Chapter 1
SPATIAL REASONING AND ONTOLOGY:PARTS, WHOLES, AND LOCATIONS
Achille C. Varzi
Columbia University[Published in M. Aiello, I. Pratt-Hartmann, and J. van Benthem (eds.), Handbook of Spatial
Logics, Berlin: Springer-Verlag, 2007, pp. 945-1038]

Spatial reasoning is no abstract business. It is, to a great extent, reasoning about entities located in space, and such entities have spatial structure. If the table is in the kitchen, then it follows that the table top isin the kitchen, and it follows because the top is part of the table. If theconcert took place at the stadium, then it didn’t take place in the theater, for concerts are spatially continuous. Even when we reason about
empty places, we typically do so with an eye to the anatomy of their potential tenants: space as such is perceptually remote and we can hardly
understand its structure without imagining what could fill the void.
This general feature of our spatial competence might suggest a deep
metaphysical truth, to the effect that concrete entities such as objects
and events are fundamentally prior to, and independent of, their spatial
receptacles. It might even suggest that space itself is just a fiction, a
picture of some kind: really there are only objects and events spatiallyrelated to one another in various ways. Such was, for instance, the gistof Leibniz’s stern relationism against Newton’s substantivalism, in spiteof the major role the idea of space plays in the sciences. At the sametime, one might argue that our understanding of the spatial structure ofobjects and events, including their spatial relationhips, depends significantly on our understanding of the structure of space per se: that the
spatial features we attribute to objects and events are somehow inheritedfrom those of the spatial regions they occupy. Thus, for example, we are

2inclined to say that ordinary objects have parts insofar as their spatial
regions have parts. We may be inclined to say that the top is part of
the table because of its salience and functional role, but we may just as
easily talk about the top half of a sphere, or its inner parts, in spite of
their lacking any cognitive or functional salience: we identify (and reify)such parts in terms of the parts of the region the sphere occupies.
This tension (not to say this ambiguity) between concrete objectoriented thinking and abstract space-oriented thinking is responsible formany of the philosophical issues that lie behind any formal theory of
spatial reasoning. On the one hand, it is natural (if not necessary) tosupplement the theory with an explicit account of what kinds of thing
may enter into its scope, an account of the sorts of entity that can belocated in space—in short, an account of what may be collected under
the rubric of “spatial entities”. On the other hand, we also want thetheory to be independent of any specific ontological biases we might
have. Whatever spatial entities we are inclined to build into the basic
furniture of the world—subatomic particles, middle-size objects of thegarden variety, large scattered entities such as crowds, forests, archipelagos, galaxies—our reasoning about their spatial properties and relations
appears to be governed by the same general principles, and it is natural to think that such principles must reflect our understanding of thestructural features of the spatial environment in which such entities arelocated. In short, although a theory of spatial reasoning may be viewed
as an example of applied logic, it may also be regarded as an example
of a formal theory whose principles do not necessarily depend on the intended domain of application except to the extent that the domain must
include entities that properly qualify as spatial entities of some sort.
The purpose of this chapter is to take a closer look at these delicate
matters. Rather than doing this in general, however, we shall look at
how the subtle interplay between purely spatial intuitions and intuitions
about concrete spatial entities shows up in the construction of a formal
theory. More specifically, we shall consider three sorts of theory, each ofwhich occupies a prominent position in recent literature: (1) mereology,
or the theory of parthood relations; (2) topology, broadly understood asa theory of qualitative spatial relations such as continuity and contiguity; and (3) the theory of location proper, which deals explicitly with therelationship between an entity and the spatial region it occupies. Arguably, such theories may be viewed as jointly contributing to an overall
appraisal of our spatial competence, and over the last few years there
has been considerable progress in each direction. At this point there issome need for a philosophical pause, and our purpose in this chapter isto go some way in the direction of a systematic assessment.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

1.

