Shieva Kleinschmidt
University of Southern California
Placement Permissivism and Logics of Location1
There are three options for how we can characterize regions (of space, time, or spacetime)2:
either (i) some regions are mereologically simple (i.e., they have no subregions3 distinct from
themselves) and all such regions are point-sized, (ii) there are some simple regions that are not
point-sized, or (iii) there are no simple regions at all. In this paper I will develop problems that
arise for the second of those options. I will show that if extended4 simple regions are possible, a
liberal recombination principle is true, and some plausible assumptions are correct, then we will
not be able to endorse a parsimonious logic of location – that is, we will not be able to endorse a
theory of location that posits at most one locative primitive. To argue for this, I will show how
the possibility of extended simple regions together with liberal recombination leads to the
possibility of what I will call “Place Cases”, and I will show that Place Cases are incompatible
with parsimonious logics of location. Though there are many ways in which one might respond
to this incompatibility, I will provide reasons for opting to reject extended simple regions.
Here is an overview of what will follow. I will begin in §1 with a presentation of logics
of location, and I will describe some of the locative relations any such logic must account for. In
§2 I will present “Place Cases”, which involve extended simple regions that contain objects
smaller than those regions. I will show that these Place Cases are incompatible with every
possible parsimonious logic of location, because such logics lack the resources to adequately
describe the locative features of these cases. In §3, I will motivate the possibility of these cases,
first by motivating the possibility of the materials I used to build the cases, and next by
motivating recombination principles used to combine those materials. In §4 I will discuss
several ways of attempting to respond to Place Cases, and I will focus on two of the most
1
I am grateful to Andrew Bacon, Matthew Davidson, Kenny Easwaran, Michael Hall, Hud Hudson, Jake Ross, Mark Schroeder,
Ted Sider, and Gabriel Uzquiano for help with this paper, and to Peter van Inwagen for comments on this paper presented at the
2012 Carolina Metaphysics Workshop. I am especially indebted to Achille Varzi, who provided extensive, helpful feedback on
multiple drafts of this paper.
2 In this paper I will focus on regions involving space, but analogous cases for temporal and spatiotemporal regions can be given.
3 For ease of exposition, I will be taking subregionhood to be parthood restricted to regions. Those who wish to resist this claim,
thinking that subregionhood is merely analogous to parthood, are welcome to substitute in talk of subregions when I talk of parts
of regions.
4 I will use ‘extended’ to mean ‘with size, and not point-sized’. Thus, the fusion of two point-sized regions will, for my purposes,
count as extended (though not simple) in spite of being discontinuous and having zero measure.
2
tempting:
rejecting the possibility of the cases via appeal to extrinsicality of shape, and
accepting the possibility of the cases and revising our logics of location in light of them. I will
argue against these responses as well as others, and I will point to the rejection of extended
simple regions as our best alternative.
1. Logics Of Location
There are many different ways in which entities5 can relate locatively to regions: an entity can be
entirely within a region, partly in a region, exactly at a region, etc. A logic (or theory) of
location aims to capture all of the different locative relations entities can bear to regions (be
these places, times, or regions of spacetime), and describe how these locative relations relate
definitionally to one another. Further, it should list the axiomatic constraints on the instantiation
and combination of these relations. The logics of location currently endorsed in print are,
almost6 exclusively, theoretically parsimonious: each says that, of all of the locative properties
and relations, (i) at most one of them is indefinable, and (ii) none has a definition that appeals to
anything beyond locative relations, mereological relations, and basic logical relations (such as
identity).7
The first question to ask is, which relations are the locative ones these logics of location
ought to account for? A rough but perhaps useful guide is this: they are relations corresponding
to combinations of answers to these two questions, for some object somehow present in some
region: “How much of the object is somehow present in the region, and how much of the region
has the object somehow present throughout it?”8 Restricting our answers to “(at least) some”
and “all”, we have four possible combinations of answers: Some/Some, Some/All, All/Some,
I will use ‘entity’ as a fully general term for anything that exists (and I will sometimes use ‘object’ in an almost equally
unrestricted sense, as a general term for anything that exists and is not a region). Our discussion will not be restricted to merely
some kinds of entities, such as material things, events, universals, etc. Though there may be disagreement on what features
entities of different sorts must have in order to bear particular locative relations to regions, the locative relations themselves
should be accounted for in any adequate theory of location.
6 Arguably, Kit Fine (2006) and Peter Simons (2014) have presented theories that are not theoretically parsimonious, for they
posit a difference between the way objects are located in space (and the way events are located in both time and space), and the
way objects exist in time (as universals might be said to exist in space). However, both theorists endorse parsimonious logics of
locative relations between objects and spatial regions, and my cases will be incompatible with those logics.
7 To read more on theories of location, see Varzi 2007, Gilmore 2014, and Kleinschmidt 2015. I will not discuss theories on
which regions and objects bearing locative relations to them are not ontologically independent from one another, nor theories on
which mereological relations are reducible to locative relations or vice versa (see Markosian 2014 for an example). These
theories are also incompatible with the possibility of Place Cases, but for reasons independent of those I will discuss.
