## 1. Preliminaries: Spacetime and Parthood



This article focuses on the recent literature on location and
mereology. On the history of these topics, see Marmodoro (2017), Harte
(2002), Sorabji (1983, 1988), Pasnau (2011), and Holden (2004), as
well as the entries on
 ancient atomism,
 medieval mereology,
 atomism from the 17th to the 20th century,
 and
 mereology.



In keeping with the recent literature, we will focus on
'entity-to-region' location relations--i.e., location
relations that paradigmatically hold between entities and
*regions*. We will ignore location relations that hold between
entities and nonregions. (Consider 'the transmitter is located
at the top of the ridge'. On its face, the top of the ridge is
not a region.)



Since our focus is on entity-to-region location relations, we will
work under the following controversial but popular assumption:



Substantivalism There are such
entities as regions.


Unless otherwise noted, we will assume that regions are *spacetime
regions*, where these are thought of in accordance with the
'Block Universe' picture, as captured by the
following:



Eternalism The past, present, and
future are all equally real.
The B-theory of Time No time is
present in any absolute, not-merely-indexical sense.
The Spacetime View There is just
one fundamental spatiotemporal arena: spacetime. Instants and
intervals of time, if there are such things, are just spacetime
regions of certain sorts. Likewise for points and regions of
space.


Relatively little depends upon the Block Universe picture, however. If
one dislikes talk of spacetime, one should be able reinterpret most of
what we say here in terms of regions of space without loss of
plausibility. Further, we assume that



Subregions as Parts If *x*
and *y* are both regions, then *x* is a subregion of
*y* if and only if *x* is a part of *y*. 


Subregions as Parts contrasts with another fairly common view
according to which regions are sets of points and have their
subregions as subsets but not as parts.



More generally, we will assume that there is just one fundamental
parthood relation, that this relation holds univocally both between
one region and another and between (say) one material object and
another, and that the given relation is a two-place relation.



All of these assumptions are controversial. Some philosophers have
explored the view that the fundamental parthood relation is
three-place (Thomson 1983; Hudson 2001; Balashov 2008; Donnelly 2010;
McDaniel 2014; Lafrance 2015) or four-place (Gilmore 2009;
Kleinschmidt 2011). Others have argued that one fundamental parthood
relation holds between regions, another between material objects
(McDaniel 2004, 2009). And some have argued that there are no
fundamental parthood relations (or other fundamental mereological
relations, such as overlapping). Of these philosophers, some propose
reductive definitions of parthood, whereas others eliminate parthood
(Dorr 2005; Sider 2013; Cowling 2014a), though the distinction between
reduction and elimination is sometimes blurry.



Nolan (2014) calls attention to a range of reductionist proposals,
which can be divided into three main strategies.



A first strategy, which we can call the *locationist* strategy,
is to define parthood in terms of the location relation together with
the subregion relation (Oppenheim and Putnam 1958; Markosian 2014).
For example, one might say that for *x* to be a part of *y*
is for *x*'s exact location to be a subregion of *y*'s exact
location. To avoid circularity, friends of the locationist strategy
should reject Subregions as Parts, in favor of either (i) the view
that regions are sets of points and *x* is a subregion of
*y* if and only if both are regions and *x* is a subset of
*y* or (ii) the view that subregionhood is primitive and
fundamental. An objection to (i) is that regions are concrete, whereas
sets are abstract, not concrete; an objection to (ii) is that positing
a fundamental relation of subregionhood seems very similar to, and
probably no more theoretically virtuous than, positing a fundamental
parthood relation (though see Nolan 2014: 105-111). There are
two main objections to the locationist strategy more generally.



First, there is the objection from the topic-neutrality of parthood.
The idea here is that entities need not be located in space or
spacetime to be parts or to have parts. For example, one might hold
that the word type 'cat' is a part of the sentence type
'the cat is on the mat', despite the fact that neither is
located. According to this objection, locational subregionhood is
unnecessary for parthood.



Second, there is the objection from interpenetration, according to
which an entity *x* is in some cases exactly located at a
subregion of the exact location of an entity *y* despite the fact
that *x* is not a part of *y*. For example, two bosons (or
tropes or...) might have the same exact location without being
parts of each other. (See Section 4 for further discussion.) If the
objection from interpenetration is correct, then locational
subregionhood is not sufficient for parthood



A more radical reduction of parthood to (among other things) a
location-like relation is developed in Bennett (2013), who proposes
that for *x* to be a part of *y* is for there to be an
entity *z* such that (i) *z* is a parthood-slot in *y*
and (ii) *x* occupies *z*. Bennett's views are discussed in
Fisher (2013), Cotnoir (2015), and Garbacz (2017).



A second strategy, which we can call the *monist* strategy, is
available to supersubstantivalists (those who hold that each material
object is identical to a region). They can say that what it is for an
entity *x* to be a part of an entity *y* is: (i) for
*x* and *y* both to be material objects, i.e., both to be
among those privileged regions that count as material objects, and
(ii) for *x* to be a subregion of *y*. (Wake 2011 and Nolan
2014 develop this strategy.) One virtue of the monist strategy is
that, unlike the locationist strategy, it does not need to invoke a
fundamental location relation. But it is vulnerable to versions of
many of the same objections, including those concerning the proper
treatment of subregionhood, the objection from topic-neutrality of
parthood, and the objection from interpenetration.



A third strategy, which we can call the *plural* strategy, is
to define parthood in terms of identity and the 'is one
of' notion from plural logic (on which see
 plural quantification),
 as follows: for *x* to be a part of *y* is for *x* to
be one of some entities *zz* that (collectively) are identical to
*y* (Bricker 2016: 270). But this strategy will be only as
attractive as *strong composition as identity*, the
controversial view that whenever some things *xx* compose a thing
*y*, *xx* are identical to *y*. (See Wallace (2011a, b)
and the essays in Cotnoir and Baxter (2014) for discussion and further
references.) (A hybrid of the first and third strategies is pursued by
McDaniel (2014: 21), who defines a three-place parthood relation in
terms of a three-place, region-relative identity relation and a
two-place subregionhood relation.)



For a defense of the view that parthood is fundamental, see Eddon
(2017).



We now turn to the logic of the parthood relation. (See the entry on
 mereology
 for much more detail.) Here we assume merely that parthood is
reflexive and transitive (a 'preorder'). In a first-order
language with identity and a primitive two-place predicate
'*P*(*x*,*y*)' for parthood, we can write:




Reflexivity of Parthood
[?]*x**P*(*x*,*x*)
  

*Each entity is a part of itself.*
Transitivity of Parthood
[?]*x*[?]*y*[?]*z*[[*P*(*x*,*y*)
& *P*(*y*,*z*)] - *P*(*x*,*z*)]
  

*If *x* is a part of *y* and *y* is a part of
*z*, then *x* is a part of *z*.*


We will also find it convenient to have predicates for overlapping,
disjointness, proper parthood, simplicity, complexity, gunkiness, and
mereological coincidence. We define them as follows:



(D1) 
Overlapping *O*(*x*,
*y*)
=*df* [?]*z*[*P*(*z*,
*x*) &
*P*(*z*, *y*)]
  

'overlaps' means the same as
'shares a part with'
(D2) 
Disjointness *D*(*x*,
*y*)
=*df* !*O*(*x*,
*y*)
  

'*x* is disjoint from *y*'
means the same as '*x* does not overlap
*y*'
(D3) 
Proper Parthood *PP*(*x*,
*y*)
=*df* *P*(*x*,
*y*) & !*x*=*y*
  

'*x* is a proper part of
*y*' means the same as '*x* is a part of but not
identical to *y*'
(D4) 
Simplicity *S*(*x*)
=*df*
![?]*y**PP*(*y*,
*x*)
  

'*x* is simple' (or
'*x* is a simple') means the same as '*x*
doesn't have any proper parts'
(D5) 
Complexity *C*(*x*)
=*df*
!*S*(*x*)
  

'*x* is complex' means the same
as '*x* is not simple'
(D6) 
Gunkiness *G*(*x*)
=*df*
[?]*y*[*P*(*y*, *x*)
- *C*(*y*)]
  

'*x* is gunky' means the same as
'each part of *x* is complex'
(D7) 
Coincidence *CO*(*x*,
*y*)
=*df* [?]*z*[*O*(*x*,
*z*) -
*O*(*y*,*z*)]
  

'*x* mereologically coincides with
*y*' means the same as '*x* overlaps exactly the
same things as *y*'


Finally, if we supplement our first-order language with plural
quantifiers and variables and the predicate
'*x*&pr;*yy*' for '*x* is one of
*yy*' (see the entry on
 plural quantification),
 we can define a fusion predicate:



(D8) 
Fusion *F*(*y*,
*xx*)
=*df* [?]*z*[(*z*&pr;*xx*
-
*P*(*z*, *y*)) & (*P*(*z*, y) -
[?]*w*(w&pr;*xx* & *O*(*z*,
*w*)))]
  

'*y* fuses *xx*' means the
same as 'each of *xx* is a part of *y* and each part
of *y* overlaps at least one of *xx*'


We will treat the expressions '*y* is composed of
*xx*' and '*y* is a sum of *xx*' as
synonyms of '*y* fuses *xx*'.



This entry addresses questions framed in terms of modal notions. Are
extended simples possible? Is it necessary that nothing is
multilocated? All talk of possibility and necessity in what follows
should be understood as talk of metaphysical possibility and
metaphysical necessity, respectively. In keeping with current
orthodoxy, we assume that being metaphysically necessary (a property
of propositions or sentences) is not to be identified with being a
logical truth, being an analytic truth, being a conceptual truth, or
being an *a priori* truth. Although metaphysical necessity is
not identified with conceptual truth--and, correlatively,
metaphysical possibility is not identified with
conceivability--one might still think that conceivability (or
something in that vicinity) is evidence for metaphysical possibility.
See Chalmers (2002) for distinctions between various sorts of
conceivability and for discussion of how each of them connects or
fails to connect to metaphysical possibility. Also see the entry on
 the epistemology of modality.



One last preliminary. For better or worse, the recent literature on
location and mereology tends to bracket considerations of vagueness
and indeterminacy (though see Eagle 2016a) and quantum theory (though
see Pashby 2016). We will do the same.


## 2. Location


### 2.1 Which Location Relation is Fundamental?



Debates about location have been framed in terms of two main
locational predicates: 'is exactly located at' and
'is weakly located at'. Philosophers tend to agree about
how these predicates apply in particular cases, and they tend to agree
that one of these predicates should be taken as primitive and used to
define the other. But there is disagreement about which to define and
which to take as primitive. (Though certain exotic possibilities
involving gunky objects (Leonard 2014) or extended simple regions
(Kleinschmidt 2016) generate pressure to posit more than one
fundamental location relation.)



Here is a typical informal gloss for 'is exactly located
at': an entity *x* is exactly located at a region *y*
if and only if *x* has (or has-at-*y*) exactly the same
shape and size as *y* and stands (or stands-at-*y*) in all
the same spatial or spatiotemporal relations to other entities as does
 *y*.[1]
 Thus, small cubes are exactly located only at small cubical regions,
large spheres are exactly located only at large spherical regions, and
so on (see Casati and Varzi 1999: 119-120; Bittner, Donnelly,
and Smith 2004; Gilmore 2006: 200-202; Sattig 2006: 48).