3

Philosophical issues in mereology

Let us begin with mereology. This is often defined as the theory of thepart-whole relation, but such a definition is misleading. It suggests that
mereology has something to say about both parts and wholes, which is
not true. As we shall see in Section 2, the notion of a whole goes beyond
the conceptual resources of mereology and calls for topological concepts
and principles of various sorts. By itself, mereology is best understoodas the theory of the parthood relation, regardless of whether the second
term of the relation may be said to qualify as a whole entity. Thus, forinstance, it is a mereological fact that my hand is part of my arm justas it is a mereological fact that it is part of my body, although it may
plausibly be argued that only my body qualifies as a whole (maximallyconnected) object; my arm doesn’t.
It is also worth pointing out that mereology has a long pedigree, whichmakes it a central chapter, not only of formal theories concerned withspatial reasoning, but of any theory in the realm of formal logic and ontology broadly understood. Its roots can be traced back to the early daysof philosophy, beginning with the Pre-Socratic atomists and continuing
throughout the writings of Plato (especially the Parmenides and theThaetetus), Aristotle (the Metaphysics, but also the Physics, the Topics, and De partibus animalium), and Boethius (In Ciceronis Topica).
Mereology occupies a prominent role also in the writings of medieval
ontologists and scholastic philosophers such as Peter Abelard, Thomas
Aquinas, Raymond Lull, and Albert of Saxony, as well as in Jungius’s
Logica Hamburgensis (1638), Leibniz’s Dissertatio de arte combinatoria(1666) and Monadology (1714), and Kant’s early writings (especially the
Monadologia physica of 1756). As a formal theory of the parthood relation, however, mereology made its way into our times mainly throughthe work of Franz Brentano and of his pupils, especially Husserl’s third
Logical Investigation (1901). The latter may rightly be considered thefirst attempt at a rigorous formulation of the theory, though in a format
that makes it difficult to disentangle the analysis of mereological concepts from that of other formal notions (such as the relation of ontological dependence). It is not until Leśniewski’s Foundations of a General
Theory of Manifolds (1916) that a pure theory of parthood was given
an exact formulation. And because Leśniewski’s work was largely inaccessible to non-speakers of Polish, it is only with the publication of
Leonard and Goodman’s The Calculus of Individuals (1940) that mereology has become a chapter of central interest for modern ontologists andlogicians. Indeed, although Leśniewski’s and Leonard and Goodman’s
theories came in different logical guises, they are sufficiently similar to4
be recognized as a common basis for most subsequent developments.
The question that interests us here is how such developments—and thevariety of motivations that lie behind them—reflect and affect our understanding of mereology as a formal theory of spatial reasoning.1

1.1‘Part’ and parthood

To this end, the first thing to observe is that the word ‘part’ has manydifferent meanings in ordinary language, not all of which correspond to
the same relation. In a way, it can be used to indicate any portion of agiven entity, regardless of whether the portion itself is attached to theremainder, as in (1), or undetached, as in (2); cognitively salient, as in(1)–(2), or arbitrarily demarcated, as in (3); self-connected, as in (1)–(3),or disconnected, as in (4); homogeneous, as in (1)–(4), or gerrymandered,
as in (5); material, as in (1)–(5), or immaterial, as in (6); extended, asin (1)–(6), or unextended, as in (7); spatial, as in (1)–(7), or temporal,
as in (8); and so on.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)

The handle is part of the mug.
This cap is part of your pen.
The left half is your part of the cake.
The cutlery is part of the tableware.
This stuff is only part of what I bought.
That area is part of the living room.
The outermost points are part of the perimeter.
The first act was the best part of the play.

All of these cases illustrate the notion of parthood that forms the focus of
mereology. Often, however, the English word ‘part’ is used in a restrictedsense. For instance, it may be used to designate only the cognitively
salient relation illustrated in (1) and (2). In this sense, the parts of an
object x are just its “components”, i.e., those parts that are available asindividual units regardless of their interaction with the other parts of x.
(A component is a part of an object, rather than just part of it; see e.g.
Tversky 1989). Clearly, the properties of such restricted relations may
not coincide with those of parthood broadly understood, so the principles
of mereology should not be expected to carry over automatically.
Also, the word ‘part’ is sometimes used in a broader sense, for instanceto designate the relation of material constitution, as in (9), or the relationof mixture composition, as in (10), or even a relation of conceptual
inclusion, as in (11):1 For a historical survey of mereology, see Henry (1991), Burkhardt and Dufour (1991), andSimons (1991c). For systematic comparisons, see Simons (1987) and Ridder (2002).