8 This is intended as a rough guide to a range of locative relations. It is not intended as a starting point if we have no
understanding of any locative relations whatsoever (the questions themselves invoke locative relations), and it is not intended to
generate a complete list. Further, different senses of ‘in’ and ‘all’ generate different relations. For more on this, see
Kleinschmidt forthcoming.
5
3
and All/All. Each of these roughly corresponds to a different kind of locative relation. Though
we may disagree about which relations should count as locative ones, these should certainly
make the cut. And the leading theories of location each provide different ways to cash out these
(or roughly these) relations, with Roberto Casati and Achille Varzi being the first to divide up
the space of locative relations in roughly this way.
Casati and Varzi (1999) believe we can define all locative relations in terms of a single
primitive, exact location. Intuitively, this is the relation a thing bears to the region that is shaped
just like it,9 the same size as it, and which the thing completely fills and does not spill out of.
(Thus, all of the entity is in the region, and all of the region has the entity present throughout it.)
They then provide the following definitions.

x is generically located at r =df there is a part of x that is exactly located at a subregion of r.

x is partly located at r =df there is a part of x that is exactly located at r.

x is wholly located at r =df x is exactly located at a subregion of r.10
Josh Parsons (2007) presents a similar but not equivalent way of accounting for this space
of relations. Parsons believes that we can define all the locative relations in terms of a single
primitive, though he believes it does not matter which relation we take to be primitive. So, on
one way of endorsing his system of location relations, we can take weak location as primitive.
An object is weakly located in a region iff the region is not free of the object. So the object can
be contained within the region or not, and it can fill the region or not. This is the easiest locative
relation to instantiate. Parsons then give these definitions:

x is pervasively located at r =df x is weakly located at every subregion of r.

x is entirely located at r =df x is weakly located at r and not weakly located anywhere disjoint
from r.

x is exactly located at r =df x is pervasively and entirely located at r.11
Generic and weak location roughly correspond to our Some/Some answer: with each
relation, at least some of our object is located within at least some of our region. Partial and
pervasive location each roughly correspond to the Some/All answer: with each, the object fills
the relevant region. Whole and entire location each roughly correspond to the All/Some answer:
9
If you think objects are extended in time as well as in space, the region they exactly occupy will be temporally extended as well.
Casati and Varzi 1999, p. 120-121. See Varzi 2007 for more on this system and how it relates to other theories of location.
11 These definitions are from Parsons 2007, p. 203.
10
4
with each, the region in question contains our object. And exact location, for both theorists,
corresponds to the All/All answer; all of the object is within the region, and the object is present
throughout the region. As their definitions show, however, they disagree about how to best
capture these relations.
Casati, Varzi, and Parsons do, however, agree that each entity can have at most one exact
location.
Many have called this into question, wanting to allow for the possibility of
multilocation, where entities (such as immanent universals, 12 subatomic particles, 13 or timetravellers14) can be, in some sense, exactly present at more than one location. Thus, when Casati,
Varzi, and Parsons say that “all” of an entity is within a region, they may mean this in in no
sense present in any region disjoint from that region. However, when the multilocation theorist
says that “all” of an entity is present within a region, they may mean simply that the parts of the
entity within that region fuse to make the entity. This allows for the possibility of the entity also
having parts present outside of the region. Hudson (2001) presents a logic of location that allows
for multilocation; his primitive, located at, is similar to exact location but is non-functional.
In what follows, for simplicity I will use Parsons’ terminology and definitions, though I
will mainly rely on our intuitive notions of the relations these systems point to. And the
arguments I give will apply to any parsimonious logic of location. The important take-away
from this section is that leading proponents of logics of location believe that we can capture all
of the locative relations using at most one locative primitive together with Mereology and basic
logic. Now on to the cases that challenge this.
2. The Problematic Cases
The Place Cases I will discuss all have this feature: each involves some object, x, and some
mereologically simple region, r, such that r fails to have a subregion completely free of x (since r
is simple, its only subregion is itself), and yet, intuitively, x does not fill r. In §2.1, I will present
such a Place Case and a counterpart of it. In §2.2 I will argue that no parsimonious logic of
location can distinguish between the case and its counterpart with respect to locative features. I
will make claims about what some of those locative features are, but what is crucial to my
12
Armstrong 1997
Parsons Unpublished 2003
14 Effingham and Robson 2007, Gilmore 2009, and Kleinschmidt 2011.
13
5
argument is merely that there are some locative differences that our parsimonious logics of
location are blind to.
2.1. Place Cases
The most straightforward example of a Place Case is the following:

Almond in the Void: There is an extended, simple
region, r, and an almond (and its parts) which is
smaller than r and seems to be entirely located in r.15
Region r is otherwise empty, and there are no other
regions.16
In this case, the almond does not fill the region, although every subregion of the region (namely,
the region itself) is not free of the almond.