Here is a typical informal gloss for 'is weakly located
at': an entity *x* is weakly located at a region *y*
if and only if *y* is 'not completely free of'
*x* (Parsons 2007: 203). Thus I am weakly located at a certain
region, *r*, whose shape, size, and position perfectly match my
own, I am weakly located at the bottom half of that region, I am
weakly located at a certain much larger region, *r*+,
that has *r* as a proper part, and I am weakly located at a
certain scattered region, *r*\*, that's made up of the bottom half
of *r* together with a small region somewhere in Siberia. In
short, if *r* is the one and only region at which I am exactly
located, then I am weakly located at just those regions that overlap
*r*.




Figure 1


![Figure 1: a shaded circle labeled 'r4'; inside the circle is a pentagon labeled 'r5'; completely outside the circle is a pentagon labeled 'r1'; partially in the circle and partially not is a triangle labeled 'r2'; a large rectangle contains everything and is labeled 'r3'. A label 'd' is inside r4 but outside r2 and r5.](index-fig1.png)


Figure 1 illustrates exact location and weak location. The dotted
lines indicate regions
(*r*1-*r*5). The shaded circle
indicates a disc-shaped object, *d*, that is located at regions.
In the diagram, *d* is *exactly* located at
*r*4 but not at *r*1,
*r*2, *r*3, or *r*5,
and *d* is *weakly* located at *r*2,
*r*3, *r*4, and *r*5
but not at *r*1.



Assuming that we now grasp the intended relations, we are in a
position to consider proposals about how the associated predicates
might be defined. Suppose that we take '*x* is exactly
located at *y*' as primitive and symbolize it as
'*L*(*x*, *y*)' (Casati and Varzi 1999).
Then, as Parsons (2007: 204) notes, the following definition of
'is weakly located at' ('*WKL*') is
natural:



(D9)
Weak Location *WKL*(*x*,
*y*)
=*df* [?]*z*[*L*(*x*,
*z*) &
*O*(*z*, *y*)]
  

*'*x* is weakly located at *y*' means
'*x* has an exact location that overlaps
*y*'* 


One consequence of (D9) is that the following principle



Exactness
[?]*x*[?]*y*[*WKL*(*x*, *y*) -
[?]*z**L*(*x*, *z*)]
  

*If a thing is weakly located somewhere, then it's exactly located
somewhere.* 


is an analytic truth (Parsons 2007). This might be seen as a drawback,
for the following reason among others (Gilmore 2006: 203; Parsons
2007: 207-9). One might think that the following situation is
possible: (i) all regions are extended and gunky and decompose into
smaller (but still extended and still gunky) regions, (ii) some
located entities are point-like and unextended, and (iii)entities are
located only at regions. A point-like entity will be too small to be
exactly located in any extended region, but it should still be weakly
located at many regions--in particular, at each in a series of
nested regions that 'converge' on the point-like entity.
So, if it's possible that (i)-(iii) are all true, then it's
possible that, contrary to Exactness, a thing is weakly located
somewhere without being exactly located anywhere.



On the other hand, suppose that we take '*WKL*' as
primitive. How might we use this predicate to define
'*L*'? This is less obvious, but the definition that
Parsons (2007: 205) proposes is



(D10) 
Exact Location *L*(*x*,
*y*)
=*df* [?]*z*[*WKL*(*x*,
*z*)
- *O*(*y*, *z*)]
  

*'*x* is exactly located at *y*' means
'*x* is weakly located at all and only those entities that
overlap *y*'*


To see how this definition works, return to the object *d* in
Figure 1. We want it to turn out that *d* is exactly located at
*r*4 and nowhere else. Given the natural assumptions
about which regions are parts of which, (D10) delivers the desired
verdicts.



For example, (D10) tells us that *d* is not exactly located at
*r*3, for the reason that *r*3
overlaps certain regions (e.g., *r*1) at which
*d* is not weakly located. In other words, *r*3
'overlaps regions it shouldn't'. The situation with
*r*5 is reversed. (D10) tells us that *d* is not
exactly located at *r*5, for the reason that there are
regions at which *d* is weakly located (e.g.,
*r*2) that *r*5 does *not*
overlap. In other words, *r*5 'fails to overlap
regions it should'. Region *r*2 has both vices:
it overlaps certain regions it shouldn't (e.g., its southeast corner),
and it fails to overlap certain regions it should (e.g.,
*r*5). Region *r*4 has neither vice.
Hence *d* counts as being exactly located at it, according to
(D10).



One potential problem with (D10), however, is that it makes the
following principle



Quasi-functionality
[?]*x*[?]*y*[?]*z*[(*L*(*x*,
*y*) & *L*(*x*, *z*)) -
*CO*(*y*, *z*)]
  

*Nothing has two different exact locations, unless those locations
mereologically coincide with each other.*


an analytic truth. As we will see in
 Section 6,
 there are many who would deny Quasi-functionality, and there are
others who deny that it is necessary.



In the spirit of van Inwagen (1981), one might take the fundamental
location relation to be lying-within, where this holds only between a
simple spacetime point, on the one hand, and a located entity, on the
other. One might then define exact location in terms of it, as
follows: '*x* is exactly located at *y*' means
the same as 'for any *z*, if *z* is simple, then:
*z* lies within *x* if and only if *z* is a part of
*y*'. This has consequences similar to those of (D10). It
has the additional disadvantage that, in the context of gunky
spacetime, it yields the result that everything is exactly located
everywhere.



Is it possible to do without any fundamental location relations? For
substantivalists, there are two main strategies.



First, there is the *supersubstantivalist strategy*.
Supersubstantivalists say that each material object is identical to
the region that is its exact location. A first attempt at a
supersubstantivalist definition of exact location is this: for
*x* to be exactly located at *y* is for *x* to be
identical to *y*. But this faces counterexamples involving
abstract entities. The empty set is identical to itself but is not
exactly located at itself. A natural repair invokes either the notion
of a region (for those who are willing to say that all regions are
exactly located at themselves) or the notion of a material object (for
those who are not): for *x* to be exactly located at *y* is
for *x* to be a region (alternatively: a material object) and
identical to *y*. Weak location can then be defined via (D9).
This account of exact location makes it hard to avoid full-blown
Mereological Harmony, as captured by (H1)-(H8) below. (See
Section 3.) As a result, every threat to Mereological Harmony,
including the arguments for interpenetration, extended simples, and
mutlilocation, is thereby a threat to the given account of exact
location. But by the same token, one might argue that every instance
of Mereological Harmony is explained by, and hence confirms, the
supersubstantivalist strategy.



A variant of the supersubstantivalist strategy is available to those
who deny that located entities are identical to their locations,
provided that they are willing to say that those entities
mereologically coincide with their locations. They can say that for
*x* to be exactly located at *y* is for *y* to be a
region and for *x* to mereologically coincide with *y*
(Hawthorne 2006: 118, note 18).



Second, there is the *relativizing strategy*. As noted above,
some philosophers hold that the fundamental parthood relation is
three-place or four-place, with the additional argument places
reserved for locations of the part and/or the whole. For example, to
allow for multilocation and mereological variation from one location
to another, Gilmore (2014b: 21) suggests taking the predicate
'*x* at *y* is a part of *z* at *w*'
as primitive and using it to define '*x* is exactly located
at *y*', as 'for some *z* and some *w*:
either *x* at *y* is a part of *z* at *w* or
*z* at *w* is a part of *x* at *y*'. The
definition of the *existence-at* relation in Thomson (1983) is an
early example of this strategy. To keep things simple, however, we will
continue to assume that the fundamental parthood relation is
two-place. So we will set the relativizing strategy aside.


### 2.2 The Pure Logic of Location



Most of the formal work on location has focused on questions about how
location interacts with parthood. But there are important questions
about location in its own right. Here we highlight two groups of
questions.


#### 2.2.1 Logical Form



We will assume that exact location, not weak location, is the unique,
fundamental location relation. We will also assume that (i) it is a
two-place relation, and (ii) both argument places in that relation are
singular. (On the distinction between singular and plural argument
places in relations, see Yi (1999).) But both (i) and (ii) have been
questioned.



For example, one might reject (i) in favor of the view that exact
location is a three-place relation that holds between a located
entity, a region of space, and an instant of time (Thomson 1983, Costa
2017). This is a natural view for those who accept Eternalism and the
B-theory but deny the Spacetime View, and instead think of space as a
three-dimensional entity that endures through, and is separate from,
time. (This picture is discussed in Skow 2015 and Gilmore, Costa and
Calosi 2016.) To allow for the possibility of motion, those who
endorse such a view will want to be able to say, of a given material
object, that it is exactly located at region *r*1, not
at region *r*2, at time *t*1, and that
the same object is exactly located at *r*2, not at
*r*1, at time *t*2. To allow for the
possibility that time is gunky and does not contain instants, one
might take exact location to be expressed by '*x* is
exactly located at region *r* within interval *s*'. A
different option is to reject (i) in favor of the view that exact
location is variably polyadic, an idea floated by Jones (forthcoming:
note 29).



Alternatively, one might agree that exact location is a two-place
relation but reject (ii) above in favor of the view that, say, the
second argument place in exact location (the 'location'
slot) is plural. One idea is that an extended object can be exactly
located at many points, collectively, without being exactly located at
any one of them individually or at the set or fusion of them. This is
suggested by Hudson (2005: 17); motivations are developed in Gilmore
(2014b: 25). In principle, one could also hold that the first argument
place in exact location (the 'occupant' slot) is
plural.


#### 2.2.2 Purely Locational Principles



If we assume that exact location is the one fundamental locational
relation, that it's two-place, and that both of its argument places
are singular, what should we say about its behavior? Here we confine
our attention to purely locational principles, i.e., principles that
can be stated in a first-order language with identity whose only
non-logical predicate is '*L*(*x*,
*y*)'.



Casati and Varzi (1999: 121) propose two principles:



 Functionality
[?]*x*[?]*y*[?]*z*[(*L*(*x*,
*y*) & *L*(*x*, *z*)) -
*y*=*z*]
  

*Nothing has more than one exact location.*
 Conditional Reflexivity
[?]*x*[?]*y*[*L*(*x*, *y*) -
*L*(*y*, *y*)]
  

*Exact locations are exactly located at themselves.*


Functionality bans multilocation, which we discuss at some length in
Section 6. It tells us that nothing bears the exact location relation
to more than one entity. Functionality leaves open the possibility of
entities (regions, numbers, ...) that do not bear that relation
to anything.



According to Conditional Reflexivity, exact locations are exactly
located at themselves. (See also Donnelly (2004: 158), who presents a
system in which Conditional Reflexivity is a theorem, though she
replaces the location predicate '*L*' with a
primitive function symbol '*r*' for 'the exact
location of'.) To appreciate the significance of Conditional
Reflexivity, suppose that Obama is exactly located at region *r*.
Together with Conditional Reflexivity, this entails that *r* is
exactly located at itself. This conflicts with a purely locational
principle endorsed by Simons (2004b: 345):



 Asymmetry of Location
[?]*x*[?]*y*[(*L*(*x*, *y*) -
!*L*(*y*, *x*)]
  

*If *x* is exactly located at *y* then *y* is not
exactly located at *x*.*


Cases in which a region is exactly located at itself do not conflict
with



 Antisymmetry of Location
[?]*x*[?]*y*[(*L*(*x*, *y*) &
*L*(*y*, *x*)) - *x*=*y*]
  

*No two entities are exactly located at each other.*


Thus the Antisymmetry of Location may salvage some of the motivation
for the Asymmetry of Location while still harmonizing with Conditional
Reflexivity. The Antisymmetry of Location is a logical consequence of
Functionality and Conditional Reflexivity (as is the view that exact
location is transitive).



