relational

universals among first-order universals. ADT aims to give a unified account of laws

across worlds by maintaining (i) that laws are to be analyzed in terms of some special

second order relational universals—call them ‘the Rs’—with respect to every world

that contains laws (though it need not be the same Rs across worlds) and (ii) that the

Rs are fundamental* with respect to the relevant worlds.

Consider now the question of whether ontological fundamentality itself ought

to be given a disjunctive or unified treatment across worlds. That is, can what it is to

be ontologically fundamental vary? I am inclined to think not. After all, the

phenomena that (negatively) informed our characterization of fundamentality*—

dependence, supervenience, priority, reduction, derivation—are all, however we wish

to explicate them, such that their conditions for obtaining do not vary from world to

world. Moreover, fundamentality’s unity across worlds is important to its role in

Metaphysical explanation. To see this, consider the role of fundamentality* in ADT.

Since the Rs are fundamental*, there is supposed to be no deeper theory of laws

available: if ADT is true then the Rs are (or at least are among) the bedrock of

explanation with respect to the ontology of laws of nature. Suppose then, for reductio,

that what it is to be fundamental* is allowed to vary among worlds. At world w, for

example, the Rs count as being fundamental* in virtue of being irreducible to anything

else, but nonetheless are posterior to some other type of entities, the Fs. At distinct

world w', by contrast, the Rs count as being fundamental* in virtue of not being

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posterior to any other kind of entity, but nonetheless are reducible to some distinct

kind of entities, the Gs. So in w, the Fs furnish a deeper explanation of laws than do

the Rs, while in w' the Gs furnish a deeper explanation. This contradicts the claim that

the Rs are among the explanatory bedrock with respect to laws of nature across

worlds. In light of these considerations, I submit that fundamentality* requires a

unified treatment across worlds.

Notice, however, that the preceding argument works only at a very general

level. It does not seem to bear, for example, on whether a more specific treatment of

ontological fundamentality such as the sparse local theory ought to be true at all

worlds if it is true at any.97 For all that has been said about the factors that inform our

notion of fundamentality*, it may still be the case that at some worlds the fundamental

level is sparse and local while at others it is abundant and global. Is there any good

reason to believe that the sparse/abundant and local/global debates concern more than

fairly weakly contingent theories? I am inclined to think so. Specifically, the

sparse/abundant and local/global debates ought to be decided uniformly across worlds

that contain and lack infinite complexity.

To begin to see why, notice that there is important Metaphysical work that

certain sparse local theories are advertised as doing that could not be done if they were

not true at infinitely complex worlds. For example, consider the Humean

Supervenience based account of laws of nature. On this account (simplifying a little),

laws are truths that supervene on regularities among point distributions of sparse

properties across time. The Humean Supervenience theorist takes his view to furnish a

97 Thank you to Jonathan Schaffer for pressing this point.

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complete ontological analysis of laws at the actual world and sufficiently nearby

worlds. Notice, however, that in a world that contains gunk the Humean

Supervenience account of laws will fail, for the gunky objects in question may well be

law-governed even though there are no point instances of sparse properties available

to furnish the requisite regularities.

The defender of the Humean Supervenience approach to laws might shrug off

this concern by (i) emphasizing that his view only claims to apply at the actual world

and nearby worlds and (ii) denying optimistically that the actual world or any

sufficiently proximate world is gunky. But this response is too quick, even if we grant

the controversial claim that the actual world is not gunky. The problem is that MG

worlds may still be very nearby. Consider, for example, a world w in which only one

very small and fleeting object, a, is gunky. So small and fleeting is a that it looks

much like a grain of sand and exists for only one millisecond. Suppose that w is just

like the actual world except that a lies on w’s counterpart of Old Orchard Beach,

Maine sometime in May 2010. The trouble for the Humean Supervenience account of

laws is that a may well be governed by numerous laws of nature even though no

intrinsic facts about a are fixed by the point-instances of the properties exemplified at

w. This means that, at w, the laws are not fixed exclusively by regularities among the

point-sized instances of properties. So, despite w’s close proximity to the actual world,

the Humean Supervenience account of laws does not hold there.

The basic thought here is that theories of fundamentality need to apply with

significant modal generality if they are to serve an ample explanatory function.

Importantly, this line of argument does not turn on the idiosyncrasies of the Humean

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Supervenience account of laws. Parallel considerations apply to any Metaphysical

theory that invokes an underlying conception of fundamentality that applies to entire

worlds and yet cannot apply to certain kinds of objects. (I will suppress the details of

how the argument works with Q objects, but the basic idea is that if we let a be a

qualitatively unique Q object then the actually sparse properties that obtain at w do not

fix the abundant facts that obtain there, even though the enormous majority of w is just

like the actual world.)

The discussion of this section motivates two conditions of adequacy on any

account of ontological fundamentality that seeks to settle the sparse/abundant and

local/global debates. First, it must respect the notions with which fundamentality* was

characterized (or at least a critical mass thereof). Second, it must be true at worlds that

contain infinite complexity if it is true at all. Let us call this second condition the

‘modal generality condition’. The puzzle of modal generality, then, concerns how any

sparse, non-global theory can apply in infinitely complex worlds without the

fundamental* items outstripping the sparse and the non-global.

3. De-Motivating the Global: Ontological Economy and MKJ

Both kinds of global theorist, sparse as well as abundant, are well positioned to

circumvent the MG and E puzzles. If the whole world is more fundamental than its

parts, then there is no MG-based threat, for no recourse to atomic parts is required.

Similarly, if certain properties that can only be exemplified at arbitrarily high levels of

mereological complexity count as fundamental, then it is no problem if they fail to be

fixed by the distribution of properties exemplified at lower mereological levels. Where

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the difference between the sparse and the abundant global theorist comes into play

with respect to the puzzle of modal generality is in the QS case. In short, the abundant

global theorist is equipped to accept the possibility of infinite qualitative complexity

and the sparse global theorist is not. If all of the properties (or all that are above some

threshold of naturalness) count as fundamental, then it is no problem if none of these

properties have a minimal supervenience base, for not-being-supervenient is no longer

considered necessary for being fundamental. The abundant global theorist can instead

emphasize the priority aspect of fundamentality* and maintain that, say, sociological

properties are not posterior to, say, microphysical properties, thus discharging the QS

puzzle. The sparse global theorist, by contrast, is just as threatened by the QS puzzle as

is the sparse local theorist, for both hold that there is a minimal supervenience base of

fundamental properties, which claim would seem to be inconsistent with QS.

So where does this leave the global/local debate? As we have just seen, the

abundant global theorist’s solution to the QS puzzle requires him to give up the not-

supervenient component of fundamentality*, which might lead one to doubt whether

he has fully succeeded in addressing the puzzle. Of course, the abundant global

theorist will hold that this component was specious anyway. While this is an

interesting disagreement, I will set it aside. What I want to emphasize is the fact that

the abundant global theorist, for reasons of explanatory power discussed in the last

section, must apply his view across both worlds that contain infinite complexity and

those that do not. But doing so commits him to the claim that even at the actual world

certain high level properties, say sociological properties, are just as ontologically

fundamental as microphysical properties. Indeed, the abundant global theorist would

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face the general problem of being unable, with respect to non-M, non-Q, non-E

worlds, to deflate the ontological status of certain high level properties like being a

low-income senior by appropriately relating them (via, say, reduction, grounding, or

supervenience) to Armstrong/Lewis sparse properties. Those with any Ockhamist

leanings will see this as an important reason to favor the sparse approach over the

abundant. Even if one were dubious about the prospects for reducing certain special

scientific properties to Armstrong/Lewis sparse properties, one is better off—all else

equal—with a theory of fundamentality that does not require every special scientific

property (or even just those above some threshold of naturalness) to be impervious to

ontological deflation. So much the worse, I say, for the abundant theorist. He can

dismantle the QS puzzle, but only by accepting a prohibitively expensive ontology.

Let us turn, then, to the sparse local/sparse global debate. Both sparse theories

seem to be in the same boat with respect to the QS puzzle. However, I will argue in the

next section for a new kind of sparse theory that dissolves QS. The new sparse view to

be suggested is consistent with the strictly local, the regional, and the global approach,

though I think the local and global approaches are best resisted. The chief strike

against the strictly local approach is that it cannot address adequately the MG puzzle

(Schaffer 2003, 2004, 2010). The chief strike against the global approach is that it

seems unable to address adequately a puzzle that arises from MKJ:

MKJ Puzzle: In knuggy/junky worlds, there is no whole that is not also a proper part.

There is thus no ‘whole’ world available to be fundamental, for every whole at such

worlds will fail to fix all of the facts.

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The sparse regional theory to be suggested presently has the advantageous feature of

allowing for both gunk and knug/junk.

4. Markedness

Let us begin by examining how the regional approach can allow for both MG and MKJ.

The idea is straightforward. Since the regional approach presupposes neither simples

nor maximal composites, it is not embarrassed by worlds that lack simples or maximal

composites. On the regional approach, once God settles the facts about what exists at a

certain fundamental “regional” level of mereology R, he settles all of the facts,

including those about whether the world is gunky or knuggy/junky. If we call the

objects at R ‘regional objects’, we can state the regional treatment of M worlds as

follows.

(Regional Gunk): If at least one of the regional objects at arbitrary world w is such that

all of its parts have proper parts, then w is gunky.

(Regional Knug/Junk): If for every collection C of regional objects at w there is some

further regional object available to be fused with the members of C, then w is

knuggy/junky.

The most pressing question for the regional view is how to determine which

mereological level counts as the fundamental regional level R, for prima facie it seems

that any choice will be arbitrary (Schaffer 2003, 2010). My response to this worry is

that R is determined by the distribution of the second order property of being

maximally connected across instances of first order fundamental properties, where a

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property instance is maximally connected at time t just in case it is neither scattered

nor in contact with any wholly distinct instance of the relevant property at t. (My

preferred way to understand property instances is as being spatiotemporally located

tropes, but they may also be understood, for present purposes, as spatiotemporal

locations of particular bearers of universals.)98 So, on the sparse regional theory, the

fundamental items at world w are the maximally connected instances of the sparse

properties at w.

I think that the problem of how to determine which mereological level is

fundamental is a deep problem for the regional theory. I am also inclined to think that

the maximal connectedness criterion furnishes the most promising response. This is

not to say, however, that it is without a serious problem of its own, namely, that it is in

tension with the following intuitively possible case. Consider a maximally connected

property bearer a such that

(i) all of a exemplifies some fundamental property F (via a maximally

connected instance), and

(ii) some non-scattered proper part b of a exemplifies some fundamental

property G (≠F) (via a maximally connected instance).

The problem here is that both a and b would seem to be at the fundamental

mereological level even though one is a proper part of the other. In light of this worry,

I think that something about the sparse regional theory described thus far has to give.

Since it is my own suggestion about how best to understand the regional aspect of the

98 Tropes are non-repeatable, non-universal features. Properties are primitive resemblance classes of tropes. On my preferred view, tropes are located in space and time and thus have sizes, shapes, and durations.

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theory that has backed us into this corner, I am inclined to make the modification to

the sparse aspect.