To make it especially clear that the object is not exactly or pervasively located at the
region, imagine a variant of the case in which there are two almonds in the region. If one
almond in the region would fill the region, then two almonds in the region should each fill the
region, and they will be superimposed. But intuitively, if the region is big enough, both almonds
could be within the region without this happening. On another variant case, an extended simple
region contains something that is, intuitively, point-sized. Something point-sized is not big
enough – or, if we worry about how to specify its size: it is not made of enough matter – to fill
an extended region. But still, the region does not have any subregions that are free of the
object.17 I will set aside these variants and focus on Almond in the Void, but each of these cases
would be sufficient to cause the problems I have in mind.
The almond case and its variants can be contrasted with their less problematic
counterparts (hereafter, ‘Place Case Counterparts’), which do not differ from the problematic
cases in terms of (I) the number and kind of entities involved, (II) the mereological properties
and relations that are instantiated, and (III) facts about which regions are free of the object. They
15
I have purposely avoided invoking non-relative size. And, if complex objects in extended simple regions seem particularly
problematic, consider a simple object in the extended simple region instead.
16 One might worry that this requires the possibility of a world containing just an almond and some regions. But I do not require
that my cases correspond to worlds; they can correspond merely to world-parts.
17 Another variant: a region that is extended, simple, and discontinuous, which contains a (zero-dimensional) grain. In this case,
it is even easier to see that the object does not fill the region: the region is discontinuous and the object is not.
6
do differ, however, in either the size of the regions or the size of the objects. The unproblematic
counterpart to Almond in the Void is:

Almond in its Shadow: There is an extended, simple region,
r’, and an almond (and its parts) which is exactly the same
size and shape as r’, and which seems to be entirely located
in r’. Region r’ is otherwise empty, and there are no other
regions.
In this case, the almond is intrinsically the same as it was before. And the region has the same
mereological features (it is simple, as before). But the region has been shrunk down to perfectly
fit the almond (or the almond has expanded to fill the region). Not only is no subregion of the
region entirely free of the almond (as before), but in this case, the almond fills the region. So,
intuitively, the locative relations in Almond in its Shadow and Almond in the Void are different.
2.2. The Conflict
In Almond in the Void, the almond is present in the extended region but does not fill it, even
though the region has no subregions that are free of the almond. We want to say that the locative
relations the almond bears to the region are different in this case than in the Almond in its
Shadow case, where the almond does fill the region it is in.
Because we are assuming a parsimonious logic of location is correct, we are allowed only
one locative relation that is not defined in terms of the others. Suppose we take this to be a
relation that is held fixed between the cases. For instance, suppose we take the primitive relation
to be weak location, or entire location. Almond in the Void and Almond in its Shadow do not
differ with respect to instantiation of either of these relations. But the mereological and basic
logical properties and relations are all also the same between the cases. Because all of our other
locative features are defined in terms of these properties and relations together with our primitive
weak or entire location relation, and these are all held fixed between the cases, the instantiation
of exact and pervasive location relations must be held fixed between the cases as well. So we
either have to say that the almond fills its region in both the Shadow and the Void case, or that it
fills its region in neither. But intuitively, it does not fill its region in the Void case, and it does
fill its region in the Shadow case.
7
If we want to avoid this result, we might take as basic a relation that we think differs
between the cases, such as pervasion or exact location. Suppose, for instance, following Casati,
Varzi, and Parsons (on one reading), we take exact location as basic and define the other locative
relations in terms of it. For instance, following Parsons, we would say something like: an object
is weakly located at a region iff that region mereologically overlaps with the region at which the
object is exactly located.18 But then we will have the result that, in the Void case, our almond is
not weakly located anywhere. Because it is not exactly located at any region, and our definition
of weak location requires the entity to be located somewhere, the almond will not even be
weakly present in any regions. It may as well be the number seven for all of the locative
relations it bears to spacetime. This is unacceptable. And we will get the same result regardless
of which relation we choose to take as primitive (be it exact location, pervasion, or Hudson’s
located at relation) and which relation we’re defining in terms of it (be it generic location, weak
location, etc), if the instantiation of the non-basic relation requires instantiation of our primitive
relation.
Suppose we want our primitive locative relation to be one that is instantiated in one of the
Void or Shadow cases and not the other, but we do not want all of our non-basic relations to each
be instantiated in exactly one of those cases. To achieve this, we can define our non-basic
relations more permissively, so that an object can stand in a non-basic relation to a region even if
the object does not stand in the primitive relation to any region. For instance, we can take exact
location as basic, and then say: an entity, x, is weakly located at a region, r, iff if x is exactly
located somewhere, then r mereologically overlaps a region at which x is exactly located. If we
say this, we can claim that, though there is no region at which our almond is exactly located in
the Void case, the almond is nonetheless weakly located in the extended simple region. This is
because the condition for weak location at a region will be vacuously satisfied: since the almond
is not exactly located anywhere, it is vacuously true that if the almond is exactly located
somewhere, then the extended simple region overlaps the region at which it is exactly located.