If we further assume that Obama is not identical to his exact location
*r*, we get the result that there are two different entities
exactly located at *r*--namely, *r* and Obama. In that
case, we have a counterexample to another purely locational principle
that some have found attractive, namely:



 Injectivity of Location
[?]*x*[?]*y*[?]*z*[(*L*(*x*,
*z*) & *L*(*y*, *z*)) -
*x*=*y*]
  

*No two entities share an exact location.*


Opponents of co-location may take this as a *reductio* of
Conditional Reflexivity. Others may take it as a reason to reject the
Injectivity of Location in favor of a weaker variant, e.g.:



 Conditional Injectivity of
Location
[?]*x*[?]*y*[?]*z*[(!*x*=*z*
& !*y*=*z*) - ((*L*(*x*, *z*)
& *L*(*y*, *z*)) - *x*=*y*)]
  

*If neither x nor y is identical to z, then if each of them is
exactly located at z, then x and y are identical to each
other.*


Conditional Injectivity is equivalent to the claim that whenever two
different entities share a given exact location, one of them is
identical to that location. This may salvage some of the motivation
for the ban on co-location, while still harmonizing with Conditional
Reflexivity.



With Conditional Reflexivity as a backdrop, Casati and Varzi (1999:
123) define a predicate for regionhood, '*R*':



(DR)
 Regionhood *R*(*x*)
=*df* *L*(*x*, *x*)
  

'*x* is a region' means the same as '*x*
is exactly located at itself'


Together with Conditional Reflexivity, (DR) entails that that if
*x* is exactly located at *y*, then *y* is exactly
located at itself and hence is a region. But the (DR) + Conditional
Reflexivity package also allows for cases in which an entity *r*
counts as a region even though nothing but *r* is exactly located
at *r*.



The regionhood predicate is potentially useful in stating restricted
versions of familiar mereological principles. One might want a
principle to the effect that any plurality of *regions* has a
fusion, while remaining neutral on the more general claim that any
plurality of entities whatever has a fusion. Likewise one might want a
principle to the effect that non-identical *regions* never
mereologically coincide, while rejecting the stronger claim that
non-identical entities in general never mereologically coincide.
(Perhaps a certain clay statue mereologically coincides with, but is
not identical to, a certain lump of clay.) The most natural
formulations of these restricted principles employ a regionhood
predicate. Though one might take this predicate as primitive,
considerations of ideological parsimony provide some motivation for
treating it as defined.



To close this section, it is worth drawing attention to another
logical consequence of Functionality and Conditional Reflexivity,
namely:



 Shortness of Location
[?]*x*[?]*y*[?]*z*[(*L*(*x*,
*y*) & *L*(*y*, *z*)) -
*y*=*z*]
  

*If x is exactly located at y and y is exactly located at z, then
y=z.*


This says that exact locations are exactly located only at themselves
(hence that 'chains' of exact location are always
'short'). To see why this follows from Functionality and
Conditional Reflexivity, suppose that *x* is exactly located at
*y*. Then by Conditional Reflexivity, *y* is exactly located
at *y*. If *y* is also exactly located at *z*, then by
Functionality, *y*=*z*.



Shortness of Location might also appeal to those who accept
Conditional Reflexivity but deny Functionality. Since Shortness of
Location does not follow from Conditional Reflexivity alone, some may
wish to adopt Shortness as an axiom.


## 3. Interaction with Parthood



Philosophers have put forward of variety of axiom systems that are
meant to capture the interaction between parthood and location. One
especially bold idea is that the mereological properties of, and
relations between, located entities *perfectly match* those of
their locations. This has been dubbed *Mereological Harmony*
(Schaffer 2009).



For a collection of axioms that makes no attempt enforce Mereological
Harmony, see the supplementary document on
 An Interpenetration-Friendly Theory of Location.




Mereological Harmony has been captured formally in different ways by
Varzi (2007), Uzquiano (2011), and Leonard (2016). Saucedo (2011:
227-228) offers the following principles:



(H1)
 *x* is mereologically simple iff *x*'s location is
mereologically simple.
(H2)
 *x* is mereologically complex iff *x*'s location is
mereologically complex.
(H3)
 *x* has exactly *n* parts iff *x*'s location has
exactly *n* parts.
(H4)
 *x* is gunky iff *x*'s location is gunky. ...
(H5)
 *x* is a part of *y* iff *x*'s location is a
subregion of *y*'s location.
(H6)
 *x* is a proper part of *y* iff *x*'s location is
a proper part of *y*'s location.
(H7)
 *x* and *y* overlap iff *x*'s location and
*y*'s location overlap.
(H8)
 The *x*s compose *y* iff the locations of the *x*s
compose *y*'s location.


Some philosophers take Mereological Harmony to be a necessary truth
(Schaffer 2009:
 138).[2]
 The remainder of this entry considers three separate threats to the
view that Mereological Harmony is necessary: interpenetration
 (SS4),
 extended simples
 (SS5),
 and multilocation
 (SS6).



There are other threats to Mereological Harmony that we will not
discuss, e.g., threats to H7 and H8 that arise from 'moderate
views about receptacles', according to which only topologically
open (alternatively: only topologically closed) regions can be exact
locations (see Cartwright 1975; Hudson 2005: 47-56; and
especially Uzquiano 2006), or threats to H4 discussed in Uzquiano
(2011).)



A case of interpenetration occurs when non-overlapping entities have
overlapping exact locations--e.g., when a ghost passes through a
wall. In such a case, the right-to-left direction of H7 fails. Similar
cases involve violations of the right-to-left directions of H5 and H6.
An extended simple is a simple entity with a complex exact location:
it violates the left-to-right direction of H1, the right-to-left
direction of the (equivalent) H2, and the left-to-right direction of
the instance of H3 that results from letting *n* = 1. A case of
multilocation occurs when a given entity has more than one exact
location. This violates Functionality, which is left implicit in
Saucedo's statement of Mereological Harmony.



The three questions that we will consider--Is interpenetration
possible? Are extended simples possible? Is multilocation
possible?--are logically independent of one another. Thus there
is room for eight specific packages of views. See Figure 2.





Figure 2| Package | Interpenetration? | Extended simples? | Multilocation? |
| 1 | yes | yes | yes |
| 2 | yes | yes | no |
| 3 | yes | no | yes |
| 4 | yes | no | no |
| 5 | no | yes | yes |
| 6 | no | yes | no |
| 7 | no | no | yes |
| 8 | no | no | no |




It is important to note that even if interpenetration, extended
simples, and multilocation are all possible, some substantive
principles linking parthood and location may still survive. For
example, the possibility of interpenetration and extended simples
poses no threat to:



Expansivity
#[?]*x*[?]*y*[?]*z*[?]*w*[[*P*(*x*,
*y*) & *L*(*x*, *z*) & *L*(*y*,
*w*)] - *P*(*z*, *w*)]
  

*Necessarily, if *x* is a part of *y*, and if *x* is
exactly located at *z* and *y* is exactly located at
*w*, then *z* is a part of *w*: 'the part's
location is a part of the whole's
 location'.*[3]
Delegation
#[?]*x*[?]*y*[?]*z*[[*C*(*x*)
& *L*(*x*, *y*) & *P*(*z*, *y*)]
- [?]*w*[?]*u*[*PP*(*w*, *x*)
& *O*(*u*, *y*) & *L*(*w*,
*u*)]]
  

*Necessarily, if *x* is complex and is exactly located at
*y*, then for any part *z* of *y*, some proper part
*w* of *x* is exactly located at some region that overlaps
 *z*.*[4]


Expansivity rules out cases like the following (Figure 3), in which
the object *a* is a part of the object *o*, but *a*'s
exact location, *r**a*, is *not* a part of
*o*'s exact location, *r*.




Figure 3


![Figure 3: Two rows of three boxes each, with arrows connecting each top row box, labeled 'a', 'b', 'c', with the corresponding bottom box, labeled 'ra', 'rb', 'rc'. The top row boxes are bracketed and labeled 'o' and bottom row boxes rb and rc are bracketed and labeled 'r'. An arrow connects 'o' to 'r'.](index-fig3.png)


The object *a* is part of the object *o*, but *a*'s
exact location *r**a* is not a part of *o*'s
exact location, *r*.
  

*Ruled out by Expansitivity*



The idea behind Delegation is that if *x* is complex, then if you
stick a pin into *x*'s exact location, you will have stuck that
pin into the exact location of one of *x*'s proper parts as well.
In slightly different terms, a complex entity can't be weakly located
at a certain region unless one of its proper parts--a
'delegate'--is weakly located at that region as well.
(Regarding the formal statement of Delegation, one might wonder why it
is not formulated with '*PP*(*u*, *y*)' in
place of '*O*(*u*, *y*)' in the consequent.
The reason for this is that Delegation is meant to be friendly to
extended simples. Suppose that a complex, spherical object, *C*,
is exactly located a spherical region, *r*. Suppose that *C*
is composed of two hemispherical simples, *h* and *h\**, and
that *r* is composed of continuum-many simple points, each
plurality of which composes a region that is a part of *r*. Then,
contrary to the proposed revision, it will not be true that for every
part *y* of *r*, some proper part of *C* is exactly
located at a region that has *y* as a proper part. Consider, for
example, the spherical region *r\** with the same center point as,
but half the volume as, *r* itself. *C* does not have a
proper part that is exactly located at *r\**, nor does it have a
proper part whose exact location has *r\** as a proper part. But,
as Delegation requires, *C* does have a proper part (*h*,
for example) that has an exact location that overlaps *r\**.)



Delegation rules out cases like the following (Figure 4), in which
*o*\* is a complex object that is exactly located at region
*r*\*, but *r*\* has a part *r**a* that
does not overlap an exact location of any of *o*\*'s proper
parts:





Figure 4


![Figure 4: Two rows the top with two boxes labeled 'b' and 'c' and the bottom with three boxes labeled 'ra', 'rb', and 'rc'. Arrows point from 'b' to 'rb' and 'c' to 'rc'. The top row boxes are bracketed and labeled 'o*' and all three bottom row boxes are bracketed and labeled 'r*'. An arrow connects 'o*' to 'r*'.](index-fig4.png)


The region *r**a* is a part of object *o*\*'s
exact location, and object *o*\* is complex, but no proper part of
*o* has an exact location that overlaps
*r**a*.
  

*Ruled out by Delegation*



Neither interpenetration nor extended simples seem to threaten
Expansivity or Delegation.



A potential threat to Delegation comes from Pickup (2016: 260), who
considers the possibility of a complex entity that is exactly located
somewhere despite the fact that none of its proper parts is exactly
(or weakly) located anywhere. One route to such entities (not
Pickup's) runs as follows: (i) some material objects (electrons,
maybe) do not have any other material objects as proper parts, (ii)
any such material object is a complex entity whose only proper parts
are universals, (iii) all material objects have locations, but (iv) no
universals have locations. Bundle theorists who are platonic realists
about universals, and who take the constituents of a given bundle to
be parts of that bundle, will face pressure to accept (i)-(iv)
and hence to reject Delegation. The same basic idea is discussed in
connection with the Burying Strategy in Section 4.1 below.