But what can be modified about the sparse conception? After all, we cannot go

abundant or minimally natural without giving up ontological economy. My suggestion

is that, instead of looking to go less sparse, we go ultra sparse. Instead of accepting the

traditional sparse ontology according to which there may be many fundamental

properties at a given world and many distinct sets of fundamental properties across

worlds, we should explore the suggestion that there is exactly one ultra-fundamental

property across all worlds. This solves the present worry by blocking the

presupposition in (ii) that there may be distinct fundamental properties.

This ultra-fundamental move might seem an ad hoc, extreme, and even

desperate means of saving the maximal connectedness criterion and the regional

theory. Why would it not be better, for example, to go global and just shrug off

worries about MKJ? I have two answers. First, the MKJ puzzle is no less compelling

than the MG puzzle, which is one of the primary weapons the global theorist wields

against the local theorist. Second, the ultra-fundamental move has a further important

benefit over and above saving the maximal connectedness criterion. Namely, it

furnishes a sparse-theoretic means of dissolving the Q puzzles without having to deny

the intuitions that underwrite our inclination to assent to the possibility of Q.

Before explaining just how the ultra-fundamental move achieves the advertised

goal, it will be good to make the proposal a bit more precise. The core idea is that

there is a single (previously unfamiliar) uniquely fundamental property that

necessarily subvenes every other property. (The property cannot be any antecedently

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familiar property, say mass, since any such property may possibly go unexemplified.)

Given the maximal connectedness criterion, the ultra-fundamental property had better

have spatiotemporally located instances. Notice, however, that it need not do much

else. For simplicity, then, I submit that the foremost function of the proposed special

property is merely to mark off certain maximally connected locations from their

surroundings. Accordingly, I suggest calling the property ‘markedness’.

On the present suggestion, we do not directly see, hear, or otherwise sense

markedness playing its marking-of-locations role, but it plays it nonetheless and it

plays it non-derivatively. Markedness does not confer detectible features like color,

texture, sound, or charge, nor is it a second order property of any property that does,

for it does not ontologically piggyback on other properties. Rather, it marks certain

locations from others even in worlds where no familiar qualitative properties are

exemplified. Indeed, markedness is what I (idiosyncratically) will call a ‘non-

qualitative’ property to distinguish it from the antecedently familiar first-order

properties of material objects, whether macro dry goods or micro particles.99

Importantly, however, markedness is not non-qualitative in the sense of being a

haecceity, a quiddity, a self-identity property, or an essence. Rather, it is ‘non-

qualitative’ in two distinct senses: first, in the sense of being necessarily more

fundamental than any traditionally ‘qualitative’ property and second, in the sense that

its chief functional role as a marker does not entail that it is empirically detectible. It is

rather a “metaphysical” property in something like the same sense that instantiation

relations, compresence relations, bare particulars, Aristotelian kinds, spacetime 99 Notice that this is a technical sense of ‘qualitative’ that does not jibe with more traditional uses according to which ‘qualitative’ means something like ‘descriptive’.

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regions, etc. are metaphysical: the proponents of these items store them at the

ontological ground floor without any expectation that their existence will be

empirically confirmed.

Obviously, markedness is not currently posited by our best physical theories—

the proposal is an entirely new philosophical proposal. Consequently, it is very

different from the kind of builder theories defended by Lewis and Armstrong, for

whom the proper source of fundamental properties is a complete and final physics.

The severity of this break with traditional builder methodology is offset by the work

that the new theory can do, however. Specifically, it can do the philosophical job of

solving the puzzle of modal generality without giving up the central Ockhamist and

explanatory intuitions that motivate the builder approach. The severity is also

mitigated by the fact that markedness is non-qualitative. It is not as though positing

markedness upsets the ability of final physics to “save the phenomena.” Positing

markedness, insofar as the empirical soundness of final physics would be concerned,

is rather more like coming down on one side of the universal/trope debate: it is simply

not an issue with which physics is concerned.

As a heuristic, we can think of the spatiotemporal distribution of markedness at

a given world as furnishing a sort of binary code for fundamental ontology: the

maximally connected markedness instances, in virtue of how they are arranged

spatiotemporally (intuitively: in terms of their shapes), fix all of the other facts. To see

in a bit more detail how the idea works, consider without loss of generality some

qualitative property F and world w. The distribution of F at w is fixed by the

distribution of maximally connected markedness instances of some particular shape

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(and perhaps size) at w, in the following sense. No maximally connected markedness

instance of the relevant shape at w fails to contain an F instance at its location, and no

F instance at w fails to be contained by the location of some markedness instance of

the relevant shape.100 The markedness instances and the F instances need not be of the

same “shape;” but the latter must be entirely contained by the former in the sense

described. The criteria for a given maximally connected markedness instance to be of

the correct shape for fixing an F instance at w will have some flexibility: those and

only those markedness instances of a certain limited range of resembling shapes play

this role with respect to F at w. In this sense, the markedness instances that fix the F

distribution at w are “multiply realizable.” This “multiple realizability” secures the

asymmetry of the supervenience of F upon markedness at w that the sparsity of

markedness requires, for while there can be a difference in markedness instance shape

(i.e. in markedness distribution) without there being an F difference at w, there can be

no F difference without a difference in markedness distribution. Moreover, it need not

be the entire shape of a given maximally connected markedness instance that matters

in determining whether it fits into the range that fixes arbitrary property F; rather, it

may be that the presence of a “tab” of a certain shape range at the perimeter of the

relevant maximally connected markedness instance is sufficient to fix a given F

instance at a given world. (As a heuristic, we may think of the shape of the tab as

being the shape of a written token of the predicate that picks out F.) This would allow

100 Despite its utility with respect to exposition, substantivalism need not be presupposed. Talk of contained locations can be paraphrased in terms of parthood and overlap: a contains b in its location just in case every part of b overlaps some part of a.

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a single maximally connected markedness instance to fix multiple qualitative property

instances in virtue of having multiple tabs of appropriate shape.

Since I would like to beg as few questions as I can manage about the ontology

of property exemplification, I will leave open whether markedness instances are prior

to the existence of any non-properties.101 For example, the stern trope theorist or

object nihilist may accept that they are while the bare particular or states of affairs

theorist may deny it. Alternatively, one might prefer to think of the bearer of

markedness as spacetime itself, with the bearer for each maximally connected instance

being some particular region. I happen not to prefer this strategy, however, because I

would like to remain neutral with respect to substantivalism about spacetime.102

Two further points about the suggested view are worthy of emphasis. First, the

supervenience relation between appropriately arranged markedness instances, on the

one hand, and qualitative properties, on the other, is world-bound. This world-bound

or ‘nomological’ supervenience does not presuppose any sort of robust realism about

laws; it is consistent with a neo-Humean regularity theory, for example. What is

presupposed is that there are the relevant regularities, namely, regularities between

arrangements of connected markedness instances—on the one hand—and the

distributions of all other (exemplified) properties, on the other. Importantly, this move

to nomological supervenience does not constitute as big a break with traditional

101 On my preferred understanding of tropes, tropes exemplify themselves. On my preferred trope-theoretic understanding of markedness, then, it is not true that markedness can exist unexemplified even if it need not be exemplified by regions or material objects, for every markedness trope will be reflexively exemplified. Whether a similar account of reflexive exemplification is available to those who favor immanent universals is a question that I will leave open. 102 Could markedness really mark locations if substantivalism were not presupposed? Yes; markedness may be understood in relationalist terms. The idea would be to emphasize distance relations not among fundamental material objects but among maximally connected markedness instances.

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builder theories (e.g. Lewis’s Humean Supervenience) as it might seem to at first. This

is because metaphysical supervenience relations among exclusively qualitative

properties are consistent with the nomological supervenience relations that the present

view posits, namely, those that hold between markedness and qualitative properties.

So the proponent of markedness can maintain, for example, both (i) that having mass

M nomologically supervenes on some range of markedness instance types at w and

some distinct range of markedness instance types at w' (where instances are typed by

shape) and (ii) that myriad higher level qualitative properties metaphysically

supervene on mass.

I have already alluded to the second point to be emphasized, namely, that a

given markedness instance might well fix more than one qualitative property in the

sense described. This allows for markedness instances to subvene properties

exemplified at lower spatiotemporal-cum-mereological levels than the level of the

instances themselves. In light of these remarks, it should be clear that markedness

necessarily is the ultimate asymmetric supervenience base without having to be

exemplified by any mereological atom and without having to be understood as a

global property of the whole world. It makes for a builder theory that is capable of

functioning from the mereological “middle” of a given world: a sparse regional theory

of ontological fundamentality.

4.1 Cost-Benefit Analysis

Before explaining how markedness helps to discharge the Q puzzles, it will be good to

address an important prima facie worry, namely, that markedness is sufficiently

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strange, idiosyncratic, “metaphysical,” and unprecedented as to be a major ontological

expense. The worry is important because ontological economy has been the chief

motivation for the sparse view over its abundant competitors. In response, I maintain

that markedness is no stranger, more idiosyncratic, or further from being empirically

confirmed than any of a number of other metaphysical items with which it competes

for various philosophical jobs and whose proponents take them to be necessarily

fundamental, for example, bare particulars, primitive compresence or instantiation

relations, arbitrarily complex distributional properties, substantival spacetime regions,

and Aristotelian kinds.103 If I am right, then markedness—qua strange and non-

empirical—is not unprecedented. Moreover, since markedness can save the sparse

non-global theory without committing us to these other more familiar pieces of

metaphysical machinery (many of which are either implicitly or explicitly endorsed by

past proponents of both local and global theories), it is no more ontologically costly

than they are. Indeed, it is less so if it truly can save the sparse, non-global approach

where they cannot, for the ontological economy relevant to motivating sparsity

concerns deflation of the non-fundamental many in terms of the fundamental few. It

does not concern which kinds of items are posited as among the fundamental few or

whether or not these items are strange, unprecedented, etc. Let us move on, then, to

the proposed solutions.

103 That markedness competes with these items is shown more clearly in my (PaperA), where I argue that instances of a property like markedness can play the role of metaphorical adhesive for bundling properties into objects, thus furnishing a theory of property exemplification that does not require commitment to any of the aforementioned items. I also argue there that markedness can furnish a novel approach to four-dimensionalism that discharges the temporal analog of the MG puzzle, which is raised in (Stuchlik 2003).

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4.2 Applying Markedness: The QS Puzzle

Markedness prompts a reexamination of the possibility of there being no minimal

supervenience base. Prior to the suggestion of markedness, it has been assumed that

there is a store of actual (and perhaps alien) qualitative properties F1…Fn (where

markedness, qua non-qualitative, is not one of F1…Fn) from which we may allow a

certain few to be included in the minimal supervenience base at a given world. The

sub-collection of F1…Fn functioning as the select few minimal subveners at w1 need

not be the one playing that role at w2. Moreover, no special one of F1…Fn would be in

the select group at every world. Considering just F1…Fn, then, it might be the case that

no Fi is in a minimal supervenience base. But the ultra-fundamental move explicitly

rejects the claim that no special property is in the select group at every world, since

markedness just is the minimal base at every world. So the proponent of markedness is

forced to reject the possibility of QS. Importantly, however, the proponent of

markedness still is able to allow that there are worlds in which none of F1…Fn is in a

minimal supervenience base, which—I conjecture—is really what those who take QS

worlds seriously have had in mind all along.