However, there are two problems with this. First: the number 7, if it is immaterial, will
satisfy this condition for being weakly located at every region. Consider any region, r: if the
number 7 is exactly located somewhere, then r mereologically overlaps that region. Since 7 isn’t
located anywhere, the conditional is vacuously satisfied. So, 7 is weakly located everywhere.
Parsons 2007, p. 204. If we follow Casati and Varzi in taking exact location to be primitive, we’ll take an entity to be
generically located in a region iff the entity has a part that is exactly located at some subregion of that region.
18
8
Here is the second problem. We can imagine a variant of the Void case in which there is
another region disjoint from the original one, which is the same size and shape and is empty.

Almond in the Void+: There
is an extended, simple region,
r, which contains an almond
(and its parts). The almond is
smaller than r, and r is
otherwise empty.
There is
exactly one region disjoint from r: an empty duplicate of r.
Using our more permissive definition of weak location, in this Void+ case the almond will count
as being weakly present in both regions, though it is clearly weakly present only in one. The
general problem is that, if we opt for a definition of a non-basic locative relation that does not
require instantiation of the basic locative relation, then we will not be able to get correct results
about which regions entities bear the non-basic locative relation to.19
So the basic moral is this: if we endorse a parsimonious logic of location, we get at most
one basic location relation. If we take that to be a relation that is held fixed between the Shadow
and Void cases, we will have to say that all the non-basic relations are the same between the
cases, too. If instead we take our basic relation to be one that is instantiated in exactly one of the
cases, then for any non-basic relation, if the non-basic relation is defined in a way that requires
instantiation of the basic relation, the non-basic relation will not be instantiated in the other case.
If instead the non-basic relation is defined more liberally, then we will not always get the right
results about which regions objects bear the non-basic relation to. More formally:
The Argument For Almonds:
19
The example just presented involved taking as basic a relation that the almond stood in to a region in only the Shadow case.
But similar difficulties arise when we take as primitive a relation the almond stand in only in the Void case. For instance,
suppose we posit a new relation like this one: being weakly located at a region without filling it. If this is basic, we can say: if
an object bears this primitive relation to a region, then it is weakly located there. And if it bears this primitive relation to a
region, then it does not fill the region. So we can claim that the almond does not fill the extended simple region in Almond in the
Void, though it is weakly present in every subregion of that region. Unfortunately, here, we have taken as basic a relation that no
object bears to any region in Almond in its Shadow. Thus, there are not any locative differences between this almond and a
genuinely immaterial object like the number 7. Any definition of a non-basic locative relation will give the same treatment to
each: either both stand in the relation to a region in Almond in its Shadow, or neither does. And we face this dilemma: for any
non-basic locative relation, if, by definition, its instantiation requires the instantiation of our basic relation, then the almond in
Almond in its Shadow will not bear the non-basic relation to any region. Alternatively, if the non-basic relation is defined more
permissively, then the almond will bear it to all relevantly similar regions in any cases like Almond in its Shadow+ (i.e., Shadowlike cases with multiple regions that are intrinsic duplicates of one another).
9
1. If a parsimonious logic of location is correct, then there is at most one basic locative
relation, and instantiation of this relation is either held fixed between the Shadow and
Void cases or it is not.
2. If a parsimonious logic of location is correct and instantiation of the basic locative
relation is held fixed between the Shadow and Void cases, then everything (including
the basic locative relation) we have available with which to define the non-basic
locative relations will be held fixed between the Shadow and Void cases.
3. If everything (including the basic locative relation) we have available with which to
define the non-basic locative relations is held fixed between the Shadow and Void
cases, then the instantiation of all locative relations will be held fixed between the
cases.
4. It’s not the case that the instantiation of all locative relations is held fixed between the
Shadow and Void cases.
5. Either (a) every non-basic locative relation is defined so that its instantiation requires
instantiation of the basic relation, or (b) there is some non-basic locative relation that
is defined so that its instantiation does not require instantiation of the basic relation.
6. If instantiation of the basic locative relation is not held fixed between the Shadow and
Void cases, then (a) is false, for it would entail that in one of these cases no non-basic
locative relations are instantiated.
7. Option (b) is false, for if it were true of some non-basic relation, we would not have
the resources in every case to pick out which regions entities bore that relation to.20
8. So, no parsimonious logic of location is correct.
Premises 1, 2, 3, and 5 are truths of logic and/or are taken to follow from stipulations about what
it is for a logic of location to be parsimonious and stipulations about Place Cases. Premises 4, 6,
and 7 are based on our intuitions about locative features in particular cases described above.