## 4. Interpenetration



In this section we consider arguments for and against the following
principle:



No Interpenetration
#[?]*x*[?]*y*[?]*z*[?]*w*[(*L*(*x*,
*z*) & *L*(*y*, *w*) & *O*(*z*,
*w*)) - *O*(*x*, *y*)]
  

*Necessarily, if *x* and *y* have exact locations that
overlap, then *x* and *y* themselves overlap.*


According to No Interpenetration, it is metaphysically impossible for
entities of any type to 'pass through one another' without
sharing parts--in the manner of a ghost passing through a solid
brick wall. There is a related principle that deserves some comment.
The related principle says that, necessarily, if *x*'s exact
location is a part of *y*'s exact location, then *x* is a
part of *y*. In symbols:



(1)

#[?]*x*[?]*y*[?]*z*[?]*w*[(*L*(*x*,
*z*) & *L*(*y*, *w*) & *P*(*z*,
*w*)) - *P*(*x*, *y*)]


This principle may seem to say basically the same thing as No
Interpenetration but to say it more simply--using the primitive
predicate '*P*' instead of the defined predicate
'*O*'. Why then focus on No Interpenetration instead
of (1)?



The reason for this is that some of the opposition to (1) will stem
from opposition to a purely mereological principle: Strong
Supplementation. It says that if every part of *x* overlaps
*y*, then *x* is a part of *y*. Those who deny this
will be very likely to deny (1), but they might still be attracted to
No Interpenetration. Let Lump1 be a lump of clay, and let Goliath be
the statue that is 'made out of' Lump1. One might think
that every part of Goliath overlaps Lump1 (and vice versa), and that
Goliath's exact location is a part of (indeed, identical to) Lump1's
exact location, but that Goliath is not a part of Lump1 (Lowe 2003).
In that case one will reject (1). But one might still go on to say
that 'ghostly-interpenetration-without-part-sharing' is
impossible, and hence that No Interpenetration is true. After all,
Goliath and Lump1 share parts, as do Goliath's top two-thirds and
Lump1's bottom two-thirds, and so on.



In general, our task here is to set aside the purely mereological
controversies (see the entry on
 mereology)
 and to focus instead on the issues that are exclusively concerned
with location and its interaction with parthood. Too much of the
controversy over (1) arises from controversy over 'pure
mereology'. By contrast, if No Interpenetration is
controversial, this is only because of what it says about the
*connections* between parthood and location.


### 4.1 For Interpenetration #1: from Universals or Tropes



Immanent realists say that a universal is in some sense 'wholly
present' in each thing that instantiates it (Armstrong 1978: 79;
Bigelow 1988; O'Leary-Hawthorne 1995; Smith 1997;
O'Leary-Hawthorne and Cover 1998; Paul 2002, 2006, 2012; Newman
2002; Hawley and Bird 2011; Lafrance 2015; Peacock 2016). This makes
it natural to hold that disjoint universals frequently
interpenetrate.



To see why, let *e* be an electron, and suppose that it
instantiates two different universals: a mass universal,
*u**m*, and a charge universal,
*u**c*. Suppose that *e* is exactly located
at region *r*. Then it will be natural for the immanent realist
to say that (i) *u**m* is exactly located at
*r* as well, or at least at some region
*r**m* that has *r* as a part, and (ii)
*u**c* is exactly located at *r* or at some
region *r**c* that has *r* as a part. (If
these universals are also instantiated elsewhere, then it will be
debatable as to whether they are *exactly* located at *r*.
Perhaps *u**m* has only one exact location, which
fuses the exact locations of its instances (Effingham 2015). Likewise
for *u**c*.) Either way, the immanent realist
will say that *u**m* and
*u**c* have exact locations that overlap. But
presumably *u**m* and *u**c*
do not overlap. Assuming that these universals are non-structural,
non-conjunctive, and perfectly natural, they are plausibly
*simple*, in which case they overlap only if they are
identical, which they are not. It would seem that a similar point can
be made in terms of
 tropes--particular,
 spatiotemporally located 'cases' of properties or
relations. For trope theorists who take tropes to be located at
spacetime regions, it will be natural to say that mass tropes and
charges tropes, for example, frequently interpenetrate. (Since
electrons have charge and mass, trope theorists will find it natural
say that if an electron is exactly located at a region *r*, then
the electron's mass trope and the electron's charge trope are both
exactly located at *r* as well, hence that those tropes'
locations overlap (by being identical). But presumably the tropes
themselves are not identical and do not overlap. Three responses to
this argument are worth considering.



The first response says: so much the worse for tropes and immanent
universals. This response uses a mereo-locational principle, No
Interpenetration, as a premise in an argument against certain
metaphysical views, namely those that posit tropes or immanent
universals. Is there some reason why mereo-locational principles
should not be used in this way? The principles of pure mereology are
often so used. For example, Lewis (1999: 108-110) rejects states
of affairs and structural universals on the grounds that they would
violate the Uniqueness of Composition, the principle that no entities
*xx* have more than one
 fusion.[5]
 Why not give the same status to certain mereo-locational principles?
Why not, for example, say that No Interpenetration is better justified
than is the view that universals or tropes are spatiotemporally
located? True, it is hard to say much by way of argument for No
Interpenetration. But the same can be said of various purely
mereological principles, such as the Transitivity of Parthood, and
this is often treated as a non-negotiable constraint that any
metaphysical position must obey.



The second response says that while immanent universals or tropes are
spatiotemporal entities that are 'in their instances',
they are not exactly located anywhere. Armstrong writes that



> 
> 
> Talk of the location of universals, while better than placing them in
> another realm, is also not quite appropriate... To talk of
> locating universals in space-time then emerges as a crude way of
> speaking. Space-time is not a box into which universals are put.
> Universals are constituents of states of affairs. Space-time is a
> conjunction of states of affairs. In that sense, universals are
> "in" spacetime. But they are in it as helping to
> constitute it. (1989: 99)
> 
> 
> 



Simplified somewhat, the response holds that (i) universals are parts
or constituents of entities that have exact locations, and in that
sense they are 'in their instances', but (ii) universals
do no themselves have exact locations and hence do not have
overlapping exact locations. Given (ii), the universals or tropes in
question no longer count as examples of interpenetration. Call this
the *Burying Strategy*, since it 'buries'
universals and/or tropes in located entities, rather than treating
them as being located.



Here is an especially vivid instance of the Burying Strategy: All
**tropes** are instantaneous, spatially point-sized, and
mereologically simple. All **spacetime points** are
instantaneous, spatially point-sized, and mereologically complex. In
particular, each spacetime point is a fusion of some tropes (each of
which is at zero distance from each of the rest), and each trope is a
part of exactly one spacetime point. Something counts as a
**spacetime region** if and only if it is either a
spacetime point or a fusion of some spacetime points. Each spacetime
region is exactly located at itself and nothing else has an exact
location.



On this view, tropes are parts of spacetime points and regions but
they do not have exact locations. Hence, even if two tropes are both
parts of the same spacetime point, they do not interpenetrate, in our
sense.



The third response to the argument from universals and tropes is to
say, 'True, universals and/or tropes can interpenetrate, but
material objects can't'. This grants the argument and rejects No
Interpenetration in favor of a weaker, restricted principle. If we
introduce a one-place predicate, '*M*', for 'is
a material object', then we can state the restricted principle
as



(2)
#[?]*x*[?]*y*[?]*z*[?]*w*[(*M*(*x*)
& *M*(*y*) & *L*(*x*, *z*) &
*L*(*y*, *w*) & *O*(*z*, *w*))
-   
 *O*(*x*, *y*)]
  

*Necessarily, if material objects *x* and *y* have exact
locations that overlap, then *x* and *y* themselves
overlap.*


This response also handles potential counterexamples to No
Interpenetration arising from regions, sets, events, portions of
stuff, holes, spirits, and other 'immaterial
entities'.



(On the location of regions, see Casati and Varzi (1999: 123), who
hold that regions are located at themselves, and Simons (2004b: 345),
who holds that nothing is located at itself. On the location of sets,
see Maddy (1990), Lewis (1991), Effingham (2010b; 2012), and Cook
(2012). On the location of events, see Casati and Varzi (1999), Price
(2007), Giordani and Costa (2013, 2016) and Costa (2017). On the
location of portions of stuff, see Markosian (1998), (2004a), and
(2015). On the location of holes and shadows, see Lewis and Lewis
(1970), Casati and Varzi (1994), Wake, Spencer, and Fowler (2007),
Donnelly, Bittner, and Rosse (2006), and Sorensen (2008). On the
location of spirits, see Thomas (2009) and Inman (2017). Sanford
(1970) discusses many of these topics, and Hudson (2005: 4) mentions
many of them briefly.)



The next two pro-interpenetration arguments count equally against No
Interpenetration and (2), but we will continue to focus on No
Interpenetration for simplicity.


### 4.2 For Interpenetration #2: from Material Objects and Conceivability



Some think that it is possible for two disjoint *material
objects* to have overlapping exact locations. Perhaps there are no
actual cases of the relevant sort. Such cases may even be physically
impossible--ruled out by the laws of nature (though see the next
section). But one might still think that these cases are
metaphysically possible.



After all, what is it that keeps material objects from
interpenetrating in the actual world? Repulsive forces, presumably.
But a standard view is that the laws governing such forces are not
metaphysically
 necessary.[6]
 And on that assumption it is natural to conclude that there are
metaphysically possible worlds in which any repulsive forces that
exist can be overridden in such a way as to allow material objects to
interpenetrate. (For more on this, see Zimmerman 1996a and Sider
2000.)



A similar line of thought is sometimes framed as a conceivability
argument. One might take cases of interpenetration to be conceivable
or intuitively possible, and one might take this to be some evidence
for their possibility. In *New Essays the Human Understanding*
(II.xxvii.1), Leibniz writes that



> 
> 
> we find that two shadows or two rays of light interpenetrate, and we
> could devise an imaginary world where bodies did the same. (1704
> [1996])
> 
> 
> 



Sanford describes such a scenario in more detail:



> 
> 
> What imaginable circumstances would tempt us to say that two
> [disjoint] things were in the same place at the same time? Imagine two
> rectangular blocks of the same size and shape each moving along a
> straight path perpendicular to the other. The blocks approach the
> intersection of the paths at the same time, and apparently neither
> slows down nor changes its direction. They seem to pass right through
> one another, and they do this without changing with respect to colour,
> texture, density, etc. We want to say that we have the same two blocks
> with which we started. And we do not want to say that either block
> passed out of existence and was shortly thereafter re-created. Thus we
> want to say that each block moved along its path without any
> spatio-temporal discontinuity. And we can say this only if we admit
> that parts of one block simultaneously occupied the same space as
> parts of the other block. (1967: 37)
> 
> 
> 



An objection suggests itself. Why think that, in this scenario, the
blocks really are disjoint while they are passing through one
another?



In response, one might just *stipulate* that what is under
consideration are two always-disjoint blocks, and then insist that the
case remains conceivable, so described.



Alternatively, one might further specify the case in such a way as to
be able to argue for the blocks' disjointness. Consider a certain
region, *r*, that is said by Sanford to the exact location of two
disjoint material objects: *p*1 (a part of Block 1)
and *p*2 (a part of Block 2). One might add that two
incompatible properties are instantiated at *r*--say, *having
mass of 2 kg,* and *having mass of
3 kg*--where each of these properties is such that if it is
instantiated by an entity *x*, then it is instantiated by
anything that mereologically coincides with *x*. Presumably this
is no less conceivable (or 'intuitively possible') than
Sanford's original case. But in this new version of the case, it is
not open to Sanford's opponent to claim that the co-located objects
are identical or mereologically coincident with each other. For the
region contains a 2 kg object and a 3 kg object, and no one object is
identical to (or mereologically coincident with) both a 2 kg object
and a 3 kg object.