Notice that this conjecture is different from the flat-footed move of rejecting

QS even as it applies to F1…Fn. The flat-footed move offers no account of why

assenting to the possibility of QS is so tempting. Notice further that, in

contradistinction to the proponent of markedness, the sparse global theorist (assuming

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he rejects markedness) can do no better than to make the flat-footed move.104 By my

lights, the best diagnosis of why the possibility of QS has seemed compelling to certain

theorists is that they either have not considered or have implicitly rejected the

suggestion that there is a single, necessary, minimally subvenient property. If it is that

one never considered such a property, then whatever intuitions were responsible for

one’s acceptance of QS can be accounted for by the markedness proponent. After all,

even he accepts that something very much like QS is possible, namely, an “infinite

descent” of properties F1…Fn, each of which supervenes on some other, so long as

markedness is not one of the Fis. If, on the other hand, it is that one implicitly rejected

such a property, then I submit that such rejection is premature. The mere

conceivability of QS worlds does not help. The conceivability of QS does no more to

justify the rejection of markedness than the conceivability of markedness does to

justify the rejection of QS. I argue here and elsewhere (PaperA) that markedness has

more than mere conceivability going for it. I am not aware of any such arguments on

behalf of QS put forth by those who accept its possibility.

4.3 Applying Markedness: The QC Puzzle

Much as in the QS case, treating the QC arm of the modal puzzle involves a re-

examination of just what QC worlds amount to. The proponent of markedness can

allow for worlds that contain arbitrarily many congunktive qualitative properties

104 It is open to the sparse global theorist to adopt markedness. On the resulting view, there is exactly one maximally connected markedness instance at every world and it overlaps the whole world. The global markedness theorist still has problems with knug/junk, however, so long as he requires a maximal composite at the fundamental level. The regional theory’s success at treating the MKJ puzzle thus blocks the attempt in (Schaffer 2010) to argue against the regional theory by claiming that once one is willing to go regional one might as well go global (64).

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without leading either to redundancy or failure of modal generality (again, the sparse

global theorist who rejects markedness cannot make this allowance). Three related

points make this clear. First, markedness can subvene congunktive properties; indeed,

it can subvene infinitely many. Recall that maximally connected markedness instances

function much like mini-versions of worlds. Just as the whole world is capable of

subvening congunktive properties for the global theorist (since the distribution of

congunktive properties is fixed by the facts of the whole world), so maximally

connected instances are capable of subvening them for the proponent of markedness

(since the distribution of congunktive properties within the spatiotemporal-cum-

mereological realm of a given maximally connected markedness instance are fixed by

its shape). Second, worlds with congunktive properties are not redundant for the

markedness proponent because the congunktive properties are not among the sparse

properties; rather, markedness alone is the sparse property. Third, there is no

congunktion-relevant modal limitation imposed with respect to qualitative properties

since the markedness suggestion ranges over all worlds. If markedness theory is true

then there is no world in which congunktive properties are exemplified without

asymmetrically supervening on markedness.

What the markedness proponent must deny, however, is that there are worlds

in which all properties are congunktive, for markedness is not a conjunctive property.

Again, this controversial take on the modal space is defensible once we recognize,

first, that conceivability is not an infallible guide to possibility, and second, that the

markedness proponent does not deny that there are worlds in which all qualitative

properties are congunktive. Moreover, it is up to the markedness theorist to say

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whether or not markedness is conjunctive. Claims to conceive of markedness as

conjunctive/congunktive are confused, for if markedness were

conjunctive/congunktive then it would fail to be the ultimate subvener in certain

worlds. The objection that one can conceive of markedness being

conjunctive/congunktive is much like the objection to Lewisian modal realism that one

can conceive of concrete objects that exist in more than one world. The Lewisian

response to this objection has two parts (Lewis 1986a). First, the objection is

uncompelling on the assumption that Lewisian modal realism is true; second, we have

good reason to think that Lewisian modal realism is true because of its utility and

elegance. Similarly, the markedness proponent’s response to the objection that

conjunctive/congunktive markedness is possible is that, first, the objection is

uncompelling on the assumption that the markedness theory is true, and second, that

we have good reason to think that it is true because of its utility and relative elegance.

5. Emergence

My principal concern has been with the sparse non-global response to the M and Q

puzzles. Still, it will be worthwhile to say something, however brief and inchoate,

about how to respond to the E puzzle. Proponents of strong emergence have offered

conceivability arguments for the possibility of strongly emergent consciousness

(McDaniel 2008) and arguments from quantum entanglement for the actuality of

emergence (Maudlin 1998, Schaffer 2010). The thought in this latter case is that, since

entangled particles considered individually are not such that their intrinsic properties

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fix the facts about their entangled state, certain properties of the state as a whole are

strongly emergent.

I have two brief responses to the consciousness argument. The first is by now

quite familiar: it is far from clear that conceivability is a reliable guide to metaphysical

possibility. The second is that even if we endorse conceivability as a guide to

possibility, it is far from clear that strongly emergent consciousness is reliably

conceivable. For all we know about the fundamental ontology of conscious states, it

may be the case that differences involving some few extremely small, mereologically

basic parts—call them ‘the Cs’—are what distinguish a conscious organism from an

otherwise qualitatively identical non-conscious organism. If so, then it is not clear that

when we think we are conceiving of a non-conscious organism that is just like some

conscious organism with respect to its most mereologically basic parts (thus showing

that consciousness does not supervene on the mereologically basic), we are not in fact

confused and merely conceiving of an organism that lacks the Cs and is thus not

relevantly like the conscious organism. One might respond that we can get clearer

conceptions by simplifying the relevant thought experiments to involve small

collections of mereologically simple items instead of more familiar complex

organisms. But this move comes at the expense of making it far less clear that we are

conceiving of anything that is truly conscious. One might then suggest that we can

conceive of strong emergence in the abstract without having to invoke consciousness

(or quantum mechanics) at all (Schaffer 2010). But if we are disinclined to accept

conceivability as a guide to metaphysical possibility in the case of properties with

which we are at least somewhat familiar, then we will not be moved by conceivability

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in the logical abstract. Again, the conceivability of emergence in the abstract carries

no more dialectical weight than does the conceivability of markedness. Compare

another realm of metaphysical dispute: the nominalist might well take her position to

hold of necessity without feeling any embarrassment from the conceivability of

Platonism.

My response to the entanglement argument invokes markedness. The thought

is that markedness is well suited to play the controversial role of non-local hidden

variable in the hidden variable interpretation of quantum mechanics. According to this

interpretation, entangled particle states behave as they do because of a “hidden”

property whose distribution varies with respect to the particles in question. If there is

such a property, then entanglement does not constitute an example of emergence, for

the property in question is available to be the requisite subvening factor. The problem

for hidden variable theories is that no empirical research has ever provided evidence

for any such property. No doubt, this is a serious problem for those who wish to

defend the hidden variable interpretation from within the framework of empirical

science. Notice, however, that markedness is not intended to be empirically detectable.

Markedness is a non-qualitative, entirely “philosophical” posit. As such, its invocation

is fair game in a priori arguments about the deep nature of ontological fundamentality,

including the argument that any phenomenon relevantly like quantum entanglement is

an example of strong emergence. While that argument invokes empirically supported

premises about quantum entanglement—that is, they are empirically supported given

the standard interpretation of quantum mechanics—it is still just as much an a priori

philosophical argument as is, say, the argument from empirically observed

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resemblances to the existence of universals (Armstrong 1978, 1989). Just as the

nominalist does not deny the phenomenon of resemblance, so the non-global theorist

will not deny that the properties of individual particles that (standard) physics

currently posits fail to subvene certain properties of the entangled state. What he will

deny is that there is nothing else in the world, mereologically below the whole

entangled state, which works as subvener. The positive proposal is that the properties

of the entangled state nomologically supervene on the shapes of the maximally

connected markedness instances at whose locations the respective particles are

contained and which came about when the particles became entangled.105

The point here is not that this proposal will be confirmed in a lab. Rather, the

point is that markedness, if it in fact obtains, is just the kind of property that would

play the role of hidden variable; so the argument from quantum entanglement to global

theories of fundamentality needs to be presented more slowly and carefully than its

proponents have appreciated. Specifically, it needs to include a sub-argument for the

premise that there is no good reason, from the standpoint of fundamental metaphysics,

to suppose that any property relevantly like markedness obtains. The fact that

physicists do not favor hidden variables like markedness is as irrelevant to the

local/global debate as the fact that they do not favor universals is irrelevant to debates

in the metaphysics of properties. The debate over whether spin (or any other property)

is a universal will not be decided by physics, but by the utility of realism about

105 I will not attempt to tell a complete story about the evolution of markedness distribution across time here. But it is important to recognize that the relevant markedness instances that “come about” when the relevant particles become entangled are not the causal result of the entanglement per se, but rather of the relevantly prior arrangement of markedness, i.e. that arrangement which fixed the facts about the pre-entangled particles, the pre-experiment lab equipment, etc.

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universals in the face of distinctly metaphysical puzzles, arguments, and explanations.

Similarly, the debate over whether quantum mechanics (or any other phenomenon)

involves emergence will not be decided by physics, but by the utility of those views

that favor emergence in the face of metaphysical puzzles, arguments, and

explanations. On that note, I will close this section with a challenge to those

metaphysicians who take quantum mechanics to involve emergence and thus to favor

the global theory. The challenge, once again, comes from GKJ. It goes like this:

(i) GKJ is possible;

(ii) the possibility of GKJ and the modal generality condition entail the falsity

of the global theory;

(iii) emergence requires the global theory, so:

(iv) the possibility of GKJ suggests that emergence is impossible; so:

(v) the most charitable interpretation of quantum mechanics from the

perspective of those interested in modally general fundamental metaphysics is

to deny that it involves emergence.

6. Conclusion

I have attempted to motivate markedness and to defend it against charges of

desperation and ad hocery. Whether or not I have succeeded, I hope at least to have

shown the markedness suggestion to be novel, tenable, and worthy of further

consideration. Independently of markedness, I hope to have advanced the debate

between the builder and heir approaches to ontological fundamentality on five fronts.

First, by showing that the logical space in which the debate is embedded is more

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complex than has been appreciated antecedently. Second, by showing that our

expectations of Metaphysical explanation dictate that the best theory of ontological

fundamentality ought indeed to hold at worlds that contain infinite complexity; third,

by emphasizing that the global theory is just as threatened by MKJ as the local theory is

by MG; fourth, by defending the regional approach against the hasty dismissal it has

received in the few instances where it has been acknowledged in the literature; and

finally, by unearthing some hidden assumptions at work in the global theorist’s

argument from quantum entanglement.

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Chapter 6

Gloppy Four-Dimensionalism (with a Pinch of Causation)

I use glop theory to develop a version of four-dimensionalism that can treat the puzzles of fission and fusion without being committed either to instances of simultaneous co-location of a spatial region by more than one material object or to instantaneous objects. The former commitment, as stage theorists have emphasized, is counterintuitive. Yet the latter commitment is also problematic since it rules out the compelling possibility of temporally atomless gunk. Historically, those versions of four-dimensionalism that avoid the former commitment bear the latter, and vice versa. This chapter thus constitutes a step forward for four-dimensionalism inasmuch as it furnishes a version that avoids both commitments. The chapter ends with a discussion of a second benefit of glop-theoretic four-dimensionalism: that it allows for a theory of causation that (i) comports with the ontological austerity of trope theory and (ii) circumvents the problem of causal preemption. 1. Introduction

In the last two chapters I argued that my favored background metaphysic, glop theory,

furnishes uniquely attractive theories of property exemplification and ontological

fundamentality. In this chapter, I will argue for a similar conclusion with respect to

persistence. Specifically, I will argue that glop theory furnishes a four-dimensionalist

account that both avoids commitment to instantaneous temporal parts—which

commitment is problematic given the possibility that time is gunky106—and accords

with intuitions about how many material objects might exist in a given region at a

given time. No prior four-dimensionalist theory has both of these features.