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

5(9) The clay is part of the statue.
(10) Gin is part of martini.
(11) Writing detailed comments is part of being a good referee.
The mereological status of these relations, however, is controversial. The
constitution relation exemplified in (9) was included by Aristotle in his
threefold taxonomy (Metaphysics, ∆, 1023b), but many contemporary
authors would rather construe it as a sui generis, non-mereological relation (see Rea 1997 and references therein).2 Similarly, the ingredientmixture relationship exemplified in (10) is subject to controversy, as theingredients may involve significant structural connections besides spatial
proximity and may therefore fail to retain important characteristics they
have in isolation (see Sharvy 1983). As for statements such as (11), it
may simply be contended that the term ‘part’ appears only in the surface
grammar and disappears at the level of logical form, for instance if (11)is paraphrased as “A good referee is one who writes detailed comments”.
For more examples and tentative taxonomies, see Winston et al. (1987),
Gerstl and Pribbenow (1995), and Iris et al. (1988).
Finally, it is worth stating explicitly that mereology is typically construed as a piece of formal ontology, i.e., a theory of certain formal
properties and relations that are exemplified across a wide range of domains, whatever the nature of the entities in question. Thus, althoughboth Leśniewski’s and Leonard and Goodman’s original theories betraya nominalistic stand, reflecting a conception of mereology as a parsimonious alternative to set theory,3 most contemporary formulations assume
no ontological restriction on the field of ‘part’. The relata can be individual entities as in (1)–(8), but also abstract entities such as propositions,sets, types, or properties, as in:
(12)
(13)
(14)
(15)

That premise is part of my argument.
The domain of quantification is part of the model.
The colon is part of the title.
Humanity is part of personhood.

(The example in (11) may perhaps be read as expressing a mereologicalrelation between properties, too.) This “ontological innocence” of mereology plays of course an important role in the appraisal of what principles
2 Actually, if the statue is identified with the lump of clay, as some would argue (e.g. Noonan

1993 vs. Johnston 1993), and if identity is treated as a limit case of (improper) parthood, as
we shall indeed suppose, then the relation of material constitution is a mereological relation.
This, however, is the subject of controversy and we shall come back to it in due time.
3 To be sure, the original calculus of individuals had variables for classes; the class-free version
is due to Goodman (1951). On the link between mereology and nominalism, see Eberle (1970).

6should hold unrestrictedly: greater generality means fewer axioms, andhere the tension between the tasks of an applied logic and those of apurely formal theory shows up most vividly. In the following we focus
primarily on the spatially salient uses of ‘part’, but it is important tokeep this tension in mind when it comes to assessing the philosophicalunderpinnings of the most controversial tenets of mereology.

1.2

Basic principles

With these provisos, let us proceed to unpack the theory. We may ideallydistinguish two sorts of mereological principles. On the one hand, there
are principles that may be thought of as purely “lexical” axioms fixing
the intended meaning of the relational predicate ‘part’. On the other,
there are principles that go beyond the obvious and aim at greater sophistication and descriptive power. Exactly where the boundary should
be drawn, however, is by itself a matter of controversy.

1.2.1
Parthood as a partial ordering.
The obvious is this:regardless of how one feels about matters of ontology, if ‘part’ stands forthe general relation exemplified by all of (1)–(8) above, then it stands
for a partial ordering—a reflexive, transitive, antisymmetric relation:
(16)
(17)
(18)

Everything is part of itself.
Any part of any part of a thing is itself part of that thing.
Two distinct things cannot be part of each other.