### 4.3 For Interpenetration #3: from Bosons



One question that has been raised in the recent philosophical
literature is whether contemporary physics provides us with examples
of disjoint fundamental particles that have the same, or overlapping,
exact locations. Hawthorne and Uzquiano apparently claim that the
answer is 'Yes'. They write that



> 
> 
> particles having integral spin--otherwise known as
> bosons--in modern particle physics ... are generally thought
> to be point-sized. Moreover, according to the spin statistics theorem,
> while fermions--point-particles with half integer
> spin--cannot be colocated, bosons are perfectly well able to
> cohabit a single spacetime point. (2011: 3-4)
> 
> 
> 



Schaffer suggests otherwise:



> 
> 
> [a] more sophisticated treatment of these cases involves field theory.
> Instead of there being two bosons co-located at region *r*, there
> is a bosonic field with doubled intensity at *r*. (2009: 140)
> 
> 
> 



Whereas Hawthorne and Uzquiano apparently take bosons to provide
*actual* examples of interpenetration, McDaniel suggests that
they at least reinforce the conceivability of such
counterexamples:



> 
> 
> what this example shows is that [disjoint] co-located material objects
> are not merely conceivable, but that a tremendously detailed
> conception of them has been formed: [disjoint] co-located objects play
> a role in the interpretation of certain physical theories. It might be
> that at the end of the day speculative physics will postulate
> co-located material objects. It seems to me that we should not
> disregard this possibility *a priori*. (2007a: 240)
> 
> 
> 



If one's goal, in constructing a theory of location, is to articulate
the necessary and *a priori* truths governing location and its
interaction with parthood, then even McDaniel's modest point still
counts against including No Interpenetration in one's theory. For if
McDaniel is right, then that principle is at best an *a
posteriori* truth, though perhaps still a necessary truth. (See
Simons (1994a, 2004a) for further discussion of bosons and for related
considerations in support of interpenetration. For further discussion
of Hawthorne and Uzquiano, see Cotnoir (2016).)


### 4.4 For Interpenetration #4: from Recombination



Sider (2000: 585-6), McDaniel (2007a), and Saucedo (2011) have
all objected to No Interpenetration on the grounds that it conflicts
with plausible 'principles of recombination', which forbid
certain sorts of necessary connections between distinct properties and
relations. McDaniel's formulation of the objection, which is similar
to Sider's, runs as follows:



> 
> 
> The state of affairs in which [a simple] object *x* occupies a
> particular region of space *R* (at *t*) is distinct from the
> state of affairs in which [some other simple] object *y* occupies
> the same region (at the same time). From the fact that the first state
> of affairs obtains, we can infer nothing about the location of
> *y*. Both states of affairs obtain contingently. If any
> recombination of distinct, contingent states of affairs yields a
> genuine possibility, as I am inclined to hold, then there are possible
> worlds at which both *x* and *y* occupy *R* (at
> *t*). (2007a: 241)
> 
> 
> 



Let *o*1 and *o*2 be two different
objects, let *r* be a region, and consider the following states
of affairs:



s1
*o*1's being simple and exactly located at
*r*
s2
*o*2's being simple and exactly located at
*r*


Then we can reconstruct McDaniel's argument as follows:



Premise 1
*s*1 is a contingent state of affairs.
Premise 2
*s*2 is a contingent state of affairs.
Premise 3
*s*1 is distinct from *s*2.
Premise 4
For any *x* and any *y*, if *x* and *y* are
each contingent states of affairs, and if they are distinct from each
other, then possibly, both *x* and *y* obtain.
*Therefore*
Conclusion
Possibly, both *s*1 and *s*2
obtain.


If it's possible for both *s*1 and
*s*2 to obtain, then it's possible for a given region
to be the exact location of two different simples. And since no two
simples can overlap, this would mean that it's possible for disjoint
things (the simples) to have identical (hence overlapping) exact
locations.



Is the argument successful? As Sider and McDaniel are well aware, the
term 'distinct from' needs to mean something other than
'not identical to', if the recombination principle,
Premise 4, is to get off the ground. After all, the state of affairs
of **o*'s being 2 kg in mass* and the state of affairs of
**o*'s being 3 kg in mass* are both contingent, and they
are not identical to each other, but presumably it is not possible
that they *both* obtain. (Presumably it's not possible that
something is both 2 kg and 3 kg in mass.)



But it is no easy matter to give 'distinct from' a meaning
that makes Premises 3 and 4 simultaneously plausible. If it means
'shares no parts or constituents with', then Premise 4
avoids the counterexample given above, but Premise 3 ceases to be
plausible, since *s*1 and *s*2 do
plausibly share a constituent, namely *r*. If '*s* is
distinct from *s*\*' is defined as '(i) possibly,
*s* obtains and *s*\* does not, (ii) possibly, *s* does
not obtain and *s* does, (iii) possibly, neither *s* nor
*s*\* obtains, and (iv) possibly, both *s* and *s*\*
obtain', then Premise 4 is trivially true, but Premise 3 begs
the question.



For additional arguments in favor of interpenetration, including a
more detailed recombination argument, see Section 1.1 of the
supplementary document
 Additional Arguments.


### 4.5 Against Interpenetration #1: from Supersubstantivalism



Supersubstantivalism is often glossed as the view that each material
object is identical to the spacetime region at which the object is
exactly
 located.[7]
 If supersubstantivalism is not just true but necessary, then it is
impossible for disjoint *material objects* to be exactly
located at overlapping regions. After all, if material object *x*
is exactly located at region *r*1 and material object
*y* is exactly located at *r*2, and
*r*1 and *r*2 overlap, then, given
supersubstantivalism, *x*=*r*1 and
*y*=*r*2, and hence *x* and *y* overlap
as well.



This gives us an argument for (2), the version of No Interpenetration
restricted to material objects. (However, (2) conflicts with the
possibility of Sanford's interpenetrating blocks, so the argument will
be controversial.) But the argument does not give us an argument for
No Interpenetration. For it wouldn't follow that it is impossible for
there to be disjoint entities of any kind to have overlapping exact
locations. Supersubstantivalism, as stated above, leaves open the
possibility that there are universals or tropes, e.g., that are
exactly located at regions but not identical to them. Hence it leaves
open the possibility that disjoint entities that are not material
objects might have overlapping exact locations.



To construct an argument for No Interpenetration, we would need to
appeal to a stronger supersubstantivalist doctrine, for example:



Supersubstantivalism+
#[?]*x*[?]*y*[*L*(*x*, *y*)
- *x*=*y*]
  

*Necessarily, each entity is identical to anything at which it is
exactly located.*


Supersubstantivalism+ entails No Interpenetration. Take any objects
*x* and *y* in any possible world, and suppose that they
have exact locations, *r*1 and *r*2
respectively, that overlap. Then, given Supersubstantivalism+,
*x*=*r*1 and *y*=*r*2, hence
*x* and *y* overlap.



For a second, related argument against interpenetration, see Section
1.2 of the supplementary document
 Additional Arguments.


## 5. Extended Simples



A simple is an entity that has no proper parts. Are there any simples?
Within the realm of spatiotemporal entities, some natural candidates
are: spacetime points, fundamental particles such as electrons (or
instantaneous temporal parts of them), and perhaps certain universals,
certain tropes, or certain sets. On the other hand, it would seem to
be an empirically open possibility that all spatiotemporal entities
are gunky.



Say that an entity is *extended* just in case it is a
spatiotemporal entity and does not have the shape and size of a point.
In this sense of 'extended', a solid cube would count as
extended, but, given natural assumptions, so would a fusion of two
point-particles that are one foot apart. Although such a fusion is
naturally taken to have zero length, it would be a scattered object
and so would not have the shape of a point.



Are there any extended simples? Could there be? Those who answer
'No' to both questions will be inclined to accept



No Extended Simples (NXS)
#[?]*x*[?]*y*[[*L*(*x*, *y*)
& *C*(*y*)] - *C*(*x*)]
  

*Necessarily, if *x* is exactly located at *y* and
*y* is complex, then *x* is complex.*


Strictly speaking, NXS does not say that extended simples are
impossible; rather, it says that simples with complex exact locations
are impossible. It leaves open the possibility that there are extended
simple regions and extended simple entities that are exactly located
at them. (For more on extended simple regions and discrete space or
spacetime, see Forrest 1995, Tognazzini 2006, Braddon-Mitchell and
Miller 2006, McDaniel 2007, Dainton 2010: 294-301, Spencer 2010,
2014, Hagar 2014, Jaeger 2014, and Kleinschmidt 2016.) And it rules
out the possibility that there is a point-sized material simple that
is exactly located at a point-sized but mereologically complex region
(e.g., a region that is the fusion of several point-sized tropes each
of which is at zero distance from each of the others).



For the most part, however, it will do no harm to treat the debate
over extended simples as a debate over NXS. We can do so if we assume
that, necessarily, a region is extended if and only if it is complex.
So, in what follows, we will operate under that assumption unless we
explicitly note otherwise.


### 5.1 For Extended Simples #1: from Conceivability



An initial argument appeals to the claim that extended simples are
conceivable and takes that to be some evidence in favor of their
possibility. To conceive of an extended simple, think of an
extended--say, cubical--object that has no proper parts. The
idea is not, or not merely, that the cube *cannot be physically
split or cut up*. Whether or not it can be split is a separate
question. The idea is that there are no proper parts of the cube.
Although the object is cubical, it has no top half or bottom half, no
left half or right half. All there is, in the relevant vicinity, is
the cube itself--together with the region at which it is exactly
located and the parts of that region.



Debates about extended simples typically focus on the question of
whether extended simple material objects are possible. But entities in
other ontological categories (tropes, universals, sets, regions) are
sometimes thought to be located. So it is worth keeping in mind that,
whatever one thinks about material objects, one might hold that
extended simples in other categories are possible. With that said, we
will focus on material objects for the remainder of this section.


### 5.2 For Extended Simples #2: from String Theory



As McDaniel (2007a: 235-6) notes, some physicists interpret
string theory as positing extended simples. McDaniel quotes a passage
from Brian Greene:



> 
> 
> What are strings made of? There are two possible answers to this
> question. First, strings are truly fundamental--they are
> "atoms," uncuttable constituents. ... As the absolute
> smallest constituents of anything and everything, they represent the
> end of the line ... in the numerous layers of substructure in the
> microscopic world. From this perspective, even though strings have
> spatial extent, the question of their composition is without any
> content. Were strings to be made of something smaller they would not
> be fundamental. Instead, whatever strings were composed of would
> immediately displace them and lay claim to being an even more basic
> constituent of the universe ... [A] string is simply a
> string--as there is nothing more fundamental, it can't be
> described as being composed of any other substance. (1999:
> 141-2)
> 
> 
> 



Can strings be treated as being identical to the spacetime regions at
which they are exactly located? Greene does not explicitly address
this question. If the answer is 'Yes', however, and if
strings are exactly located only at complex regions, then string
theory would not be committed to extended simples after all. For an
argument that string theory does not posit extended simples, see Baker
(2016).