Interestingly, it will turn out that gloppy persistence is consistent with both (i) a

version of stage theory that does not require instantaneous parts and (ii) a version of

worm theory that does not require temporary material co-location in order to address

106 An item is gunky just in case all of its parts have proper parts. The term ‘gunk’ is first used in this technical sense in (Lewis 1991). The observation that temporal gunk poses a prima facie problem to certain versions of four-dimensionalism (namely, stage theories) is due to Joshua Stuchlik (2003).

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puzzles about constitution and fission. For present purposes, I will look upon this

neutrality as a merit of the general gloppy approach. Whether the glop theorist prefers

to go the stage or worm route ought then to be a matter either of intuitive pull or of

some theoretical desiderata distinct from those discussed here or in the extant

literature on persistence.107

In the final two sections, I discuss the relationships among glop-theoretic four-

dimensionalism and various forms of causation. For reasons that will emerge, one

might be tempted to object to gloppy persistence on the grounds that it is committed to

immanent causation. I will argue that no such commitment holds. I will then argue that

gloppy persistence yields a trope-based theory of causation that respects the strict

ontological economy of trope theory and avoids worries about preemption without

requiring a three-dimensionalist framework.

2. Some Background on Persistence

I argued in Chapter 3 that three-dimensionalist theories of persistence rely on a

problematic notion, that of being ‘wholly present at distinct times’. The problem took

the form of a dialectical dilemma. Either the three-dimensionalist must take on

independently implausible commitments (e.g. to presentism (Merricks 1999) or to the

actuality of temporally extended simples (Parsons 2000)) in order to explicate her

view in any clear way or else her view seems to collapse into that of her four-

107 Interestingly, Mark Moyer (2008) has recently argued that worm theory and stage theory collapse into the same theory. I am not convinced that Moyer is correct, but if he is then his conclusion reflects favorably upon the glop theorist’s neutrality.

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dimensionalist competitor.108 The upshot was that four-dimensionalism is the most

promising framework for theories of persistence.

According to the four-dimensionalist, objects persist by having appropriately

related temporal proper parts at distinct times. There are two chief versions of four-

dimensionalism, the worm version (‘worm theory’) and the counterpart version (‘stage

theory’). According to the former, ordinary object terms pick out diachronic bundles

(or ‘sums’ or ‘fusions’) of appropriately related temporal parts. The appropriate

relation might involve causation, spatiotemporal continuity/contiguity, psychological

continuity, or a variety of other factors. According to stage theory, ordinary object

terms pick out counterpart-related instantaneous temporal parts called ‘stages’, and our

intuitions about ordinary object persistence are recovered by facts about how certain

stages are counterpart-related to one another. The relevant notion of counterpart is,

roughly, a temporal analog of David Lewis’s modal notion of counterpart. Just as

Lewis holds that (i) you are possibly F just in case you have a counterpart that is F at

some possible world and (ii) that you and your counterparts in other possible worlds

are numerically distinct, so the stage theorist holds (iii) that you were (or will be) F at

past (future) time t just in case you are counterpart related to some person stage that is

F at t and (iv) that your present stage and your non-present stages are numerically

distinct. The chief difference between worm theory and stage theory, then, consists in

how each treats the reference of singular terms.

108 Here I am primarily thinking of Fine (2006), who takes pains to explicate ‘wholly present’ in a new framework. As discussed in Chapter 3, I can only make sense of the explication Fine gives if his wholly present items cannot undergo change. If his wholly present items cannot undergo change, however, then his view is not at odds with the version(s) of four-dimensionalism that I propose in this chapter.

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On stage theory, the counterpart relation is typically indexed with a sortal

notion. For example, a stage theorist will hold that certain stages are person-

counterparts whose collective existence underwrites our intuition that, say, Barack

Obama is a persisting object. The detailed story of how to distinguish the person-

counterpart relation from other sortal counterpart relations may well vary among

different stage theorists, but the story typically will involve many of the considerations

that the worm theorist invokes in telling her story about identity over time for a

particular kind of object (e.g. causation, spatiotemporal continuity, psychological

continuity, etc.). Moreover, the stage theorist is free to accept that worms exist. For

example, she might hold that the mereological sum of Obama’s person-counterpart

related stages is a worm that is very much like the worm that the worm theorist

identifies with Obama. The key difference is that the stage theorist does not identify

this worm with Obama. More perspicuously, the stage theorist denies that our token

utterances of ‘Obama’ refer to the relevant worm.

As it is typically construed, and as I have introduced it above, stage theory

entails that there are instantaneous temporal parts: objects with zero-measure temporal

existences. The worm theory, by contrast, typically is not construed as carrying this

entailment.109 Whether a given theory of persistence is committed to instantaneous

temporal parts is significant because of the following worry (Stuchlik 2003). Virtually

109 This difference between stage and worm theory is not always kept clear. For example, Lowe and McCall (2006) sometimes speak as though four-dimensionalism simpliciter requires instantaneous objects.

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every persistence theorist takes her favored view to be necessarily true if true.110 Now,

it seems possible that time is gunky—that every unit of time has a proper sub-unit.111

But if time is gunky then there are no instantaneous temporal parts. So it seems that

there are worlds in which stage theory cannot be the correct theory of persistence, if it

in fact entails that there are instantaneous temporal parts. So far, then, there is prima

facie reason to favor worm theory over stage theory.

However, there are important arguments, due primarily to Ted Sider, for the

stage view over its worm competitor. Sider (1996, 2001) argues that the stage theory,

unlike the worm theory, has the power to solve the familiar puzzles of material

constitution and fission without doing violence to our pre-theoretic intuitions about

how to count material objects. Consider, for example, the standard lump and statue

case. A lump of clay is sculpted into a statue that sits in a studio for a while before the

unsatisfied artist decides to squash it. Intuitively, the lump is not identical to the statue,

for they differ with respect to certain modal properties (e.g. the property of possibly

surviving squashing), yet there is a temporal interval during which the two seem to co-

exist and to exactly co-occupy a single spatial region. The worm theorist (continuing

to assume that the lump and statue are distinct) seems forced to accept (temporary)

spatiotemporal co-occupation by distinct physical objects—he seems forced, that is, to

accept that two objects are both entirely in the very same spatial region at a particular 110 There are exceptions. David Lewis, for example, is a four-dimensionalist who allows that, in the outer sphere of modal space, there are worlds where objects persist in accord with three-dimensionalism. 111 (Stuchlik 2003) describes temporal gunk as being such that each interval has a sub-interval. I prefer to describe temporal gunk as pertaining to temporal units rather than intervals since, strictly speaking, the claim that every interval has a sub-interval is consistent with the claim that time is pointy. It should be emphasized that I have a robust metaphysical notion of unit in mind, not merely a measure theoretic notion. On my robust metaphysical notion, a unit of time is more like a heap of sand than it is like the number of grains in the heap. Thanks to Jesse Alama for helpful correspondence on this issue.

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time. The stage theorist, by contrast, will maintain that at most one object, namely a

certain stage, occupies the relevant spatial region at any given time during the relevant

interval. The stages in question are statue-counterpart related to one another during the

interval in question, and lump-counterpart related to certain other earlier (pre-

sculpting) and later (post-squashing) stages. The stage theorist thus does not have to

accept the counterintuitive claim that distinct material objects might co-occupy a

single spatiotemporal region at a given time.112

Similar remarks apply to fission cases. Suppose Mike enters a fission chamber

at time t, whereupon he splits into spatially distinct material objects Mike1 and Mike2.

Before t, Mike1 and Mike2 have temporal parts in the very same place at the very same

time, say, time t - 1; yet Mike1 and Mike2 are distinct objects, so if ‘Mike1’ names a

worm and ‘Mike2’ does as well, then two distinct material objects are in the same

place at t minus 1. On Stage theory, the temporal part of Mike1 at t - 1 is numerically

identical to the temporal part of Mike2 at t - 1; so there is no conflict with our

intuitions about counting material objects (though there is, prima facie, a conflict with

our intuition that ‘Mike1’ utterances refer to something that lasts longer than an

instant).

In the next four sections, I show how glop theory provides a background

metaphysic for both a version of stage theory that is compossible with gunky time and

a version of worm theory that accords with our intuitions about how to count material

objects within a region at a time. Our intuitions about reference will not be as neatly

112 As is well known, some philosophers (e.g. David Wiggins (1968)) do not find the putatively counterintuitive claim sufficient for undermining the worm theory. I will not argue directly against those philosophers here, but will simply presuppose that (even temporary) co-location of distinct physical objects is best avoided.

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accounted for. However, I will set that concern aside since it is not a problem unique

to gloppy four-dimensionalism.

3. Gloppy Persistence

According to glop theory, the qualitative facts at any given time are fixed by the

spatial distribution of markedness, a non-qualitative fundamental property the tropes

of which mark certain locations from others. (Tropes are here taken to be concrete and

world-bound.) Since markedness locally grounds qualitative properties in this way, the

underlying metaphysical framework for markedness is called ‘glop theory’, where

‘glop’ functions as an acronym for grounding local ontological primitive. The term

also works as a mass noun for quantities of markedness: mark tropes just are quantities

of glop. ‘Qualitative’ properties are here taken to be all of the familiar properties of

science and everyday life, save for spatiotemporal properties like size, shape, and

duration.

Glop theory furnishes interesting new theories of property exemplification and

ontological fundamentality. The former is a version of trope bundle theory, the latter a

version of the general view according to which the world is ontologically built up

from basic local property exemplifications. Mark tropes serve to bundle qualitative

tropes into objects in the following way. If some qualitative tropes are in the same

location as a mark trope, then they are in the same bundle. The tropes that play this

important role are maximally connected mark tropes called ‘bits’, where a mark trope

is maximally connected just in case it is not scattered and not in contact with any other

mark trope. Bits also serve as the fundamental ontological base across worlds

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inasmuch as the distribution of bits at a given world fixes all of the qualitative facts at

that world. The distribution of markedness into bits whereby some locations contain

bits and some do not thus serves as a sort of binary code of fundamental ontology.

This background sketch is intended to bring the reader up to speed on the basic

contours of glop theory, even if she has not read Chapters 4 and 5. But there remains

an important question for glop theory, namely, what factors guide change in the

distribution of markedness across time? The answer to this question parallels the

answers to the puzzles of property exemplification and fundamentality: the most basic

metaphysical mechanism is the spatiotemporal distribution of markedness. More

specifically, the spatial distribution of markedness at arbitrary world w and time t

obtains “in virtue of” the immediately prior (distinct) spatial arrangement of

markedness having been so arranged for the length of time that it was so arranged.