As it turns out, virtually every theory put forward in the literatureaccepts (16)–(18), though it is worth mentioning some misgivings thathave occasionally been raised.
Concerning reflexivity, one might observe that many legitimate sensesof ‘part’ do not countenance saying that a whole is a part of itself.
For instance, Rescher (1955: 10) cited the biologists’ use of ‘part’ forthe functional subunits of an organism as a case in point. This is oflittle import, though. Taking reflexivity as constitutive of the meaningof ‘part’ amounts to regarding identity as a limit (improper) case ofparthood. A stronger relation, whereby nothing counts as part of itself,
can obviously be defined in terms of the weaker one, hence there is noloss of generality (see Section 1.2.2). Vice versa, one could frame amereological theory by taking proper parthood as a primitive instead.
This is merely a question of choosing a suitable primitive.
The transitivity principle, (17), is more controversial. Several authors
have observed that many legitimate senses of ‘part’ are non-transitive.
Examples would include: (i) a biological subunit of a cell is not a part ofthe organ of which that cell is a part; (ii) a handle can be part of a doorSpatial Reasoning and Ontology: Parts, Wholes, and Locations

7

and the door of a house, though a handle is never part of a house; (iii) myfinger is part of me and I am part of the team, yet my finger is not part of
the team. (See again Rescher 1955, Cruse 1979, and Winston et al. 1987,
respectively; for other examples see Iris et al. 1988, Moltman 1997, andJohansson 2004 inter alia). Arguably, however, such misgivings stem
again from the ambiguity of ‘part’. What counts as a biological subunitof a cell may not count as a subunit, i.e., a distinguished part of the organ,
but that is not to say that it is not part of the organ at all. Similarly,
if there is a sense of ‘part’ in which a handle is not part of the houseto which it belongs, or my finger not part of my team, it is a restrictedsense: the handle is not a functional part of the house (a “component”),
though it is a functional part of the door and the door a functional partof the house; my finger is not directly part of the team, though it isdirectly part of me and I am directly part of the team. It is obvious that
if the interpretation of ‘part’ is narrowed by additional conditions (e.g.,
by requiring that parts make a functional or direct contribution to thewhole), then transitivity may fail. In general, if x is a φ-part of y and y isa φ-part of z, x need not be a φ-part of z: the predicate modifier ‘φ’ may
not distribute over parthood. But that shows the non-transitivity of ‘φpart’, not of ‘part’. And within a sufficiently general framework this caneasily be expressed with the help of explicit predicate modifiers (Varzi
2005). In any event, it seems clear that spatial parthood is transitive:
whether we construe this as a restricted notion or identify it with thegeneral notion of parthood, (17) holds.
Finally, concerning the antisymmetry postulate (18), two sorts ofworry are worth mentioning. On the one hand, some authors maintain that the relationship between an object and the stuff it is made ofprovides a perfectly ordinary example of symmetric parthood: according to Thomson (1998), for example, a statue and the clay it is madeof are part of each other, yet distinct. This is highly controversial andthere is a large philosophical literature devoted on this topic (see e.g.
the papers in Rea 1997). For the moment, let us simply observe thatthe example trades once again on the ambiguity of ‘part’. We havealready mentioned that material constitution is best regarded as a sui
generis, non-mereological relation. Whether this relation may obtainbetween two spatially coincident objects is an interesting question, but
we should postpone its discussion to where it belongs: the theory of spatial location (Section 3.3). On the other hand, one may wonder about
the possibility of unordinary cases of symmetric parthood relationships.
Sanford (1993: 222) refers to Borges’s Aleph as a case in point: “I saw
the earth in the Aleph and in the earth the Aleph once more and theearth in the Aleph ...”. In this case, a plausible reply is simply that8
fiction delivers no guidance to conceptual investigations: conceivability
may well be a guide to possibility, but literary fantasy is by itself noevidence of genuine conceivability (van Inwagen 1993: 229). Still, one
may observe that the possibility of mereological loops is not pure fantasy. In view of certain developments in non-well-founded set theory(Aczel 1988), one might indeed suggest building mereology on the basis of a notion of parthood that may violate (18). This is particularly
significant insofar as set theory itself may be reformulated in mereological terms—a possibility that is explored in the works of Bunt (1985)
and especially Lewis (1991). At present, however, no systematic studyof non-well-founded mereology has been put forward in the literature.
Moreover, we are interested here in mereology as a tool for spatial reasoning, and in this regard the possibility of symmetric loops does indeed
appear to be pure fantasy. In the following we shall therefore confine
ourselves to theories that accept the antisymmetry postulate along withreflexivity and transitivity: parthood is a partial ordering.