### 5.3 For Extended Simples #3: from Recombination



As we noted in
 Section 4.4,
 Sider, McDaniel, and Saucedo have argued for the possibility of
interpenetration by appeal to principles of recombination. They have
also marshaled these principles in support of the possibility of
extended simples (Sider 2007; McDaniel 2007b; Saucedo 2011). Here is
the core of McDaniel's argument:



> 
> 
> (NNC): Let *F* and *G* be accidental, intrinsic properties;
> let *R* be a fundamental relation; let *x* and *y* be
> contingently existing non-overlapping entities. Then it is not the
> case that, necessarily, *R**x**y* only if
> (*F**x* if and only if *G**y*)...if we accept
> the Humean premise that there are no necessary connections between the
> accidental, intrinsic properties of regions of space and the
> accidental, intrinsic properties of material objects, then we should
> hold that there are no necessary connections between the mereological
> structure of a material object and the mereological structure of the
> region it occupies. Specifically, it is not true that, necessarily, a
> material object is a simple if and only if it occupies a simple (read:
> pointsized) region of spacetime. It follows that extended material
> simples are possible. (2007b: 135-137)
> 
> 
> 



If we let *o* and *r* be contingently existing,
non-overlapping entities, then McDaniel's argument can be
reconstructed as follows:



(3)
*Exact location* is a fundamental relation.
(4)
*being simple* is an accidental, intrinsic
property.
(5)
*being a simple region* is an accidental
intrinsic property.
(6)
Object *o* and region *r* are contingently existing,
non-overlapping entities.
(7)
For any accidental, intrinsic properties *F* and *G*,
any fundamental relation *R*, and any contingently existing
non-overlapping entities *x* and *y*, it is not the case
that, necessarily if *Rxy* then (if *F**x* then
 *G**y*).[8]
*Therefore*
(8)
It is not the case that necessarily, if *o* is exactly
located at *r*, then: if *o* is simple then *r* is a
simple region.


Supersubstantivalists will deny that exact location is a fundamental
relation. Others might take aim at McDaniel's recombination principle.
Let *F* be the property *being round* and let
*G* be the property *not being square*. These may seem to
be accidental and intrinsic. But if they are, and if exact location is
fundamental, then McDaniel's principle tells us that it is possible
for a round object to be exactly located at a square region, which
some might take as a *reductio* of (7). McDaniel (2007b) and
Skow (2007), however, take this to show that the given shapes are not
intrinsic properties. For a more detailed discussion of recombination
principles such as (7), and a defense of the view that shapes are
intrinsic, see Kleinschmidt (2015).



For further arguments in favor of extended simples, including a more
detailed recombination argument, see Section 2.1 of the supplementary
document
 Additional Arguments.


### 5.4 Against Extended Simples #1: Supersubstantivalism



According to supersubstantivalism, each material object is identical
to the spacetime region at which it is exactly located. Those who take
supersubstantivalism to be a necessary truth are likely to be hostile
to the possibility of extended simples--or at least to the
possibility of extended simple material objects. For they are likely
to endorse the following argument:



(9)
Necessarily, all material objects are regions.
(10)
Necessarily, all extended regions are complex.
*Therefore*
(11)
Necessarily, all extended material objects are complex. In other
words, it is impossible for a material object to be an extended
simple.


Most defenders of extended simples will resist this argument by
denying (9). They tend to think of material objects as being
non-identical to, and indeed mereologically disjoint from, the regions
at which they are exactly located (Markosian 1998; McDaniel 2007b,
2009; Parsons 2007; Saucedo 2011). A different option is to reject
(10) in favor of the view that there are, or at least could be,
extended simple spacetime regions (Braddon-Mitchell and Miller
2006).



The above argument yields only a weak, restricted version of NXS, the
ban on extended simples. It leaves open the possibility of extended
simples that are not material objects.



NXS does follow from Supersubstantivalism+, according to which it's a
necessary truth that each entity is identical to anything at which it
is exactly located. For suppose that *x* is exactly located at
*y* and that *y* is complex. Then, given
Supersubstantivalism+, *x*=*y*, hence *x* is complex
too.


### 5.5 Against Extended Simples #2: from Qualitative Variation



A second argument against extended simples arises from problems about
simples whose intrinsic properties vary across space or spacetime:



(12)
If it is possible for there to be an extended simple, then it is
possible for there to be an extended simple that exhibits intrinsic
qualitative variation across spacetime--e.g. an extended simple
that is white in one region and grey in another. (Colors are just
placeholders; presumably more realistic examples are available.)
(13)
Necessarily, for any *x*, if *x* varies across spacetime
with respect to its intrinsic properties, then there are incompatible
intrinsic properties *F* and *G* and parts
*x*1 and *x*2 of *x* such that
*x*1 instantiates *F* and *x*2
instantiates *G*.
(14)
Necessarily, for any *x*, if there are incompatible intrinsic
properties *F* and *G* and parts *x*1 and
*x*2 of *x* such that *x*1
instantiates *F* and *x*2 instantiates *G*,
then *x* is not simple.
*Therefore*
(15)
It is not possible for there to be an extended simple.


The argument is valid and (14) is clearly true. If *x* has a part
that instantiates *F* and a part that instantiates *G*,
then, provided that these properties are incompatible, the parts in
question are not identical. So *x* has at least two parts. So at
least one of them must be a *proper* part of *x*, in which
case *x* is not simple.



One might resist the argument by denying (12) and saying that a simple
can be extended but only if it is qualitatively homogeneous across
spacetime. (Spencer 2010 denies the possibility of heterogeneous
extended simples. Jaeger 2014 responds; Spencer 2014 replies.) But
most friends of extended simples grant (12) and reject (13). This can
be done in at least four ways.


#### 5.6.1 The relativizing approach



The relativizing approach says that if a simple, *o*, is white in
one region and grey in another, then *o* bears the being white at
relation to the first region and the being grey at relation to the
second; it need not have a part that instantiates the property
whiteness or a part that instantiates the property greyness.
Alternatively, one might treat instantiation as a three-place
relation, and say that *o* instantiates the property whiteness at
one region and instantiates the property greyness at another.



Both versions of this approach are based on theories of persistence
and change that attempt to reconcile eternalism and the B-theory of
time with endurantism, the view that objects are wholly present at
multiple times and do not have temporal parts (for more on
endurantism, see
 section 6.3).
 Accordingly, the standard criticisms of the relevant theories of
persistence and change (on which see Haslanger 2003) would seem to
apply equally to the relativizing approach to the problem of
qualitative variation for extended simples. The main criticism is
that, by appealing to relations to regions, the relativizing approach
makes an object's color distribution (or whatever) a dyadic relation
or an extrinsic property, when it is in fact (says the critic) an
intrinsic property, involving only the object, its parts, and the
intrinsic properties of these things.


#### 5.6.2 The stuff-theoretic approach



Markosian (1998, 2004a, 2015) distinguishes between things (e.g.,
material objects) and stuff (e.g., matter). Stuff, according to
Markosian, comes in portions, but no portion of stuff is a thing. Both
things and portions of stuff can have parts, but any part of a thing
is itself a thing, and any part of a portion of stuff is itself a
portion of stuff. Crucially, each thing is *constituted by* a
portion of stuff. With this framework in place, Markosian offers the
following account. There can be extended simple things, but there
cannot be extended simple portions of stuff. When an extended simple,
*o*, is white in region *r**w* and grey in
region *r**g*, this isn't because *o* itself
has a part that is exactly located at *r**w* and
instantiates whiteness and another part that is exactly located at
*r**g* and instantiates greyness; rather, it's
because the portion of stuff that constitutes *o* has such parts.
This preserves the simplicity of the extended thing. It may also avoid
the complaint, facing the relativizing approach, that it makes the
color distribution of an extended simple an extrinsic matter. The
relation between a thing and the portion of stuff that constitutes it
is apparently more intimate than the relation between a thing and the
region at which it is exactly located. Moreover, the stuff-theoretic
approach, unlike the relativizing approach, holds that when an
extended object is white in one region and grey in another, the
monadic, intrinsic properties whiteness and greyness (not merely the
dyadic relations *being white at* and *being
grey at*) are both instantiated. However, some
may doubt that a complex entity could constitute a simple one.


#### 5.6.3 Distributional properties



Parsons (2000) proposes that if a simple is white in one region and
grey in another, then it has a fundamental, intrinsic,
*distributional property*. Some distributional properties, such
as *being black all over,* are *uniform*.
Others, such as *being polka-dotted* are
*non-uniform*. When a simple has a non-uniform distributional
property, this fact is not grounded in its having parts, configured in
a certain way, that each have simpler, uniform properties. Nor is it
grounded in the simple's standing in incompatible relations (*being
white at* and *being grey at*)
to different spacetime regions. Rather, it is an ungrounded fact about
the simple. This apparently avoids the worries about extrinsicness
that face the relativizing approach, and it makes no appeal to
Markosian's unorthodox notion of constitution.



As McDaniel (2009) notes, however, Parsons is apparently forced to
treat facts of the form **x* is *F* at *r** as
unanalyzed and ungrounded, which one might take to be a drawback. In
Figure 5, the objects *o*1 and *o*2
presumably have the same maximally determinate distributional color
property: roughly, *being grey in one half and white in
the other.*





Figure 5


![Figure 5: Two rectangles, one, titled 'Object 'o1', with the left half white (labeled 'r1w') and right half grey (labeled 'r1g'), both halves bracketed and labeled 'r1'. The second, titled 'Object o2', with the left half gray and the right half white, the whole labeled 'r2'](index-fig5.png)


Call this property **P**. Further, object
*o*1 is grey at region *r*1*g*.
What grounds this fact? Not the fact that *o*1 has
*P*. After all, *o*2 also has *P*, but it is
not grey at *r*1*g*. Not the fact that
*o*1 has *P* and is exactly located at
*r*1. For it is possible that there be a thing that
has *P* and is exactly located at *r*1*g*
but is not grey at *r*1*g*: this is the case,
e.g., in the possible world that results from deleting
*o*1 and putting *o*2 in its place
(without rotating it clockwise or counterclockwise). Thus, in addition
to his basic distributional properties, Parsons is apparently forced
to treat relations to regions (e.g., *being grey
at*) as being basic as well.



The relativizing approach has no such problem: it treats relations to
regions as basic and analyzes distributional properties in terms of
them. The stuff-theoretic approach avoids the problem too. It analyzes
facts of the form *x* is *F* at *y* as: *either
*x* or the portion of stuff that constitutes *x* has a part
that is exactly located at *y* and is *F**.


#### 5.6.4 Tropes



McDaniel (2009) attempts to avoid all the above problems by appeal to
a theory of tropes developed by Ehring (1997a,b). The idea is that a
simple can be grey in one region and white in another region by
instantiating a greyness trope that is exactly located at the first
region and instantiating a whiteness trope that is exactly located at
the second region. This permits an analysis of facts of the form
**x* is *F* at *y** as **x* instantiates
an *F*-ness trope that is exactly located at *y**; it
makes no appeal to stuff or portions of stuff; and, since it invokes
*monadic* color tropes, it (arguably) refrains from treating
the facts about a simple's color distribution as being fundamentally
relational and extrinsic.



For further arguments against extended simples, see Section 2.2 of the
supplementary document
 Additional Arguments.


## 6. Multilocation



To say that an object is multilocated is to say that it has more than
one exact location: '*x* is multilocated' means
'[?]*y*1[?]*y*2[*L*(*x*,
*y*1) & *L*(*x*, *y*2)
& *y*1[?]*y*2]'. We
consider a series of putative *examples* of multi-location in
 Section 6.3.