The relevant ‘in virtue of’ relation at play is world-bound or ‘nomological’

supervenience, though this need not entail any robust realism about laws. It is quite

compatible, for example, with neo-Humean regularity theories. In any given world,

markedness is arranged across spacetime in such a way that asymmetric, world-bound

supervenience relations obtain between earlier and later temporally continuous (or

contiguous, if time is discrete), distinct spatial arrangements of markedness. The idea

that changeless intervals might bear metaphysical weight is borrowed from Sydney

Shoemaker (1969), who suggests merely that it is possible for there to be some type of

‘in virtue of’ relation that obtains between two states of affairs and which turns at least

in part on the mere length of time that one of the states of affairs obtains. Taking a

much stronger position than Shoemaker, the glop theorist rests his account of change

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on this possibility: the duration of a spatially static distribution of glop is responsible

for there being a “change” whereby a distinct spatially static distribution of glop

follows. Admittedly, this is a radical proposal. It should not be surprising, however,

for those who have taken glop theory seriously up to this point. After all, in the case of

synchronic property exemplification, it is the spatial arrangements of bits (i.e. their

sizes/shapes/durations) that is responsible for bundling tropes into objects; and in the

case of ultimate supervenience fundamentality, it is the spatiotemporal distribution of

markedness that works as the fundamental ontological base.113 In the diachronic case,

then, it is the duration of global, spatially static distributions of markedness that does

the primary metaphysical work. Let us turn to the details.

The spatiotemporal distribution of mark tropes over a maximal interval of

static global spatial arrangement is called a ‘History’. The spatiotemporal distribution

of mark tropes during sub intervals of History intervals are called ‘sub-Histories’.

Consider a static global spatial distribution of markedness (i.e. glop), g. Suppose g

begins at some time t1 and ends at distinct time t3. For all that has been said so far, it is

unclear whether g is a History or a sub-History. In order for it to be a History, it must

be the case that the intervals immediately preceding t1 and immediately following t3

are each different, with respect to spatial distribution of glop, from the interval t1-t3,

for histories are maximal intervals of static global spatial distribution. For example,

suppose that g is a History. The global spatiotemporal distribution during the interval

t1-t2 would then be a sub-History of g.

113 For details about the identification of size, shape, and duration tropes, see Chapter 4, section 3.

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Consider without loss of generality the bits of a given History h in arbitrary

world w. Call this a ‘bit-History’. Bit-Histories are individuated in part by bits, which

are world-bound tropes. So each bit-History “necessarily” will be local for a particular

class of qualitative properties, in the sense that it will be local for just those properties

in every world in which it exists. Call this class that bit-History’s ‘qualitative profile’.

Glop theory allows that two bit-Histories that are identical with respect to the

spatiotemporal distribution of markedness might differ with respect to the qualitative

profiles for which they are local, though only if the two bit-Histories in question are

not world-mates. Bit-Histories that are identical with respect to both their distribution

of markedness (which, notice, involves duration) and their qualitative profiles are of

what I will call the same ‘Career type’.

The central thesis of gloppy persistence is then that later Career types will (in

many worlds asymmetrically) nomologically supervene on immediately earlier Career

types:

(Careers Subvene) For any two pair of temporally continuous/contiguous career

types Cbefore/Cafter and C'before/C'after at arbitrary world w, there can be neither a

qualitative difference nor a difference in the spatiotemporal arrangement of

markedness between Cafter and C'after unless there is a difference in the spatiotemporal

arrangement of markedness between Cbefore and C'before.

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(Careers Subvene) is thus the guiding dynamic principle of glop theory. I will close

this section by making two additional details of the view explicit and addressing two

potential worries.

First, bits cannot survive for more than one History. Just as bits are spatially

maximally connected, so they are temporally maximally persistent for just one

History. Notice how strict this take on bit duration is. It entails that any change in the

global spatial distribution of glop from time t1 to t2 marks the end of the duration of

every bit in the distribution at t1 and the beginning of the duration of every bit at t2,

even if some t1 bits exactly spatially resemble and are in exactly the same location as

some t2 bits.114 These facts about bit duration also bear on qualitative tropes. Since bits

are metaphysically prior to qualitative tropes, it cannot be the case for any bit b and

qualitative trope Q that if Q is local to b then Q can out-persist b, on pain of violating

priority. That is, since bits only last for one career, qualitative tropes only last for one

career. This take on trope duration is in line with tropes being singly spatially located

and world-bound.

Second, not all Histories are instantaneous. Static global spatial distributions of

glop may be static for more than an instant. Indeed, differences among interval lengths

of Histories are of central importance to gloppy persistence. We will also see that

allowing for non-instantaneous Histories is important. I will close this section by

considering two potential worries.

114 I mention above (and discuss in detail in Chapter 4) that spatial tropes (size and shape) are numerically identical to duration tropes. However, this numerical identification for tropes does not entail numerical identification of properties. A t1-t2 bit might be exactly the same size and shape as a t2-t4 bit, even though the former is of a lesser duration. In this case, the latter trope and the former both are members of one size property and one shape property, but two different duration properties. Still, it is the case that there are only the two tropes, as opposed to there being six.

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To see the first worry, notice that, as stated, (Careers Subvene) rules out cases

of what we might call ‘varied on/off’ career alternation. These are cases in which one

career type follows another, then the first repeats, then a third type follows, then the

first repeats again, etc. Suppose that such varied on/off alternation occurred: some

career token of type Ca is directly followed by a token of a distinct career type Cb,

which in turn is directly followed by a second Ca token, which in turn is directly

followed by a Cc token, which in turn is directly followed by a third Ca token, which in

turn is directly followed by a Cd token (d≠c≠b≠a), etc. In this case, we have certain

pairs of continuous careers, e.g. Ca/Cb and Ca/Cc, which run counter to (Careers

Subvene), for the Cb-type career and the Cc-type career differ in their markedness

distributions while the two Ca-type careers do not. This preclusion of varied on/off

patterns gives rise to the first worry. After all, surely such patterns among qualitative

properties are possible. There are worlds with intervals during which all that changes,

say, is the color of a particular sphere from red to green to red to blue. So it would

seem to be a serious problem for a putatively modally general metaphysic if it cannot

allow for them. Fortunately, the problem does not arise for gloppy persistence. What

gloppy persistence precludes is varied on/off patterning for glop distribution, not for

qualitative distributions. This difference is crucial. It is up to the glop theorist to say

how markedness can and cannot be distributed. What the glop theorist ought not

modally constrain is the qualitative arrangement of the world over time.

The second worry arises from the fact that glop theory does not require that

Histories be instantaneous. Not requiring this allows for the possibility that there are

non-zero periods of time during which glop distribution is static, which entails the

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further possibility that glop distribution does not change across time continuously. The

worry is then that glop theory conflicts with the possibility of continuous changes in

the distribution of qualitative properties.115 Once again, however, the putative worry is

not a problem for glop theory. That the distribution of markedness is static over some

duration does not entail that the qualitative distribution is static over that duration. Just

as a given bit may be local to more than one qualitative trope at a time, so it may be

local to a variety of tropes across its temporal existence. Indeed, it may be local to a

variety of tropes of some highly specific kind across its temporal existence. For

example, a single bit b, located at region r during Career c, may be local to a series of

exactly resembling mass tropes M1-Mn, such that only one Mi exists during each c/n-

length sub-interval of c, and such that each Mi exists in a different spatial sub-region

of r. Intuitively, this is a case of a mass trope “moving” around within the local range

of some one bit. Given this flexibility, it is easy to account for possibilities like

continuous qualitative change and continuous motion.

4. Further Worries and Responses

In this section I anticipate and address some important questions and further worries.

1. What is the supervenience at work in (Careers Subvene) really doing other than

fitting a convenient pattern? How does it explain anything? Why should one believe

that the relevant patterns obtain?

Response: Suppose physical theory T (particle physics, say) holds that property F (a

quark flavor, say) is in the minimal supervenience base at a world w. Is it a worthwhile 115 Recall from previous chapters the important distinction between qualitative and non-qualitative properties.

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question to ask the proponent of T why F plays this role at w? I’m not convinced that it

is. T will be attractive just to the extent that F’s playing the role in question explains

other phenomena, solves puzzles, unifies, etc. Similarly, the supervenience patterns

upon which gloppy persistence is founded possess important explanatory value. They

(unsurprisingly) do not explain themselves; but they play a role in explaining property

exemplification, vindicating ultimate supervenience fundamentality, and—I will

argue—they furnish an attractive theory of persistence.

2. Aren’t changeless intervals inconsistent with classical physics, e.g., with constant

velocity? First, it is not clear that consistency with classically understood constant

velocity ought to be a success condition for a contemporary metaphysical theory.

Some contemporary scientists and philosophers hold that motion is discrete,

particularly at the micro levels at which glop theory is pertinent in the actual world.

Second, and more to the point of the objection, gloppy persistence is not inconsistent

with continuous motion of the objects that concern the classical physicist (or even the

objects that concern the non-classical physicist). What it is inconsistent with is

continuous motion of mark tropes. But this latter inconsistency is no bar to constant

velocity. As noted above, there could be a bit type that remains motionless for an

interval I even as the tropes for which it is local vary continuously across the sub-

intervals of I. Continuous qualitative variation during I-length intervals might simply

be part of the supervening qualitative profile that makes the bit type in question the bit

type that it is. In this sense, one can recover continuous variation even while there is

no bit-type relevant change during I.

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3. The global aspect of careers and (Careers Subvene)—insofar as it entails absolute

simultaneity— is inconsistent with non-classical physics, e.g., with special relativity.

Response: The presupposition of this objection is false: gloppy persistence does not

entail absolute simultaneity. Gloppy persistence is a theory about spacetime, not

(mere) time. Histories just are markedness distributions at certain cross-sections of

atemporal R4 along the time axis. This understanding of Histories is consistent with

various observers disagreeing about whether various events are simultaneous. Also,

(Careers Subvene) can be modified without detriment to account for probabilistic

worlds. Gloppy persistence need not presuppose determinism, though that

presupposition makes for clear introductory exposition.

4. What about the possibility of “initial Histories,” that is, states of the world that

were not preceded by any other states? Does glop theory accept initial Histories, or

does it rule them out?

Response: Glop theory neither requires nor precludes initial Histories. This is as it

should be. There are worlds with initial Histories as well as worlds that lack them. In

the former worlds, it is just a primitive fact that the initial History in question is initial.

To whatever extent this is puzzling or worrisome, it does not seem to uniquely affect

glop theory.

5. What about intervals that are too short to be History-length but that come one after

another as proper sub-Histories? What explains why these sub-Histories are the way

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they are? It can’t be nomological supervenience, because some of these sub-Histories

will be the same length as one whose endpoint coincides with the endpoint of the

whole History. Response: These short, sub-histories are like undetached proper parts

of bits, which, recall, are not metaphysically prior to the relevant whole mark tropes,

but rather come primitively arranged. The world comes wholesale arranged into

maximally connected mark tropes; likewise, it comes wholesale arranged across time

into Histories. Sub-Histories are not metaphysically prior to whole Histories. By

contrast, careers are metaphysically prior to the multi-career intervals that they

subvene. This middle road approach mirrors the glop theorist’s treatment of the

possibility of gunk as discussed in the preceding chapter.

6. Does gloppy persistence presuppose a block universe view of time? Is glop theory

inconsistent with a growing block view of time?