20
We may consider is locatively excluded from to be a locative relation, and one that does not require the instantiation of any
other locative relation in order to itself be instantiated. In that case, feel welcome to restrict our domain of locative relations in
premises 5, 6, and 7 to what we might call “positive” locative relations, which involve at least some of an object being within at
least some of a region. (That is: we’re excluding relations corresponding to answers of “none” that we may have given to the
questions at the start of §1.)
10
3. Motivation
There are multiple routes to the possibility of my cases. The two pieces of motivation I will
discuss will involve these components: (A) positing the possibility of a continuous extended
simple n-dimensional region, and the possibility of a continuous object that is smaller than the
extended simple region, and (B) endorsing some principle(s) telling us how possibilities can be
recombined to generate other possibilities.21
I will forego discussion of motivation for extended simple regions, and simply direct the
reader to literature on that topic.22 In addition to requiring that extended simple regions are
possible, (A) also requires the possibility of some objects that are smaller than some of those
extended simple regions. Here is one route to thinking that such objects are possible. It is
plausible that the number of spatial dimensions is contingent; for instance, even if space is, say,
three-dimensional, it seems possible for it to be four-dimensional. If there are extended simple
regions, and the dimensionality of space is contingent, it also seems plausible that the
dimensionality of the smallest simple regions is contingent. For instance, we may believe that
simple regions are extended a Planck-length in every dimension there is. Finally, we it is
plausible that, for any world, there is some possible object as small as that world’s smallest
region. These claims jointly entail the possibility of some possible object that is smaller than
some possible simple region, and these are the components of (A). Let us now turn to options
for (B).
First, consider Lewis’s Patchwork Principle23 (roughly formulated):

The Patchwork Principle: Necessarily, for any two possible chunks of spacetime and
their inhabitants, and any way of making them adjacent (that is, any way of arranging
them so there is zero distance between them without their being superimposed), there is
some world in which chunks of spacetime just like those, with inhabitants just like those
had by the originals, are adjacent in that way.24
This entails that if, possibly, there is a normal region (with whatever mereological
features you think regions tend to have) that contains an almond and is otherwise empty, and
21
It is important to note that, since we only require the analytic possibility of Place Cases to cause problems for parsimonious
logics of location, these principles of recombination need only be true with respect to generation of analytically possible cases.
This makes the principles much harder to reject. See §4.3 for more discussion of this.
22 See Tognazzini 2006, McDaniel 2007, Dainton 2010, and Spencer 2010. See Gilmore 2014 for an overview of arguments for
extended simple regions.
23 Lewis 1986, pp. 87-89.
24 A more formal statement of this version of Lewis’s principle can be found in Hawthorne, Scala, and Wasserman, 2004. They
call this “A Combinatorial Principle for Objects”. Also, note that in appealling to distance relations I may be presupposing that
any spaces this principle appeals to or produces have a metric. I am happy to work with a version of the principle that is
restricted in this way.
11
possibly there is an empty extended simple region (say, of 1 cubic foot), and there’s a way for
these regions to be arranged so that there is zero distance between them, then there is some world
in which a normal region containing an almond is zero distance from an empty simple region of
1 cubic foot.25
So Lewis’s principle gives us is that, if our ingredients of our Almond in the Void case
are possible, then possibly they are right next to one another. We may now supplement Lewis’s
principle with:

The Pushing Principle 26 : If some object, x, is contained within an otherwise empty
region, r1 (i.e., r1 is not free of x, and every region disjoing from r1 is free of x), and r1
is adjacent to an empty region, r2, and there is an empty continuous path from where x is
in r1 to the interior of r2, and both this path and r2 are large enough, of the right shape,
and of the right orientation relative to r1 for x to fit through the path and into r2,27 then
possibly, x moves28 from r1 to r2 without undergoing intrinsic changes.
As Joshua Spencer (2010) says, if an object has a large enough chunk of empty space in front of
it, then regardless of the mereological structure of the space and of the object, it does not seem
that anything would (at least, in all such cases) stop us from pushing the object into the new
region.29 And it does not seem that the region or object would have to undergo intrinsic changes
due to this move. Applied to the almond next to an extended simple region, it’s hard to see what
would stop you, at least in all such cases, from simply pushing the almond into the empty simple
region.
Endorsing this principle together with Lewis’s principle allows us to generate Almond In
The Void. However, we might worry about motion through regions with unusual mereological
structures. That is, we might not think it is clear how motion across extended, simple regions
would work.30 And it does not seem that the motion is actually doing any work in generating
these possibilities anyway. Instead, the key idea is that it is the size and shape of regions and
objects that determine whether they can stand in location relations. So, instead of appealing to
Note that the adjacency in Lewis’s Patchwork Principle does not guarantee contact; for instance, two open three-dimensional
regions can be at zero distance from one another with a plane between them. For contact, entities must be adjacent and have the
right match of topological features.
26 This is a generalisation of what Spencer (2010, p. 180) appeals to in his argument that extended simple objects can be located
at extended simple regions.