The debate over multilocation concerns



Functionality+
#[?]*x*[?]*y*1[?]*y*2[[*L*(*x*,
*y*1) & *L*(*x*, *y*2)]
- *y*1=*y*2]
  

*Necessarily, nothing has more than one exact location.*


Opponents of multilocation accept Functionality+. Friends of
multilocation typically want to affirm something stronger than the
negation of Functionality+. They typically accept the possibility of
an entity that is exactly located at each of two regions that do not
even overlap.



Earlier we glossed '*x* is exactly located at
*y*' as '*x* has (or has-at-*y*) the same
size and shape as *y*, and stands (or stands-at-*y*) in all
the same spatiotemporal relations to things as does *y*'.
Thus spheres are exactly located only at spherical regions, cubes only
at cubical regions, and so on. When an entity is said to be
multilocated, then, it is said to stand in this relation to *each
of several regions*: informally put, it has the same size, shape,
and position as region *r*1; it has the same size,
shape, and position as region *r*2; and so on. No
claim is made to the effect that the object is exactly located at the
*fusion* of *r*1, *r*2,
..., or at any *proper parts* of any of these regions.



To clarify the idea of multilocation in an informal way, it may be
useful to consider Figure 6, inspired by Hudson (2005: 105) and
Kleinschmidt (2011: 256).




Figure 6


![Figure 6a: Two rows of two circles each. The top row circles are grey and bracketed with a label of 'o1'. The bottom row circles are white and have dotted borders; the left one has a label of 'r1' and the right a label of 'r2'. The bottom row circles are also bracketed with a label of 'r3'. An arrow points from the top row to the bottom row.](index-fig6a.png)

A scattered, singly located
  

object




![Figure 6b: The top row has a single grey circle with the label 'o2'. Arrows point from it to two circles on the bottom row; each of which is white with a dotted border; the left is labeled 'r3' and the right, 'r4'.](index-fig6b.png)

A non-scattered, multilocated
  

object





The object *o*1 is scattered: its shape is that of the
sum of two non-overlapping circles. It is not multilocated. Rather, it
has just one exact location: the scattered region
*r*3. It is not exactly located at any proper part of
that region, such as *r*1 or *r*2.



The object *o*2 is multilocated. It has two (and only
two) exact locations. It is exactly located at the circular region
*r*3; and it is exactly located at the circular region
*r*4, which does not overlap *r*3. It
is not exactly located at their fusion, and it is not located at any
of their proper parts. Since *o*2 is exactly located
at *r*3, which is circular, *o*2 is
circular, at least *at *r*3*. For parallel
reasons, *o*2 is circular at *r*4. By
contrast, *o*1 is not circular *simpliciter*,
nor is it circular at any region.



Everything we have said so far is neutral with respect to whether
either of the material objects is simple. It may be that both objects
are simple, or that both are complex, or that *o*1 is
simple and *o*2 is complex, or *vice versa*. This
is worth emphasizing, since questions about the possibility of
extended simples and questions about the possibility of multilocation
are sometimes run together.



It is natural to think that if these two objects were visible, they
would be visually indistinguishable. Indeed, it is tempting to think
that there would be no empirical difference between
*o*1 and *o*2. For those with
verificationist leanings, this may lead to the belief that there is no
difference at all between *o*1 and
*o*2 and hence that there must be something defective
about the initial set-up of the case.


### 6.1 For Multilocation #1: from Conceivability



As with interpenetration and extended simples, one might offer a
conceivability argument for the possibility of multilocation. One
could claim that multilocation is conceivable and take this to be
evidence that multilocation is possible. Since this argument does not
appear to raise any issues that are specific to multilocation, we will
move on.


### 6.2 For Multilocation #2: from Recombination



It is natural to think that the extant recombination arguments for
interpenetration and extended simples can be adapted to yield an
argument for multilocation. If one slightly alters the passage (quoted
in
 Section 4.4)
 in which McDaniel presents his recombination argument for
interpenetration (2007a: 241), one gets the following, parallel
argument for multilocation:



> 
> 
> The actual state of affairs in which an object *x* is exactly
> located at a particular region *r* is distinct from the merely
> possible state of affairs in which object *x* is exactly located
> at some other, disjoint region *r*\*. From the fact that *x*
> is exactly located at *r*, we can infer nothing about what, if
> anything, is exactly located at *r*\*. Both states of affairs are
> contingent: they possibly obtain and possibly fail to obtain. If any
> recombination of contingent states of affairs yields a genuine
> possibility, then there are possible worlds at which *x* is
> exactly located both at *r* and at *r*\* and hence is
> multilocated.
> 
> 
> 



This argument appears to have the same virtues and vices as McDaniel's
original argument for interpenetration.



(Some recombination arguments for interpenetration and extended
simples cannot be adapted to yield an argument for multilocation. See
Section 3.2 of the supplemetary document
 Additional Arguments.)


### 6.3 For Multilocation #3: from Examples



A third kind of argument for the possibility of multilocation simply
points to examples. Entities that have been taken to be multilocated
include: immanent universals, enduring material objects, enduring
tropes, four-dimensional perduring objects, backward time travelers,
fission products, transworld individuals, works of music, and an
omnipresent God.


#### 6.3.1 Immanent universals



As we have noted, immanent realists say that universals are
spatiotemporal entities that are in some sense 'wholly
present' in the things that instantiate them. This seems to be
the view under discussion in the following passage from Plato's
*Parmenides*, 131a-131b (in Hamilton and Cairns 1961:
925):



> 
> 
> Parmenides
> Do you hold, then that the form as a whole, a single thing, is in
> each of the many, or how?
> Socrates
> Why should it not be in each, Parmenides?
> Parmenides
> If so, a form which is one and the same will be at the same time,
> as a whole, in a number of things which are separate, and consequently
> will be separate from itself.
> 
> 



One natural way to translate immanent realism into the terminology of
exact location is via the following principle:



(16)
Necessarily, for any *x*, any *y*, and any *z*, if
*x* is exactly located at *y* and *x* instantiates
*z*, then *z* is exactly located at *y*.


To see how this leads to multilocation, suppose that some monadic
universal *u* is instantiated by an entity *e*1
that is exactly located at region *r*1 and by a
different entity, *e*2, that is exactly located at
region *r*2, disjoint from *r*1. Then,
given (16), *u* itself is exactly located both at
*r*1 and at *r*2 (Paul 2006; Lafrance
2015).



(16) is not inevitable, even for immanent realists. Some of them might
prefer to say that a monadic universal is exactly located only at the
*fusion* of the exact locations of its instances (Bigelow,
1988: 18-27, can in places be read as embracing this, and
Effingham 2015 argues that this is what immanent realists should say).
On this view, a simple monadic universal might be scattered but would
not be multilocated. Others (Armstrong 1989: 99) prefer to say that
universals do not have exact locations at all, though they are parts
or constituents of things that have exact locations or of spacetime
itself. This was dubbed the 'Burying Strategy' in
 Section 4.1.[9]


#### 6.3.2 Enduring material objects



The debate over persistence through time centers around two rival
views, endurantism and perdurantism. Endurantists often say that a
persisting material object is temporally unextended and in some sense
'wholly present' at each instant of its career.
Perdurantists often say that a persisting material object is a
temporally extended entity that has a different temporal part at each
different instant of its career and is at most partially present at
any one instant.



(Informally, an instantaneous temporal part of me is an object that is
a part of me, is made of the exactly same matter as I am whenever it
exists, and has exactly the same spatial location as I do whenever it
exists, but exists at only a single
 instant.[10])



Recently, a number of philosophers have suggested that the traditional
endurantism versus perdurantism dispute runs together a pair of
independent disputes about persistence: a mereological dispute
concerning the existence of temporal parts, and a locational dispute
concerning exact locations (Gilmore 2006, 2008; Hawthorne 2006; Sattig
2006; Donnelly 2010, 2011b; Eddon 2010; Rychter 2011; Calosi and Fano
2015). Stated somewhat loosely, the mereological dispute is between
the following views:


* Mereological perdurance:
there are persisting material objects, and each such object has a
different temporal part at each different instant at which the object
exists.
* Mereological endurance:
there are persisting material objects, but none of them has a
different temporal part at each different instant of its career.
(Perhaps none of them have any instantaneous temporal parts--or
any temporal parts aside from themselves--at all.)



To frame the locational dispute, it will be useful to have one further
piece of terminology. Say that *y* is a *path* of *x*
if and only if *y* is a fusion of the exact locations of *x*
(Gilmore 2006: 204). Informally, a path of an object is a region at
which the object's complete career is exactly located.



We can then state the locational dispute as follows:


* Locational perdurance:
there are persisting material objects, and each of them has exactly
one exact location--its path.
* Locational endurance: there
are persisting material objects, and each of them has many different
exact locations, each such location being instantaneous or
'spacelike'. Typically, each of these exact locations will
count as an instantaneous temporal part of the object's path.



Philosophers on both sides of this dispute can agree about which
spacetime regions are the paths of which material objects. They will
disagree about which spacetime regions are the exact locations of
which objects. The locational perdurantist will say that material
objects are exactly located only at their paths. The locational
endurantist will say that a persisting material object is exactly
located at many regions, each of them a slice of its path. The
interaction between the two disputes about persistence is summarized
in Figure 7 (from Gilmore 2008: 1230).




Figure 7
![A 2 by 2 grid of images. The first column is titled 'Locational Perdurance', the second, 'Locational Endurance'; the first row is titled 'Mereological Perdurance', the second 'Mereological endurance'. The top left image is titled 'A singly located persisting thing with instantaneous temporal parts'. The top right image is titled 'A multilocated persisting thing with instantaneous temporal parts'. The bottom left image is titled 'A singly located thing without temporal parts'. The bottom right image is titled 'A multilocated persisting thing without temporal parts'.](index-fig7.png)


Locational endurance entails multilocation: it says that some material
objects are exactly located at many different regions. Mereological
endurance, which merely rejects temporal parts, does not entail
multilocation. Thus one might reject temporal parts while retaining
Functionality. This is the position of Parsons (2000, 2007). It
corresponds to the lower left-hand box in Figure
 7.[11]


#### 6.3.3 Enduring tropes



Whether or not one takes material objects to be multilocated in the
manner of locational endurantism, one might hold such a view about
tropes. Ehring (1997b, 2011) defends the view that tropes endure,
though he does not say whether he takes them to be multilocated in our
sense. Consider some simple, spatially unextended trope *e*, and
suppose that it persists. Let the temporally extended, one-dimensional
spacetime region *r* be *e*'s path. Then there are at least
two natural things one could say regarding *e*'s locations. One
could say that (i) *e* has just one exact location, *r*, in
which case *e* is a temporally extended simple but not
multilocated. Alternatively, one could say (ii) *e* is exactly
located at each point in *r* and nowhere else, in which case
*e* is multilocated but not extended.


#### 6.3.4 Multilocated 4D worms



Hudson (2001) defends a version of mereological perdurantism according
to which each ordinary material object is exactly located at a great
many (mostly overlapping) four-dimensional spacetime regions. Hudson
offers his view as a solution to
 the problem of the many
 among other things.



(Hudson also argues that, given his view, the fundamental parthood
relation should be taken to be a three-place relation holding between
two objects and a region. See Gilmore (2009) and Donnelly (2010) for
discussion.)