Response: I like the non-growing block view, and I agree that it seems prima facie to

be the most congenial view of “becoming” for glop theory. However, general glop

theory can be made consistent with a growing block view. To see how, we must be

clear about the sense in which markedness arrangement comes “wholesale”. Again,

the correct understanding of this notion is that according to which undetached proper

parts of mark tropes are not metaphysically prior to—not metaphysically more

fundamental than—maximally connected mark tropes. One incorrect understanding of

“wholesale” is that according to which all of the markedness for a given world comes

at one “time.” This understanding is in fact incoherent for all worlds in which change

in markedness distribution across time occurs. What glop theory is inconsistent with is

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a growing block view whereby the future is wholly unconstrained by the past. But,

since the preceding is not the most attractive version of the growing block view, there

is no deep rift between glop theory and the growing block view in general. Any given

History comes primitively into existence as a whole; but not all Histories come into

existence at once.

7. What about intervals where there is no glop at all? Are these possible? If not, why

not?—they sure seem to be possible. If so, how do they fit in with gloppy persistence?

Response: Gloppy persistence allows for non-gloppy intervals, but only under certain

constraints. We should think of a non-gloppy interval as a sort of null career. On this

way of looking at things, any non-gloppy interval—qua (null) career—will have to

accord with (Careers Subvene). This means that a non-gloppy interval of duration D

will always have to follow some one kind of career and will always have to be

followed by some one kind of career. Again, this all entails some modal constraints on

glop arrangement, but I do not see that it entails any significant modal constraint on

qualitative property arrangements.

8. Is Glop theory consistent with presentism?

Response: I am not entirely sure what to say in response to this question because it is

not clear to me whether presentism is consistent with the claim that gunky time is

possible. I think something rather like a form of presentism can be had in the glop

theoretic framework, however. What I have in mind is a version of glop theory

according to which “past” and “future” careers do not exist. I have not worked out the

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full commitments of this kind of view and so I am not entirely sure that it can even be

made coherent; but if it can then it would probably be the closest that the glop theorist

can get to a version of presentism.

9. Plausibly, action at a distance is metaphysically possible. What does glop theory

say about this possibility?

Response: Gloppy persistence is consistent with but does not entail action at a

distance. It is thus neutral with respect to certain important debates within the

philosophy of physics.

5. Recovering Ordinary Objects: Adopting Casullo’s “Double Bundle” Strategy

In many worlds—and presumably all nearby worlds—bits will be extremely tiny and

bit types may well have tokens of sufficiently close proximity to one another as to

seem as though some bit of the relevant type is “moving” extremely quickly. In such

worlds, careers will be very short. However, the temporal juxtaposition of many

careers “gives rise to” the ordinary qualitative world that we experience as time

passes. I have argued elsewhere that trope bundle theories are superior to their chief

realist competitors, bare particular theories and universal bundle theories (see Chapters

1, 3, and 4). In this section I apply the bundle theoretic approach to the problem of

change across time and explain how it can recover our intuitions about the ordinary

qualitative world.

The version of bundle theory that I favor and which furnishes a plausible view

of change for ordinary objects is based in part on a similar theory outlined in (Casullo

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1988).116 A given ordinary object at a given time t is a mereological sum, or bundle, of

certain appropriately related intra-career tropes that exist at t, where two items are

intra-career just in case they exist during the same career. Call each intra-career

bundle of this kind a ‘B’. The tropes in the bundle in question will meet the necessary

and jointly sufficient conditions for being an object of the relevant sort at some

time.117 The appropriate relations between the relevant qualitative tropes as well as the

relations that constitute and determine the relevant bit arrangement will largely be

spatiotemporal, but may involve other factors as well. Our intuitions about people,

trees, and other paradigmatically ordinary objects are then recovered as follows. Items

that have agency or lose leaves in the autumn, etc.—that is, intuitively ordinary

objects that persist through qualitative change—are accounted for by facts about

mereological sums, each called a ‘BB’, of appropriately related inter-career Bs, where

two items are inter-career just in case they exist at disjoint careers. Again the

appropriate relations will likely involve spatiotemporal considerations (such as

continuity/contiguity), but may well further involve psychological continuity,

causation, etc. So just as familiar qualitative macro-properties metaphorically “appear”

when we “zoom out” from the level of some isolated bit to a level of many bits, so

qualitative change “appears” when we “zoom out” from the level of some isolated

career to the temporal level of many continuous/contiguous careers.

116 Aspects of Casullo’s system have predecessors in Bergmann (1967) and Castaneda (1977). 117 It is not incumbant upon the glop theorist to be able to articulate, for some ordinary object of sortal S, the necessary and jointly sufficient conditions for being an S. Nor am I claiming that we can state or that we attend to these conditions when employing sortal terms and concepts, or that there are no vague cases.

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The proposal is thus a two part bundling system. The first part’s bundles (the

Bs) involve qualitative tropes for which intra-career bits are local; the second part’s

(the BBs) are inter-career bundles of the first part’s bundles. This system is attractive

because it circumvents many worries for bundle theory that have been raised in the

literature.118 For example, it allows for a bundle theoretic treatment of ordinary objects

that can account for the intuition that ordinary objects persist through qualitative

change. This intuition is underwritten by facts about qualitatively variant inter-career

B’s. Since it is possible for distinct inter-career B’s to exemplify different properties,

and since BBs are just bundles of inter-career Bs, it will be the case that different

properties and different proper parts are had by a given BB bundle at different times.

BB bundles, then, are the key to recovering our pre-theoretic intuitions about

“ordinary objects.” As we will see shortly, however, the theory’s ability to recover our

intuitions about ordinary object persistence is quite compatible with the claim that

ordinary object terms pick out unchangeable temporal objects. It is thus open to the

benefits that endorsing this claim brings, for example, the benefit of not having to

maintain that distinct material objects can occupy exactly the same place at a given

time.

6. Stages and Worms

It was advertised at the outset that gloppy persistence can furnish both (i) a version of

stage theory that does not require that all stages be instantaneous and (ii) a version of

worm theory that can address the constitution and fission puzzles without entailing 118 See Van Cleve (1985) for a good discussion of (and one example of!) failed attempts to refute bundle theory.

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that two material objects might be exactly co-located at a given time. In both the stage

and worm cases, the version that gloppy persistence furnishes will be rather different

from the traditional theory. The differences are to be expected, however, given that the

traditional versions do not have the attractive features just mentioned.

Let us consider gloppy worm theory first. To get a version of worm theory, we

simply stipulate that bits—which persist for the duration of some career—are worms.

On this view, bits persist throughout a given career by having appropriately related

temporal parts; the bits just are the mereological sums of the parts in question. This

version of worm theory strays importantly from traditional worm theory in the

following respect. The present version, unlike the traditional version, does not hold

that people, trees, planets, etc. are worms that persist through change. Rather, people,

trees and the like are identified with certain bundles of appropriately arranged tropes

during some single career (that is, people, trees and the like are Bs in the sense

introduced in the last section). In this respect, the gloppy version of worm theory is

very much like stage theory; both hold that ordinary object terms pick out objects that

have very short lifespans. Notice, however, that it is just this feature that allows the

gloppy version of worm theory to circumvent the puzzles of constitution and fission

without taking recourse to intuitively implausible counting schemes. Since the statue

can only be squashed if one career has given rise to a distinct career, and since change

in career entails that an entirely new batch of tropes replaces an earlier batch (‘entirely

new’ in the sense of numerical distinctness), the tropes whose mereological sum just is

the lump/statue before the squashing are a numerically different worm than the tropes

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whose sum just is the lump after the squashing. At any one time, there is only ever one

object—the lump/statue—at the relevant region.

Why is this one object appropriately described as a “lump/statue”? Much like

the stage theorist, the gloppy worm theorist holds that career-sized worms (the Bs) are

sortal-related to other career-sized worms (thus forming BBs). The sortal relation in

question is not necessarily a counterpart relation, but its conditions for being

exemplified are much the same as those of the stage theorist’s counterpart relation.

Again, these conditions, depending on the object in question, will likely involve

psychological continuity, spatiotemporal continuity, etc.—but, importantly, they will

not involve numerical identity. While the proper temporal parts of (career-sized)

worms are parts of some one worm, it is not the case that numerous career-sized

worms are statue-related so as to jointly compose some one statue. Rather, statues just

are career-sized worms. So it is correct to call our one object—the relevant single-

career worm—both a ‘lump’ and a ‘statue’ because it is lump-related to certain

numerically distinct single-career worms and statue-related to others. As in the case of

stage theory, the gloppy worm theorist can allow that there are mereological sums of

single-career worms; what he does not allow is that these are the referents of our

singular terms.

One might worry at this point that one of the crucial claims made above,

namely, that the statue can only be squashed if one career has given rise to a distinct

career, does not square with the liberality of gloppy persistence with respect to intra-

bit qualitative variation. After all, glop theory allows for qualitative change during a

single career, so why couldn’t there be a single career during which a certain statue

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went through the qualitative change of being deformed? To rebut this worry we must

recognize that glop theory is not best construed as allowing that just any qualitative

change can occur during a single career. It must allow some liberality in order to be

consistent with certain phenomena, for example, continuous motion of fundamental

particles, but it need not be construed as allowing for comparatively highly complex

changes such as a clay statue being squashed. It is important to notice that the puzzles

in question concern only these sorts of highly complex objects. The statues in the

puzzles are never understood as being such that they would be destroyed were the

location of the fundamental particles that compose them to change slightly. I am

comfortable assuming that if the puzzles were restructured so as to require double-

counting on the part of the gloppy worm theorist, then they would no longer be

compelling puzzles.

Let us move on then to gloppy stage theory. To get a version of stage theory,

we hold that bits are stages instead of worms. This sounds simple enough, but prima

facie there is a serious problem with going this route. Recall that bits exist for exactly

one career and that careers are not necessarily instantaneous, lest the gloppy stage

theorist fail to account for temporal gunk. The prima facie problem, then, concerns

how to account for criteria of identity over time for bits if bits are understood as

stages. After all, the stage theorist’s account of “identity” over time is given in terms

of relations among stages. So it seems as though there is nothing left with which to

give a diachronic account of stages themselves.119

119 Stuchlik (2003) takes this challenge to be an insurmountable obstacle for stage theory. I will try to show that such an assessment is too quick.

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This challenge can be met, however, by introducing a temporal-cum-

mereological hierarchy for careers. Base level careers we already have; they are just

what we have until this point been calling ‘careers’: worldwide static arrangements of

mark tropes that last for some entire duration D, where D is the maximal interval

during which the mark tropes in question are static. Let us use ‘C’ as a schematic

expression that ranges over base level careers. Consider now the two temporal parts of

some C that result from temporally bisecting it.120 Let us call these half-Cs ‘level 2

careers’ and use the schematic expression ‘C2’ to range over them. Repeating this

process produces higher levels of careers. Halves of a C2 will be quarter-Cs that we

will call ‘level 3 careers’ and use ‘C3’ to range over, etc.

So far we have considered worldwide arrangements of bits, but a similar

hierarchical model can be applied to individual bits. We might try to do this by

providing a mapping from a given career to its bits. The information that would

underlie such a mapping would be spatial information, for the temporal facts about

bits in a given career are insufficient to distinguish one bit from another. However,

since glop theory does not hold that bits are prior to arrangements or that arrangements

are prior to bits, there is no simple function that will give us the mapping we need.