27 It is spectacularly difficult to give fully general, minimally sufficient conditions for an object’s being able to fit through a path
and into a new region. For example, see discussion related to Leo Moser’s Moving Sofa Problem (Moser 1966, Wagner 1976).
28 Feel free to read “possibly, x moves” in terms of temporal and modal counterparts if you prefer.
29 Spencer 2010, p. 180.
30 We might attempt to capture motion within extended simple regions via appeal to differences in distance relations between the
object and regions outside of the one containing it, but this will not work in all cases (for instance, it will not capture motion of an
almond within an extended, simple region that is the only region in its world).
25
12
the Lewisian Patchwork Principle and the Pushing Principle, we might just endorse what I call
Possible Placement Permissivism, which says:

Possible Placement Permissivism: For any x, l, r1, and w1, if x bears locative relation l to
region r1 in world w1, than for any region r2 and world w2, if r2 in w2 is both empty and
the same size and shape that r1 is in w1, then there is some world, w3, in which there is
an intrinsic duplicate of x (from w1) that bears l to an intrinsic duplicate of r2 (from w2).
That is, if an object bears some locative relation to some region, then for any possible, empty
region of the same size and shape, an intrinsic duplicate of the object can bear the same locative
relation to an intrinsic duplicate of the empty region. So, if an almond in my refrigerator is
contained within a region of 1 cubic foot, and possibly, there exists an empty extended simple
region of the same size and shape, then possibly, an intrinsic duplicate of my almond is
contained within an intrinsic duplicate of the extended simple region. Thus, Almond in the Void
is possible.
Lewis’s principle allowed us to recombine possible entities, generating possibilities
where intrinsic duplicates of those entities are adjacent to one another. Similarly, Possible
Placement Permissivism also allows us to recombine possible entities, generating possibilities
where intrinsic duplicates of those entities bear particular locative relations to one another. The
central idea behind Possible Placement Permissivism is this: whether an object can fit in a region
is simply determined by how big the region and object are, and perhaps also by whether there is
anything causally preventing the object from being in the region (such as a territorial God who
keeps regions like this empty in all possible worlds). Which proper parts an object has, and
which proper subregions a region has, are not relevant to whether the object can fit in the region.
4. Responses
There are many ways we may attempt to respond to Place Cases. In this section, I will describe
several of the most tempting options, and give reason for favoring the rejection of the analytic
possibility of extended simple regions.
4.1. Revising Our Logics of Location
One initially tempting option may be to simply accept the possibility of Place Cases. That is,
perhaps we would like to accept the possibility of extended simple regions that can contain
13
objects that are smaller than they are, and revise our logic of location accordingly. This response
will require giving up any parsimonious logic of location we may have hoped to endorse: we
will have to take more than one locative relation to be primitive, or we will have to appeal to
something outside of locative relations, Mereology, and basic logic in giving our definitions
(though it is not clear what else we might appeal to that will do this work).
Here is an example of how we might opt for an additional locative primitive. Suppose
we take both weak location and exact location as primitive, and we do not endorse any principles
that require that if an object is weakly located in a region, then it is exactly located in some
superregion of that region. Then, in the Almond in the Void case, our logic of location will allow
for the almond to be weakly located in the extended simple region, but is not exactly located at
the region. We can say that the almond is exactly located at the region contining it in the
Shadow case, but not in the Void case, while claiming it is weakly located at that region in each
case. We can get similar results with other choices about which locative relations are primitive.
There are some restrictions on which relations we can take as primitive to get the right
results in Place Cases and their counterparts, and if we think multilocation is possible, we will
need at least three locative primitives to get the correct results in all cases. 31 However, the most
pressing problem for revising our logics of location in this way is due to questions about
recombination.
If we have multiple primitive locative relations, we face a dilemma: either we can freely
recombine instantiation of these primitive relations (as we might allow free recombination of
color and shape), or we cannot. That is: we must decide whether the instantiation of any of our
primitive relations places any restrictions on instantiation of any other primitive locative relation.
If not, we will have some very odd consequences: for instance, if weak location and pervasive
location are each primitive and place no restrictions on each other, it will be possible for an
object to be pervasively located at a region while only weakly located at some of the subregions
of that region. Similarly, if exact location and weak location are each primitive and place no
restrictions on each other, it will be possible for an object to be exactly located at a region but not
weakly located there. These results are at least as bad as the worst predictions our parsimonious
logics made about Place Cases. Further, the consequences are so odd that we do not want to
allow for even their analytic possibility. Obviously, we will want some restrictions on how our
primitive locative relations can be co-instantiated, and these will have to be analytically
31
For more on this, see Kleinschmidt forthcoming.
14
necessary restrictions. The problem with this is that it is not clear how we will explain or ground
these restrictions. With a parsimonious logic of location, we have definitional relations between
all of our locative relations, and this explains why facts about how one relation is instantiated
restrict how other relations are instantiated. Without these definitional connections, we lose our
ground for these analytically necessary restrictions. Thus, revising our logic of location by
positing multiple primitive location relations is not only less parsimonious, it also either entails
the analytic possibility of some very strange cases, or it requires analytically necessary
connections that we no longer have the grounds to posit.