1.2.2
Other mereological concepts.
It is convenient at thispoint to introduce some degree of formalization. Let us use ‘P’ for thebinary predicate constant ‘... is part of ...’. Taking the underlying logic
to be a standard predicate calculus with identity, the above minimal
requisites on parthood may then be regarded as forming a first-ordertheory characterized by the following proper axioms for ‘P’:
(P.1)
(P.2)
(P.3)

Pxx
Pxy ∧ Pyz → Pxz
Pxy ∧ Pyx → x = y

Reflexivity
Transitivity
Antisymmetry(Here and in the following we simplify notation by dropping all initial
universal quantifiers. Unless otherwise specified, all formulas are to beunderstood as universally closed.) A number of additional mereologicalpredicates can then be introduced by definition. For example:
(19)
(20)
(21)
(22)
(23)

EQxy =df Pxy ∧ Pyx
PPxy =df Pxy ∧ ¬Pyx
PExy =df ¬Pxy ∧ Pyx
Oxy =df ∃z(Pzx ∧ Pzy)
Uxy =df ∃z(Pxz ∧ Pyz)

Equality
Proper Parthood
Proper Extension
Overlap
Underlap

An intuitive model for these relations, with ‘P’ interpreted as spatial
inclusion, is given in the diagram of Figure 1.1.
Note that ‘Uxy’ is bound to hold if we assume the existence of a“universal entity” of which everything is part. Conversely, ‘Oxy’ would
always hold if we assumed the existence of a “null entity” that is part

9

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

Uxny

Oxny

PPxny

n=1

+

–

–

–

–

n=2

+

+

–

–

–

n=3

+

+

+

–

–

n=4

+

+

–

+

–

n=5

+

+

–

–

+

x2
x1

x3
x4 = yx5

Figure 1.1.

EQxny PExny

Basic mereological relations. (Shaded cells indicate parthood.)of everything. In the domain of spatial entities, the latter assumption
is of course implausible (Geach 1949).4 The former assumption may bechallenged, too (Simons 2003, Varzi 2006), but it seems reasonable, if
not obvious, in case the only spatial entities countenanced by the theory
are regions of space. We shall come back to these issues in Section 1.4.
Note also that the definitions imply (by pure logic) that EQ, O, and U
are reflexive and symmetric; in addition, EQ is also transitive—an equivalence relation. By contrast, PP and PE are irreflexive and asymmetric,
and it follows from (P.2) that both are transitive. Since the following
biconditional is also a straightforward consequence of the axioms:(24)

Pxy ↔ (PPxy ∨ x = y)it should now be obvious that one could in fact use proper parthood as
an alternative starting point for the development of mereology, using theright-hand side of (24) as a definiens for ‘P’. This is, for instance, theoption followed in Simons (1987), where the partial ordering axioms for‘P’ are replaced by the strict ordering axioms for ‘PP’:
(25)
(26)
(27)

¬PPxx
PPxy ∧ PPyz → PPxz
PPxy → ¬PPyx

Ditto for ‘EP’, which was in fact the primitive relation in Whitehead’s(1919) semi-formal treatment of the mereology of events. Other options
may be considered, too. For example, Goodman (1951) used ‘O’ as aprimitive and Leonard and Goodman (1940) used its opposite:(28)

Dxy =df ¬Oxy

4 In other contexts one may feel differently:

Disjointness

see Martin (1965) and Bunt (1985) for theories
with a null individual, and Bunge (1966) for a theory with several null individuals.

10
However, the relations corresponding to such predicates are weaker thanPP and PE and no biconditional is provable from (P.1)–(P.3) that would
yield a corresponding definiens of ‘P’ (though one could of course define‘P’ in terms of ‘O’ or ‘D’ in the presence of further axioms; see below ad(45)). Thus, other things being equal, ‘P’, ‘PP’, and ‘PE’ appear to be
the only reasonable options. Here we shall stick to ‘P’.
Finally, note that Identity could itself be introduced by definition, due
to the following corollary of the antisymmetry postulate (P.3):(29)

x = y ↔ EQxy

Accordingly, the theory could be formulated in a pure first-order language by assuming (P.1) and (P.2) and replacing (P.3) with the following
variant of the standard axiom schema for ‘=’ (where φ is any formula):(P.30 )EQxy → (φx ↔ φy)Indiscernibility

One may in fact argue on these grounds that parthood is in some sense
conceptually prior to identity (as in Sharvy 1983: 234), and since ‘EQ’
is not definable in terms of ‘PP’ or ‘PE’ without resorting to ‘=’, theargument would also provide evidence in favor of ‘P’ as the most basic
primitive. As we shall see in Section 1.3.2, however, the link between
parthood and identity is philosophically problematic. In order not tocompromise our discussion, in the following we shall therefore continue
to work with a language with both ‘P’ and ‘=’ as primitives.