Instead of a function, however, we can simply use an index to pick out a given bit

within a given career. For a given career C, let us use ‘C(b)’ to pick out some bit b that

is part of C. This notation allows us to straightforwardly apply the hierarchical model

to individual bits as follows. For a given higher level career (or ‘career-part’ as we

might call it) Cn of some base level career C, let us use ‘Cn(b)’ to pick out the temporal 120 I am happy to allow arbitrary chopping up of careers; I focus on bisecting for expository convenience.

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part b′ of some bit b such that b exists during C and b′ exists during all of and only Cn.

The “criteria of identity over time” for a given bit b are then just the obtaining of a

C(b)-counterpart relation among the C2(b)s that mark off the temporal parts that result

from temporally “bisecting” the bit in question. The “criteria of identity over time” for

C2(b)s are then just the obtaining of a C2(b)-counterpart relation among the C3(b)s, etc.

Again, it is easy to see how to carry on the process. In gunky time this is an infinite

system. Importantly, however, there is no vicious regress. Indeed, such an infinite

system seems to be precisely what is needed in order to account for the possibility of

gunky time.

It should be mentioned that in order to get this gloppy version of stage theory,

one has to allow for a modification of Sider’s original stage theory since his version

rules out the notion of a “stage-counterpart” relation. However, we should keep in

mind that—in spirit if not in letter—this modification is quite consistent with Sider’s

version of the theory. What Sider rules out is that there could be a sortal-indexed

counterpart relation that obtains among stages, where the relevant sortal concept is

stage-hood itself (as opposed, for example, to the sortal concept statue). But it is no

surprise that the modification the gloppy stage theorist endorses should be

implemented given that Sider only allows for one type of stage-interval (namely, an

instantaneous one) while the gloppy stage theorist allows for infinitely many.

Correspondingly, there is an important difference between Sider’s stages and the sort

of stages for which I am suggesting a stage-counterpart relation. Sider’s stages cannot

persist. Glop theoretic stages, like statues, can persist. This difference is important for

the following reason: since Sider’s stage theory is an account of persistence according

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to which, for some sortal S, S persistence occurs just in case certain stages are

appropriately S-counterpart related, the only way available to modify Sider’s theory to

account for persisting stages is for there to be a stage-counterpart relation.

Moreover, the C-type individuation of stages may well take into account not

just the quantitative aspects (duration) of the Cs in question, but also the qualitative

profiles of the bits that exist at the relevant C. This further detail is especially

important in the case of careers that have qualitative variation across time. The

hierarchy I introduced for careers and their parts is merely a matter of temporal

chopping up. But this does not entail that the criteria that determine whether a given

Cn-counterpart relation obtains must be limited exclusively to quantitative temporal

considerations. Such criteria may well take into account the qualitative profiles of the

relevant Cn(b)s.

Sider, in personal correspondence, has suggested that stage theory can account

for the possibility of gunky time if we conceive of stages as “shortish segments” of

worms. But either these shortish segments can persist through change (including

change in location) or they cannot. If they cannot, then Sider’s proposed modification

to stage theory takes the significant hit of no longer jibing with even the mere

possibility of continuous variation across time. But if Sider’s shortish segments can

persist through change, then the problem of how to say what it is for a stage to persist

remains, since “shortish segments” still persist—and it looks like the stage view just

collapses into a worm view. Sider can deploy a hierarchy of longer and shorter-lived

stages as I suggest, but he would then face the problem of how to give sortal-indexed

counterpart relations for perhaps wildly qualitatively variant “blips” that might occur

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over some extremely small interval. For example, suppose that a particle is in one

location and then appears in a distinct location for a very short time before reappearing

at the first location. Suppose further that while in the deviant location it is a different

color and slightly different shape. Is this deviant item to be subsumed under the same

sortal as the rest of the particle’s temporal parts? (For the expository purposes of the

example, I have applied the sortal ‘particle’ uniformly, but the question is whether this

way of speaking is correct.) What criteria could possibly decide this question? This

problem is not obviously intractable, but it is far from clear what the non-gloppy stage

theorist should say. The gloppy suggestion saves stage theory on both counts: it

provides a non-worm treatment of stage persistence, and it allows for continuous

motion and qualitative deviation since qualitative change can occur even while

markedness (and thus stage) arrangement is static; moreover, there is a built-in story to

tell about the counterpart relations even among wildly qualitatively variant intra-career

“blips”: they are all subsumed under one sortal just in case they are all local to one bit.

Having thus made good on the promise to develop two attractive versions of

four-dimensionalism from the glop-theoretic perspective, I want now to address a

potential objection, namely, that glop theoretic four-dimensionalism requires

commitment to immanent causation.

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7. Immanent Causation Avoided

Let us say that a ‘immanently’ causes b just in case (i) a causes b and (ii) a and b are

the numerically same continuant item (whether an object, event, state, etc.).121

Immanent causation is highly contentious. All else equal, it would be preferable for

one’s theory of persistence not to be committed to it. There is a potential objection to

gloppy persistence, then, that arises from the claim that it requires immanent

causation. In this section, I will consider two ways to motivate this claim and will

attempt to discharge both.

To see the first way to motivate the claim that gloppy persistence requires

immanent causation, recall that every bit is maximally connected and such that it

cannot survive changes to its size or shape. It also cannot survive for more than one

(base) career. Nonetheless, it can persist. The thought, then, is that if a given bit did

not immanently cause itself to persist, then we would have no way of addressing the

following dilemma.122 Suppose that, at time t1, a certain bit b1 exists. At the next

immediate moment, t2, an instantaneously acting annihilation machine annihilates b1;

simultaneously (i.e. still at moment t2), an instantaneously acting creation machine

creates a new bit b2 in exactly the spatial region that b1 occupied at t1. Suppose that b1

and b2 are local for just the same distributions of just the same qualitative properties.

The intuition that this sort of example is supposed to evoke is that b1 and b2—despite

being exactly alike qualitatively—cannot possibly be numerically identical and cannot

possibly be temporal parts of the numerically same bit. After all, b1 was annihilated at

121 Here I am following the terminology of (Zimmerman 1997). Zimmerman attributes this usage of ‘immanent’ to Lotze, Broad, and Johnson. 122 Worries of this general sort for non-causal theories of identity across time are found in (Morreall 1980) and (Swoyer 1984).

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the moment at which b2 was created, so b1 and b2 must be distinct; and, given that b1 is

distinct from b2, the bit whose temporal parts would have been b1-at-t1 and b1-at-t2

cannot possibly be numerically identical to the bit whose temporal parts are b1-at-t1

and b2-at-t2. Still, b1 and b2 are spatiotemporally continuous and qualitatively identical.

What beyond spatiotemporal and qualitative continuity, then, is required in order for

some items to be temporal parts of the same bit? A natural answer is that the temporal

parts must be causally linked; and if they were so linked then the kind of causation at

work would have to be immanent causation. The dilemma, then, is that unless one

accepts immanent causation as the ground for explaining why spatiotemporally

continuous, qualitatively identical temporal parts are parts of some one bit, one has no

explanation for why b1 and b2 are not temporal parts of some one bit.

Gloppy persistence is equipped to treat the dilemma without requiring

immanent causation. On glop theory, annihilation and creation machines—like any

other material object—must be bundles of tropes that are local to bits of glop. They

cannot operate without there being some change in the global glop arrangement. But a

change in the global glop arrangement entails the “creation” of a new base career.

Therefore, given the story as told, t1 and t2 are not included in the same base career.

So, according to glop theory, b1 and b2 cannot be temporal parts of some one bit.

One might suggest in response that annihilation and creation machines can

conceivably operate without resulting in a change in the global distribution of glop.

Indeed, instead of machines, we might consider immaterial deities who have the

powers of instantaneous material annihilation and creation. Two responses to this

suggestion come to mind. First, glop theory is naturalistic in the sense that it is not

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concerned with extra-spatiotemporal entities. The mere conceivability of the

aforementioned deities might thus be understood by some as being a threat to the

modal generality of glop theory, but it is no more of a threat than is the mere

conceivability of universals, enduring objects, extended simples, bare particulars, or

any other items that glop theory rejects. Second, it is far from clear that the sort of

annihilation/creation thought experiment at hand can get off the ground if geared

toward glop theory because of glop theory’s world-bound treatment of bits. Crucial to

the thought experiment is the spatiotemporal continuity/contiguity of b1 and b2. But

the typical gloss of what it is for b1 to be annihilated is counterfactual: in nearby

worlds where it is not the case that the annihilation mechanism (whether material or

not) is implemented at t2, b1 exists at t2. But this counterfactual is false according to

glop theory. In order for it to be true, b1 would have to have existed at some world at

t2. But b1 only exists at the one world, wherein it does not exist at t2. So even if one

invokes deities as the preferred harbingers of annihilation/creation, the

annihilation/creation thought experiment does not go through without begging the

question against glop theory.

The second worry about commitment to immanent causation involves the

rotating homogeneous sphere (or disc) cases due to Armstrong (1980), Kripke

(unpublished lectures), Hawley (1999), and Zimmerman (1998). The gist of this

second worry is that, unless the glop theorist takes recourse to immanent causation, he

seems to have no way to account for the intuitive difference between a stationary

spherical bit and a rotating spherical bit. Pace Hawley and Zimmerman, I think that

the right move in response to the general rotating sphere thought experiment is to deny

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that it is possible for there to be a difference, in the way that is usually assumed (i.e.

that exactly one is ‘rotating’), between the two spheres, and I will defend that general

response below.123 What is most important for present purposes, however, is to notice

that even if we set aside the ‘no-difference’ response as it applies in the general case

where the rotating spheres are material objects, what is really at issue is whether there

can be rotating homogeneous spheres of markedness, and here the answer is simply

‘no’. Even if one holds dearly the intuition that a solid homogeneous sphere could

rotate, one presumably has no intuitions as to whether a spherical mark trope could

rotate. The fact of the matter is that it cannot: in order for it to rotate, the mark tropes

that are its undetached proper parts would have to change location; but no mark tropes

can survive changes in location. Still, for the sake of being thorough and for the sake

of independent interest, it will be worthwhile to discuss where the general rotating

sphere argument goes off course.

Those who maintain that two homogeneous spheres can be such that exactly

one is rotating at some non-zero velocity have, I want to contend, been too liberal in

their understanding of homogeneity. Toward seeing this, let us compare the sphere

case with Mark Scala’s (2002) star-shaped homogeneous object at time t. Suppose that

it rotates clockwise during an interval I that lasts from t to tn. In order to parallel the

sphere case, I has to be non-zero. If it were not, then instantaneous velocity would

123 Zimmerman rejects this sort of move as arising only from bad philosophical faith. But, as a matter of contingent fact, I never had the intuition that the putative possibility of the rotating homogeneous sphere was genuine. I never thought of the putative possibility until after spending a lot of time thinking about the philosophy of persistence. So I am not “giving up” the putative intuition just to save my favored view. It might be the case that my favored view and antecedent thoughts on persistence have affected whether or not I happen to have the putative intuition, but that is not to reject the intuition out of bad faith.