4.2. Appeals to Extrinsicality of Shape
One may instead hope to reject the possibilty of Place Cases. For instance, one who thinks that
shape and size are extrinsic believes something along these lines: necessarily, what it is for an
object to have a given size/shape, s, is for the object to be exactly located at a region with
size/shape s.32 My Place Cases seem to require there to be an object that is smaller than every
region it is contained within, and which has a shape that is unlike that of any region in its world.
Such an object does not seem to be exactly located anywhere. But if size and shape are extrinsic,
how can an object have a size when it does not have an exact location? It seems that this would
require an object to have a size that is not inherited. Thus, it seems we cannot endorse both the
extrinsicality of size and shape and the possibility of Place Cases.
In actuality, the situation is far less grim. Instead of producing any incompatibility, Place
Cases can be taken to simply leave us with a size and shape deficit: objects without exact
location may simply lack non-relative size and non-relative shape. So, for instance, in a case
where an extended simple region contains something we would have thought was zerodimensional, we cannot say the inhabitant of the region is point-sized. This view also brings
with it a kind of size-difference and shape-difference blindness: we will not be able to posit any
differences in the sizes and shapes of an almond and a palm tree when both are located within an
32
The extrinsicality of shape has recently been argued for by Kris McDaniel (2007) and Brad Skow (2007), using a Humean
principle of liberal recombination which says roughly: there are no necessary correlations between the intrinsic, inessential
features of ontologically independent relata of fundamental relations. This same principle has recently been used by Kris
McDaniel (2007) and Raul Saucedo (2010) to argue for the possibility of extended simples and for the possibility of some
mismatches of the mereological structure of regions and the mereological structure of the entities located at them (such as when a
simple object is located at an extended region that is not simple, or when a composite object is located at an extended simple
region). If the extrinsicality of shape is incompatible with the possibility of Place Cases, it seems there will be some tension for
the Humean who wants to endorse both arguments. However, I argue in Kleinschmidt 2015 that one can endorse the argument
for extended simples and mereological mismatches without also endorsing the argument for the extrinsicality of shape.
15
ocean-sized extended simple region.
These are very strange results, but they are not
incompatible with the extrinsicality of shape. Place Cases do not require us to claim that objects
contained within extended simple regions have non-relative sizes and shapes. Rather, they
merely require that some objects are present within regions but do not fill any regions. This
seems to require, at most, facts about relative size.
Here is how we might attempt to meet the modest goal of predicating relative size- and
shape-facts in these cases. If we claim that objects can inherit size and shape properties from
regions they are exactly located at, perhaps non-located objects can inherit relative size and
shape properties from regions they bear other locative relations to. For instance, perhaps for an
object to be no larger than size s is just for the object to be contained within a region of a size no
larger than s. And perhaps for an object to be larger than size s is for the object to fill but not be
contained within a region of a size larger than or equal to s.
So, if we believe shape is extrinsic, Place Cases produce some very strange results.
Nonetheless, the extrinsicality of shape is not incompatible with Place Cases, so endorsing the
extrinsicality of shape is not sufficient to give us the tools needed to reject the possibility
(especially the analytic possibility) of Place Cases.
4.3. Rejecting Essential Components of Place Cases
There are other ways one may hope to reject the possibility of Place Cases. For instance, one
may believe that some of the ingredients used to generate Place Cases are metaphysically
impossible, or that the principles of recombination used to generate Place Cases are false.
However, there is a problem with any response to Place Cases that involves merely rejecting
their metaphysical possibility: Logics of Location involve claims about how locative relations
relate definitionally to one another, and those are analytic claims. Thus even the mere analytic
possibility of a violation of one of these connections would be problematic. So, to defend our
Logics of Location from the problems raised by Place Cases, merely rejecting the metaphysical
possibility of Place Cases will not be enough; we must reject the analytic possibility of the cases.
This is harder: for instance, even if one thinks the principles of recombination I presented are not
good guides to metaphysical possibility, it is much harder to claim they are not plausible as ways
of generating analytic possibilities.
16
One may attempt to reject the analytic possibility of Place Cases because they think that
any possible extended simple regions must be same size as the smallest possible simple objects.
For instance, one may take the smallest objects and all extended simple regions to be of a Planck
length in some fixed number of dimensions. Thus, we will not be able to constuct a case where
an extended simple region contains an object smaller than it. However, there are two difficulties
with this response to Place Cases. First, though such a claim may be plausible when restricted to
metaphysical possibility, it is not clear why we would want to accept it when taken to be about
analytic possibility. Second, there are cases that generate many of the same worries our Almond
in the Void case, while taking all of the smallest regions and objects to be of the same size.