1.3

Decomposition principles

Let M be the theory defined by the three basic principles (P.1)–(P.3).
M may be viewed as embodying the common core of any mereologicaltheory. Not just any partial ordering qualifies as a part-whole relation,
though, and deciding what further principles should be added to (P.1)–(P.3) is precisely the question a good mereological theory is meant toanswer. It is here that philosophical issues begin to arise.
Generally speaking, such refinements may be divided into two maingroups. On the one hand, one may extend M by means of decomposition
principles that take us from a whole to its proper parts. For example, one
may consider the idea that whenever something has a proper part, it has
more than one—i.e., that there is always some mereological difference (aremainder) between a whole and its proper parts. This need not be truein every model for M : a world with only two items, only one of which ispart of the other, would be a counterexample, though not one that could
be illustrated with the sort of geometric diagram used in Figure 1.1. On
the other hand, one may extend M by means of composition principles

11

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

y•
x•

y

y •

•

x •
z •

x•

•

•z
•

•

Figure 1.2. Three unsupplemented models. (Parthood relationships are represented
by connecting lines going uphill.)that go in the opposite direction—from the parts to the whole. For
example, one may consider the idea that whenever there are some things
there exists a whole that consists exactly of those things—i.e., that there
is always a mereological sum (or fusion) of two or more parts. Again,
this need not be true in a model for M , and it is a matter of controversy
whether the idea should hold unrestrictedly.

1.3.1
Parts and remainders.
Let us begin with the first sortof extension. And let us start by taking a closer look at the intuition
according to which a whole cannot be decomposed into a single properpart. There are various ways in which one can try to capture this basic
intuition. Consider the following (from Simons 1987: 26–28):(P.4a )
(P.4b )
(P.4)

PPxy → ∃z(PPzy ∧ ¬z = x)
PPxy → ∃z(PPzy ∧ ¬Pzx)
PPxy → ∃z(Pzy ∧ ¬Ozx)

Weak Company
Strong Company
Supplementation

The first principle, (P.4a ), is a literal rendering of the idea in question:
every proper part must be accompanied by another. However, there isan obvious sense in which (P.4a ) only captures the letter of the idea,
not the spirit: it rules out the unintended model mentioned above (seeFigure 1.2, left) but not, for example, an implausible model with an
infinitely descending chain in which the additional proper parts do notleave any remainder (Figure 1.2, center).
The second principle, (P.4b ), is stronger: it rules out both models as
unacceptable. However, (P.4b ) is still too weak to capture the intended
idea. For example, it is satisfied by a model in which a whole can be
decomposed into several proper parts all of which overlap one another(Figure 1.2, right), and it is may be argued that such models do not dojustice to the meaning of ‘proper part’: after all, the idea is that theremoval of a proper part should leave a remainder, but it is by no means
clear what would be left of z once x (along with its parts) is removed.