196

have to be built into the example, which would constitute an intrinsic qualitative

difference-maker between rotating and non-rotating objects. Let us suppose, then, that

I is 10 seconds. Let us consider just some early fraction of the spinning process and

just some proper part of the star, say its northernmost triangular tip that has some non-

zero volume (suppose the star is 10 inches long and the tip in question has a “height”

of one-quarter inch). Call this proper part of the star at t ‘Tip’. Now, the three-

dimensionalist will say that the star persists through rotation by having its proper parts

such as Tip persist through various changes in location. For example, Tip occupies a

different region (let us presuppose substantivalism for ease of exposition) at t than it

does one second after t. The three-dimensionalist can give this treatment of the

rotating precisely because she maintains that Tip at t and Tip at t-plus-one-second are

numerically identical. The four-dimensionalist cannot say this, for he denies that Tip

exists at t-plus-one-second. Nevertheless, the four-dimensionalist can explain what is

going on when a star-shaped quantity of homogeneous “stuff” rotates: all that happens

is that various closely related star-shaped regions take turns being occupied by tropes

of the properties that characterize the stuff in question. Why do we consider all of

these various momentary tropes to be exemplified by temporal parts of the same star?

Well, that question is open to being answered quite differently by different four-

dimensionalists, but the answer will likely have something to do with the overlap of

the regions in question, as well as with their all being the same or close to the same

shape, etc. If the region R occupied by Tip at t is not such that, at the next moment

following t, some R-shaped region that nearly-exactly overlaps R but is slightly

“rotated” clockwise is, at this next moment, occupied by tropes of the properties that

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characterize the homogeneous stuff in question, then the four-dimensionalist and the

three-dimensionalist alike will be inclined to deny that the star is rotating in the

relevant sense. What distinguishes the four-dimensionalist and three-dimensionalist

views of the matter is that only the latter numerically identifies the occupant of R with

the occupant of its relevantly nearby region. Crucially, the four-dimensionalist does

not need to identify the relevant occupants in order to distinguish the rotating star from

a material object exactly qualitatively alike that is not rotating.

Now consider the two spheres, one ‘rotating’ and the other not. What do we

mean by the claim that the one sphere is rotating? Do we just mean that it is composed

of proper parts analogous to Tip (except, of course, that they would be wedges instead

of tips) that can survive changes in location? If so, then the question is being begged

against the four-dimensionalist who denies that rotating is possible for the sphere, for

he did not need such persisting parts in order to give his account of ‘rotating’ for the

sort of object (the star) that everyone agrees is capable of rotating. Does ‘rotating’

mean merely that the proper parts of the sphere at one time occupy regions such that

appropriately closely overlapping, appropriately similarly-shaped regions will, at

appropriately close moments, be occupied by tropes of the same kind that make up the

proper parts in question? If so then it follows from homogeneity that both spheres are

spinning, which contradicts the premise of the thought experiment.

I think that what has happened is this: when we try to imagine any sphere

rotating, we somehow index at least one proper part of the sphere and imagine it being

located in distinct regions across time. But it seems to me extremely difficult to

imagine the indexed proper part changing location unless we are somehow slipping in

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a qualitative means of distinguishing it from the other proper parts of the sphere,

which would violate homogeneity (and which can be mirrored in thought for the

stationary sphere by considering various regions at different times), or unless we are

assuming that it survives change in location, which would beg the question against the

four-dimensionalist.

Hawley (1999) rejects the “no-difference” response because, according to her,

it leads to an unacceptable indeterminacy with respect to the angular velocity of the

spheres:

If there is no fact of the matter about whether a given disc is rotating, then there is no fact of the matter about what would have happened if someone had touched the disc, or had splashed paint onto it. For each disc, it is true that if someone had measured the angular velocity of the disc, then she would have obtained some determinate result. But in neither case is there some determinate result that would have been obtained had someone measured the angular velocity of the disc. The result of any possible measurement of angular velocity is undetermined (1999).

Hawley saddles the proponent of the no-difference objection with the claim that there

is no fact of the matter as to whether the spheres (or discs) are rotating. In doing so,

she presupposes that there would be some difference between measurements of

rotating and stationary homogeneous spheres, and that there would be some difference

between touching (or splashing paint upon) a rotating sphere and touching a stationary

sphere. I think that this presupposition betrays a failure to respect the premise of

homogeneity. We have no idea what it would be like to touch a homogenous material

object, whether rotating or not; nor do we have any idea of how possibly to measure

the angular velocity of such objects, if they are spherical. Hawley mistakes the claim

(i) that touching or measuring both spheres would yield the same result for the claim

(ii) that there is no fact of the matter as to whether the spheres are rotating. One can

199

accept that it is a determinate matter whether the spheres are rotating without

accepting that measuring or touching them would yield a result that is at all like the

result one would get upon measuring or touching a rotating inhomogeneous sphere.

Hawley makes a further mistake in holding that that the no-difference proponent is

committed to the claim that there is no determinate result that would obtain upon

measurement. What the no-difference proponent really holds is that measurement of

the sort of quantity that concerns us in the case of rotating inhomogeneous objects

simply cannot be had in the case of homogeneous spheres. It is not that the feature in

question is indeterminate; it’s that it cannot be measured, or at least not in the way

Hawley has in mind, namely, by tracking a speck of dust that lands on one of the

spheres. If a speck of dust were to land on a truly homogeneous sphere, what reason

do we have for thinking that it could possibly “move” the way it would have had it

landed upon an inhomogeneous sphere? And if we begin by attaching the speck to one

of the spheres and then seeing whether it “moves,” then we are violating homogeneity

from the get-go.

In summary, if the four-dimensionalist could not account for the sense in

which the spinning star and the stationary star are different, then he would have a

major problem on his hands. That there even exists such a difference between the two

spheres is much less clear. And if there is really no difference, then the second

motivation for holding that gloppy persistence is committed to immanent causation is

uncompelling. In the next section, we will see a further reason why avoiding

commitment to immanent causation is beneficial.

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8. Prospects for Trope-Based Singular Causation

Douglass Ehring (1997) has developed a theory of causation that features trope

persistence as a singularist causal mechanism. Ehring’s brand of singular causation is

attractive for two reasons. First, like other singularist mechanism or ‘process’ theories

of causation, it is well equipped to deal with puzzles about preemption that continue to

plague non-singularist theories. Second, and most importantly, it gives an

ontologically economical account of the causal relation: at the fundamental level,

causation just is trope persistence. This economy comports very well with much of the

general motivation for trope theory since the proposed causal relata do not outstrip the

austere trope ontology. However, Ehring argues that in order for trope persistence to

underwrite a theory of singular causation, it must be understood within a three-

dimensionalist framework. The argument is that, since four-dimensionalist accounts of

trope persistence all presuppose that certain causal relations obtain between temporal

parts of tropes, any theory of causation that is based upon a four-dimensionalist

approach to trope persistence will be circular. I think that the premise of the argument

is false. There is a four-dimensionalist theory of causation that turns on trope

persistence and yet does not turn on the presupposition that causal relations obtain

among the temporal parts of the tropes in question. The theory I have in mind, of

course, is gloppy persistence. That gloppy persistence does not entail commitment to

immanent causation is thus significant not merely because immanent causation is

controversial, but also because a four-dimensionalist theory of trope persistence that

does not turn on causal relations between temporal parts can furnish a non-circular

201

trope-based theory of singular causation. I will not attempt to give such a theory in

detail here, but I will offer a quick sketch of how it might go.

To get from general gloppy persistence to a four-dimensionalist version of

trope-based singular causation, one needs simply to stipulate that the ‘in virtue of’

relation that obtains between earlier and later careers ala (Careers Subvene) is a

‘causal’ relation. This version is importantly different from Ehring’s in two respects.

First, it is compatible with four-dimensionalism. Second, while Ehring holds that the

singular causal relation is trope persistence and that the relata are individual tropes at

single times, the glop-theoretic version of singular causation holds that the singular

causal relation is temporal contiguity and that the relata are careers. Trope persistence

is still of central importance, however, and the trope ontology is not outstripped,

because careers just are worldwide distributions of certain persisting tropes, namely,

bits. Following Shoemaker’s idea, the gloppy causal theorist holds that the very length

of time that a career lasts is causally relevant.

As noted, the details of gloppy causation will have to be explored elsewhere,

but I want to close this section with an important observation about the dialectic

surrounding Ehring’s defense of the three-dimensionalist approach to trope-based

causation. That dialectic centrally involves causal preemption. The point I want to

emphasize is that glop-theoretic approaches to causation of the sort that I sketch above

are immune to worries about preemption. Before explaining the immunity, however, it

will be useful to discuss a reason that the friend of glop-theoretic-singular-causation

ought to care about preemption. Recall that the central thesis of gloppy persistence,

(Careers Subvene), features a robustly neo-Humean element inasmuch as it turns on

202

the existence of regularities among temporally contiguous career types. The reason the

friend of glop ought to care about preemption is that her opponent might well argue

(as Ehring does) that Neo-Humean theories of causation get the wrong results in

preemption cases. Here is the schematic argument. Suppose c1 causes e, preempting c2

from causing e. The neo-Humean story about c1’s causing e is regularity-based (in the

gloppy case, the story is that there are regularities among career types). The problem,

however, is that in the abstract preemption case described, the regularity that obtains

between c1 and e is matched by the regularity that obtains between c2 and e. So, the

anti-Humean concludes, the neo-Humean cannot pair c1 and e as causal relata without

also (incorrectly) pairing c2 and e as causal relata.

It is easy to see that the glop-theoretic theory of causation is immune to

worries about preemption, for it has a built-in version of the general response to

preemption worries according to which causal relata do not “survive” across

counterfactual cases.124 In the gloppy case, this response is built-in via the claim that

tropes are world-bound. Given that c1 causes e, it is simply not the case—as is

assumed in order to run preemption—that c2 would have caused e had c1 not

occurred. Some event similar to c2 would have caused some event similar to e, but

numerical identity would not hold. Ehring’s focus on preemption in arguing for his

124 More precisely, this general response applies to so-called ‘cutting’ cases of preemption, in which the preempting event temporally cuts off the preempted event before it can be causally efficacious. The distinction here is with so-called ‘trumping’ cases of preemption (Schaffer 2000), in which preempting and preempted events either occur simultaneously or take “effect” simultaneously. Whether the neo-Humean has a good response to trumping worries is a matter of live debate. I am inclined to think that there are good responses, but I will set the issue aside here.

203

three-dimensionalist brand of trope-based singular causation falls to the wayside once

glop theory is in the picture, for the possibility of preemption is simply ruled out.125

One final remark on the present response to worries about preemption is worth

making. One way of stating the glop-theoretic response to preemption is that glop

theory builds into its causal relata what David Lewis (2000), in responding to

preemption worries, calls ‘modal fragility’. While Lewis is concerned not to commit

himself to modal fragility for the reason of not wanting to flout folk semantics (recall

Ehring’s worry discussed in footnote 21 below) by having to deny that, say, the JFK

assassination could not possibly have occurred a day earlier, the glop theorist

circumvents this worry even while committing himself to modal fragility. There

simply are no folk semantic intuitions about bits or careers.

125 Ehring considers and quickly dismisses the general claim that every effect has its causal history essentially, of which the gloppy-singular-causation-theoretic claim that careers are world-bound causal relata can be understood as an instance. He contends that “…surely it is false that the Kennedy assassination would not have occurred had just one molecule moved differently in the distant causal past of Oswald’s efforts, but his actions were otherwise the same.” The glop theorist will reject this contention straightaway, for it is a consequence of glop theory that any possible world which differs from this one with respect to the movement of some one molecule will not be a world in which any event numerically identical to the Kennedy assassination occurs.

204

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