Consider:

Out of Place Tiles:
The smallest simple regions and
simple objects are all extended one unit in each of n
dimensions. However, the objects do not line up neatly
with any of the regions. No object fills any region, though
each object has multiple regions that fail to be free of it.
This case does not require that some world contains extended simple regions and objects that are
smaller than those regions. But it will generate the same problems for logics of location that our
Place Cases have raised: it entails that there is some region such that all of its subregions fail to
be free of an object (since the region is simple, it has itself as its only subregion), yet the object
does not fill the region.
To respond to Place Cases and Out of Place Tiles, one may simply reject the analytic
possibility of such cases by rejecting the analytic possibility of extended simple regions. For
instance, one may adopt the response I prefer and claim that the very notion of extension of an
entity can only be understood in terms of distance relations between distinct parts of that entity.
Thus, ‘extended simple’ involves a contradiction in terms.
To conclude this discussion, I will present what I take to be the strongest objection to this
response, and mention two replies to that objection. Here is the objection: one may argue that
we do not actually need extended simple regions to generate many of the worries that Place
Cases raise. For instance, consider this case involving gunky regions (i.e., regions with no
simple parts).33
33
This is a variant on a Stoic mixing case from Daniel Nolan (2006), though it should be noted that I am using the case for
purposes other than those for which it was intended.
17

Grains in Gunk: Region r is a one-dimensional, gunky region such that every subregion
of r has a continuous, extended subregion. Uncountably many red and white point-sized
grains are distributed across r. Between any two red grains there is a white grain and
between any two white grains there is a red grain, and indeed every continuous, extended
subregion of r contains some white and red grains. Further, the fusion of all of the grains
fills r: it is the same size as r, has the same shape, and is completely contained within r
while not spilling outside of r.34 There is also a fusion of all and only the red grains, Red,
and a fusion of all and only the white grains, White.
Note that, if every subregion of r has some continuous, extended subregion, then r will not have
any subregions that are completely free of Red. Nonetheless, Red does not fill r: if it did, and we
treated Red and White in the same way, both would fill r and, contrary to intuition, Red and
White would be superimposed. So, one may think, Grains in Gunk causes many of the problems
for parsimonious logics of location that Place Cases do. In fact, I have presented this case as an
objection to Parsons’ logic of location35, and Matthew Leonard has developed the objection and
applied it to several logics of location36.
There are two responses one may give to this case. First, even if Grains in Gunk is
possible, it is not strong enough to rule out every possible parsimonious logic of location.37 For
instance, Peter van Inwagen38 has pointed out that we can construct Whiteheadian points and use
these to describe the different locative features of our red and white grains; that is, we can talk of
each point-sized grain as being contained within a unique, infinite series of subregions of r. We
can then claim (though this would be contentious) that an object fills a region, R, only if, for
each infinite series of embedded subregions of R, the object has a part contained in every
member of that series. This sort of logic is incompatible with Place Cases, but is compatible
with Grains in Gunk. So Grains in Gunk is not as problematic as Place Cases are.39
Second, we may simply take Grains in Gunk to be analytically impossible. One of the
crucial claims needed for the case to generate problems was the stipulation that every subregion
of our gunky region had at least one continuous, extended subregion (that is, it has no thoroughly
discontinuous subregions). But Peter Forrest (1996), Frank Arntzenius (2008), and Jeff Russell
(2008) have presented worries that gunk without thoroughly discontinuous parts is in conflict
with some analytically true axioms of Mereology. One option in responding to Place Cases,
34
One may believe that a plurality of points, no matter how dense and continuous, cannot fill a gunky region. This is one way to
reject the possibilty of this case.
35 See Parsons 2007, p. 207, fn. 7, and the corresponding section of his paper.
36 Leonard 2014.
37 Thank you to Andrew Bacon and Peter van Inwagen for pointing this out in discussion.
38 Peter van Inwagen presented this point in comments on this paper given at the 2012 Carolina Metaphysics Workshop.
39 For more discussion on this topic, see Kleinschmidt forthcoming.
18
then, is to say that extended simple regions are analytically impossible, and that gunky regions
do not produce similar worries because any such regions, as a matter of analytic necessity, must
have thoroughly discontinuous subregions.
So, we have seen why we may be tempted to posit the analytic possibility of Place Cases,
and the way in which these cases are incompatible with parsimonious logics of location. And we
have discussed the shortcomings of attempting to respond to this incompatibility positing by
multiple locative primitives in our logic, appelaing to the extrinsicality of shapes of objects, and
claiming that any extended simple regions must always be the same size as the smallest objects.
However, we still have the option of rejecting the analytic possibility of extended simple regions,
a response that is most effective when supplemented with the rejection of certain kinds of gunk.
Further, the explanation of the analytic impossibility of extended simple regions, via appeal to an
understanding of extension that involves non-zero distance between distinct parts, seems natural.
Thus, absent strong reason to opt for an alternate understanding of extension, I recommend this
option for defense of our parsimonious logics of location.
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