#### 6.3.5 Backward Time Travelers



Suppose that Suzy is born at time *t*0 and dies 90
years later, at time *t*90. At time
*t*40, the forty-year-old Suzy steps into a time
machine and disappears, reappearing (with her time machine) out of
thin air many years earlier. Suzy then enters the nursery in the house
where she was raised and, at *t*1, sees a one-year-old
baby, whom she knows to be herself. (The character Suzy originates in
Vihvelin 1996.)



If one is an endurantist, one will find it natural to describe this
situation as involving multi-location in space or spacetime: Suzy is
exactly located at *r**A*, a region with the size
and shape of an adult; Baby Suzy is exactly located at
*r**B*, a disjoint but simultaneous region with
the size and shape of a baby; and Suzy=Baby Suzy. Hence there is a
single thing (namely, Suzy/Baby Suzy) that is exactly located at two
different, simultaneous regions, *r**A* and
*r**B* (MacBride 1998, Keller and Nelson 2001;
Gilmore 2003, 2006, 2007; Miller 2006; Carroll 2011; Kleinschmidt
2011; Effingham 2015). All of this holds even if one rejects talk of
regions of spacetime in favor of talk of regions of space.



To be sure, even endurantists can deny that the situation involves
multilocation. Those who accept mereological endurantism and
locational perdurantism will say that Suzy does not have temporal
parts but is still four-dimensional in at least the following sense:
she is exactly located at only one region: her four-dimensional path
(Parsons 2000, 2007, 2008). If one is a presentist--one who says
that only the present time and its contents exist--one might say
that, at *t*1, Suzy is exactly located only at a
fusion of *r**A* and *r**B*,
not at either of those regions themselves (Markosian 2004c).



However, if Suzy is exactly located only at the sum of
*r**A* and *r**B* at
*t*1 then she is not shaped like a human being at
*t*1; rather, she is a scattered object shaped like
the sum of two human beings. It might seem doubtful that Suzy can
exist at a time without being even approximately human-shaped at that
time. This would speak in favor of the view that Suzy is multilocated,
rather than scattered, at
 *t*1.[12]


#### 6.3.6 Fission



Dainton (2008: 364-408) argues that if a person were to undergo
fission (in roughly the manner of an amoeba), the person would
continue to exist after the fission in bilocated form: she would be
wholly present in two different places at once. (This view is also
discussed by Johnston 1989). Dainton's view is naturally understood as
involving multilocation, either in spacetime or in space at a time:
after the fission, the person is exactly located at two different
regions (see Gilmore, 2008: 1246, for an objection to Dainton's
view).


#### 6.3.7 Modal Realism with Overlap



McDaniel (2004) develops a version of concrete modal realism that
posits multilocation; he dubs the view *Modal Realism with
Overlap* (MRO). Like Lewis's (1986) modal realism, MRO treats
possible worlds as concrete, spatiotemporal entities and holds that
spacetime regions are 'worldbound': no spacetime region
that is part of one world overlaps any spacetime region that is part
of another world. Unlike Lewis's view, however, MRO claims that
material objects are wholly present in more than one world.
Specifically, MRO entails multilocation. It says that if Lewis is a
philosopher in world *w*1 and a plumber but not a
philosopher in world *w*2, then he is exactly located
at two or more regions (one of which is a part of *w*1
and another of which is a part of *w*2).


#### 6.3.8 Works of Music



Tillman (2011) develops a view about the ontology of musical works
according to which *Beethoven's 9th Symphony* (the
symphony itself, as opposed to any particular performance of it) is
multilocated. The view has it that a work of music is exactly located
at a spacetime region *r* if and only if some performance of that
work is exactly located at *r*. When combined with facts about
the relevant performances, this principle yields the result that
Beethoven's 9th is exactly located at spacetime regions
confined to the 19th century Vienna and also at other
regions confined to 20th century New York; it is
multilocated. Tillman argues that this view is preferable to other
forms of 'musical materialism'. He does not say whether it
is preferable to the view that works of music are abstract objects
with no spatiotemporal location.


#### 6.3.9 God



On some interpretations, the doctrine of divine
 omnipresence
 entails that God is exactly located at each region of space or
spacetime (Hudson 2009, Inman 2017). The relevance of multilocation to
issues in Christian theology, including the real presence of Christ in
the eucharist and the resurrection of the dead, is discussed in Pruss
(2009, 2013), Hudson (2010, 2014), Baber (2013), Effingham (2015), and
Pickup (2015).



For an additional argument in favor of multilocation, see Section 3.1
of the supplementary document
 Additional Arguments.


### 6.4 Against Multilocation #1: from Definition



As we noted in
 Section 2,
 if one defines '*x* is exactly located at *y*'
as '*x* is weakly located at all and only those entities
that overlap *y*'--as Parsons (2007) claims that we
are free to do--then one can argue as follows:



(17)
Necessarily, for any *x* and any *y*, *x* is
exactly located at *y* if and only if for any *y*\*, *x*
is weakly located at *y*\* if and only *y* overlaps *y*\*
(by the definition of 'is exactly located at').
*Therefore*
(18)
So, necessarily, for any *x*, any *y*1, and
any *y*2, if *x* is exactly located at
*y*1 and *x* is exactly located at
*y*2, then *y*1 mereologically
coincides with *y*2 (from (17)).
(19)
Necessarily, for any *y*1 and any
*y*2, if something is exactly located at
*y*1 and something is exactly located at
*y*2 and *y*1 mereologically coincides
with *y*2, then
*y*1=*y*2. (Uniqueness for
Locations)
*Therefore*
(20)
Necessarily, for any *x*, and *y*1 and any
*y*2, if *x* is exactly located at
*y*1 and *x* is exactly located at
*y*2, then *y*1=*y*2
(from (18) and (19).)


To see that the inference from (17) to (18) is valid, suppose that
object *o* is exactly located at regions
*r**a* and *r**b*. Since
*o* is exactly located at *r**a*, *o*
is (by (17)) weakly located at all and only the entities that overlap
*r**a*. Likewise, since *o* is exactly
located at *r**b*, *o* is weakly located at
all and only the entities that overlap *r**b*. So
*r**a* overlaps a given entity if and only if
*o* is weakly located at that entity; and
*r**b* overlaps a given entity if and only if
*o* is weakly located at that entity. Hence
*r**a* and *r**b* overlap
exactly the same entities: in other words, they mereologically
coincide. The rest of the argument is self-explanatory.



The argument may persuade some. However, those who are initially
inclined to take the possibility of multilocation seriously may see
this argument as a reason to doubt the first premise and the
associated definition (Gilmore 2006: 203; Effingham 2015). (A parallel
argument runs through an alternative definition: '*x* is
exactly located at *y*' means 'for any *z*, if
*z* is simple, then *z* lies within *x* if and only if
*z* is a part of *y*'. As we noted in
 Section 2,
 this definition has the additional vice that in the context of gunky
spacetime, it yields the result that everything is exactly located at
every region.)


### 6.5 Against Multilocation #2: from Supersubstantivalism



When taken to be a necessary truth, supersubstantivalism yields
arguments against the possibility of interpenetrating material objects
and extended simple material objects. Given the symmetry and
transitivity of identity, it also yields an argument against the
possibility of multilocated material objects. Since, necessarily, a
material object is identical to each of its exact locations, and since
nothing can be identical to two different entities, no material object
can have two different exact locations.



This leaves open the possibility of multilocation for entities (e.g.,
universals) that are not material objects. This possibility is ruled
out by Supersubstantivalism+, according to which it is necessary that
any entity is identical to anything at which it is exactly
located.


### 6.6 Against Multilocation #3: from Change



Extended simples face a problem arising from qualitative variation.
Multilocated entities face a similar problem. One might think that if
multilocation is possible, then there should be a possible situation
involving a thing *o*1 exactly located at a region
*r*1 and a thing *o*2 exactly located
at a disjoint region *r*2, such that
*o*1 is white, *o*2 is grey, and
*o*1=*o*2. (Again, change in color is
just a placeholder.) Since it's impossible for a single thing to be
both white and grey, one might continue, no such situation is
possible. This might lead one to conclude that multilocation itself is
impossible.



Friends of multilocation reject the initial (conditional) premise,
though not always for the same reason. Some say that multilocation is
possible but only for entities, such as universals or tropes, that do
not vary qualitatively between locations. Others say that
multilocation is possible even for entities that do vary between
locations. These philosophers can choose among the roughly same range
of options that were available to friends of heterogeneous extended
simples, including the relativizing approach, the stuff-theoretic
approach, and distributional
 properties.[13]



For further arguments against multilocation, see Section 3.3 of the
supplementary document
 Additional Arguments.


## 7. Further Issues



We conclude by listing some important questions about which we have so
far said little:


* **Topology of Receptacles**: What topological
constraints are there on the sorts of regions that can be the exact
locations of material objects? For example, is it a necessary truth
that material objects are exactly located only at
	+ three-dimensional regions?
	+ topologically closed regions, regions with an 'outermost
	skin' of points?
	+ topologically open regions, regions without such a skin? (Cartwright 1975; Hudson 2005; Bays 2003; Uzquiano 2006; Smith
2007; Wilson 2008)
* **Topology More Generally**: How do parthood and
location interact with topological notions such as *contact*,
*connectedness*, and *boundary*? (Zimmerman 1996a,
1996b; Casati and Varzi 1999; Donnelly 2004; Hudson 2005: 47-86;
Varzi 2007; Smith 2008; Wilson 2008)
* **Atomism and Gunk**: Can gunky objects inhabit
spacetimes that are composed of simple points? Can simple and/or
point-like objects inhabit gunky spacetimes? (Kant 1756 [1992]; Hudson
2001: 84-90; Simons 2004a; McDaniel 2006; Parsons 2007;
Arntzenius 2008; Russell 2008; Fowler 2008; Uzquiano 2011; Giberman
2012)
* **Relativity**: How are debates about parthood and
location affected by the transition from a pre-relativistic to a
relativistic setting? (Balashov 1999, 2008, 2010; Gibson and Pooley
2006; Gilmore 2006, 2008; Sattig 2006, 2015; Calosi and Fano
2015)
* **Quantum Theory**: How are debates about parthood
and location affected by quantum theory (Pashby 2013, 2016)?
* **Grounding**: How do parthood and location interact
with the notions of metaphysical dependence or grounding (Brzozowski
2008)?
* **Indeterminacy**: How do parthood and location
interact with vagueness and metaphysical indeterminacy? (McKinnon
2003; Hawley 2004; Smith 2005; Donnelly 2009; Barnes and Williams
2011; Carmichael 2011; Eagle 2016a)
* **Personal Ontology**: Are *we* extended
simples? (Lowe 1996, 2000, 2001; Olson 1998)
* **Social Groups**: Are social groups, such as teams
and clubs, located in spacetime? (Effingham 2010b; Hindriks 2013)
* **Topic-Neutrality**: Do things have
non-spatiotemporal locations, such as positions in relations,
propositions, or abstract quality spaces? (Simons 2004b; Cowling
2014b; Gilmore 2014a)
* **Locational Pluralism**: Is there more than one
fundamental location relation? (Fine 2006; Leonard 2014; Kleinschmidt
2016)
* **Meta-metaphysics**: Are there considerations about
location or its interaction with parthood that bear on issues in
meta-metaphysics, or *vice versa* (Sider 2011)?
* **Ordinary Concerns**: How do esoteric philosophical
questions about parthood and location interact with ordinary concerns
of non-specialists (Giberman 2018)?