12
It is only the third principle, (P.4), that appears to provide a fullformulation of the idea that nothing can have a single proper part. According to this principle, every proper part must be “supplemented” byanother, disjoint part, and it is this last qualification that captures thenotion of a remainder.
Should (P.4) be incorporated into M as a further fundamental principle on the meaning of ‘part’ ? Most authors (beginning with Simons
himself) would say so. Yet here there is room for disagreement. In fact,
it is not difficult to conceive of mereological scenarios that violate not
only (P.4), but also (P.4b ) and even (P.4a ). For example, in Brentano’s(1933) theory of accidents, a soul is a proper part of a thinking souleven though there is nothing to make up for the difference (see Chisholm
1978; Baumgartner and Simons 1994). Similarly, in Fine’s (1982) theory of qua objects, every basic object (John) qualifies as the only properpart of its incarnations (John qua philosopher, John qua husband, etc.).
Now, such putative counterexamples are controversial and, more importantly for our present concerns, they appear to be of little significance if
mereology is to be thought of as a theory of space. The spatial relations
illustrated by our initial examples (1)–(7) all seem to satisfy (P.4) and, afortiori, (P.4a ) and (P.4b ). Nonetheless there are counterexamples also
in the realm of truly spatial mereologies. The best illustration comesfrom Whitehead’s (1929) theory of extensive connection: on this theory, a topologically closed region includes its open interior as a properpart in spite of there being no boundary elements to distinguish them—the domain only consists of extended regions. Whether the omission ofboundary elements such as points, lines, and surfaces is a reasonable
thing to do when it comes to the task of modeling our understandingof space, and whether in the absence of such elements the distinction
between open and closed regions is still legitimate, are questions thatevery theory of space must of course address. In Section 2.4 we shallsee that answering in the affirmative involves serious philosophical andtechnical complications. But we shall also see that several theories areavailable to do the job, including theories that occupy a prominent rolein the current literature on qualitative spatial reasoning. One may relyon the intuitive appeal of (P.4) to discard such theories as implausible,
but one may as well turn things around and regard the adequacy of suchtheories as a good reason not to accept (P.4) unrestrictedly. As things
stand, it therefore seems appropriate to regard such a principle as providing a minimal but substantive addition to (P.1)–(P.3), one that goesbeyond the mere lexical characterization of ‘part’ provided by M . For
future reference, let us label the resulting mereological theory MM (forMinimal Mereology).

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

Figure 1.3.

x •

•y

z1 •

• z2

13

A weakly supplemented model.

1.3.2
Supplementation, extensionality, identity.
There isanother way of expressing the supplementation intuition that is worth
considering. It corresponds to the following axiom, which differs from
(P.4) in the antecedent:(P.5)¬Pyx → ∃z(Pzy ∧ ¬Ozx)

Strong Supplementation

Intuitively, this says that if an object fails to include another amongits parts, then there must be a remainder. It is easily seen that (P.5)
implies (P.4), so any theory rejecting at least (P.4) will a fortiori reject(P.5). (For instance, on Whitehead’s boundary-free theory of extensive
connection, a closed region is not part of its interior even though eachpart or the former overlaps the latter.) However, the converse does nothold. The diagram in Figure 1.3 illustrates a model in which (P.4) istrue, since each proper part counts as a supplement of the other; yet(P.5) is false.
The theory obtained by adding (P.5) to (P.1)–(P.3) is thus a proper
extension of M M . Let us label this stronger theory EM , for Extensional
Mereology, the attribute ‘extensional’ being justified precisely by theexclusion of countermodels that, like the one just mentioned, contain
distinct objects with the same proper parts. In fact, the following is atheorem of EM :(30)

∃zPPzx → (∀z(PPzx → PPzy) → Pxy)from which it follows that no composite objects with the same properparts can be distinct:(31)(∃zPPzx ∨ ∃zPPzy) → (x = y ↔ ∀z(PPzx ↔ PPzy))(The analogue for ‘P’ is, of course, already provable in M , since parthood is reflexive and antisymmetric.) Thus, EM is truly an extensional
theory incorporating the view that an object is exhaustively defined byits constituent parts. This goes far beyond the intuition that lies behind
the weak supplementation principle (P.4). Does it go too far?
On the face of it, it is not difficult to envisage scenarios that would correspond to the diagram in Figure 1.3. For example, we can obtain a counterexample to (P.5) by identifying x and y with the sets {{z1 }, {z1 , z2 }}14and {{z2 }, {z1 , z2 }} (i.e., with the ordered pairs hz1 , z2 i and hz2 , z1 i, respectively), interpreting ‘P’ as the ancestral of the improper membership
relation (i.e., of the union of ∈ and =). But sets are abstract entities;
can we also envisage similar scenarios in the spatial domain?
Here is a case where the answer may differ crucially depending onwhether we are interested in modeling a domain of concrete spatial entities or just the domain of the regions of space that they occupy. In thelatter case there is little room for controversy: spatial regions are extensional, if anything is, unless of course we favor a Whitheadian conceptionof space. In the former case, however, the answer is controversial. There
are two sorts of objection worth considering. On the one hand, it issometimes argued that sameness of parts is not sufficient for identity, assome entities may differ exclusively with respect to the arrangement of
their parts. For example, it is sometimes argued that: (i) two words canbe made up of the same letters, as with ‘fallout’ and ‘outfall’; (ii) thesame flowers can compose a nice bunch or a scattered bundle, dependingon the arrangements of the individual flowers; (iii) a cat can survive theannihilation of its tail, but the amount of feline tissue consisting of the
cat’s tail and the rest of the cat’s body cannot survive the annihilationof the tail, hence they have different properties and must be distinct byLeibniz’s law in spite of their sharing exactly the same ultimate mereological constituents. (See Hempel 1953: 110, Eberle 1970: §2.10, andWiggins 1968, respectively; variants of (iii) may also be found in Doepke
1982, Lowe 1989, Johnston 1992, and Baker 1999, inter alia.) On theother hand, it is sometimes argued that sameness of parts is not necessary for identity, as some entities may survive mereological change. If a
cat survives the annihilation of its tail, then the tailed cat (before theaccident) and the tailless cat (after the accident) are one and the samein spite of their having different proper parts (Wiggins 1980). If any ofthese arguments is accepted, then clearly (31) is too strong a principle
to be imposed on the parthood relation. And since (31) follows from
(P.5), it might be concluded that EM is on the wrong track.
Let us look at these objections separately. Concerning the necessityof mereological extensionality, i.e., the left-to-right conditional in theconsequent of (31):(32)

x = y → ∀z(PPzx ↔ PPzy)

it is perhaps enough to remark that the difficulty is not peculiar to extensional mereology. The objection proceeds from the consideration that
ordinary entities such as cats and other living organisms (and possibly
other entities as well, such as cars and houses) survive all sorts of gradualmereological changes. Yet the same can be said of other types of changeSpatial Reasoning and Ontology: Parts, Wholes, and Locations

15

as well: bananas ripen, houses deteriorate, people sleep at night and eat
at lunch. How can we say that they are the same things, if they are notquite the same? Indeed, (32) is just an instance of the identity axiom(ID)

x = y → (φx ↔ φy)and it is well known that this axiom calls for revisions when ‘=’ is givena diachronic reading. Arguably, any such revisions will affect the caseat issue as well, and in this sense the above-mentioned objection to (32)can be disregarded. For example, if the basic parthood predicate werereinterpreted as a time-indexed relation (Thomson 1983), then the problem would disappear as the tensed version of (P.5) would only warrant
the following variant of (32):(320 )

x = y → ∀t∀z(PPt zx ↔ PPt zy)

Similarly, the problem would disappear if the variables in (32) weretaken to range over four-dimensional entities whose parts may extend
in time as well as in space (Heller 1984, Sider 2001), or if identity itself
were construed as a contingent relation that may hold at some times
but not others (Gallois 1998). Such revisions may be regarded as anindicator of the limited ontological neutrality of extensional mereology.
But their independent motivation also bears witness to the fact that
controversies about the necessity of extensionality stem from larger andmore fundamental philosophical conundrums and cannot be assessed byappealing to our intuitions about the meaning of ‘part’.
The worry about the sufficiency of mereological extensionality, i.e.,
the right-to-left conditional in the consequent of (31):(33)∀z(PPzx ↔ PPzy) → x = y

is more to the point. However, there are various ways of resisting suchcounterexamples as (i)–(iii) on behalf of EM . Consider (i)—two words
made up of the same letters. Insofar as we are dealing with truly spatial
entities, this is best described as a case of different word tokens madeup of distinct tokens of the same letter types. There is, accordingly, no
genuine violation of (33) in the opposition between ‘fallout’ and ‘outfall’(for instance), hence no reason to reject (P.5) on these grounds. (Besides,
even with respect to abstract types, it could be pointed out that ‘fallout’and ‘outfall’ do not share all their proper parts: the string ‘lo’, forinstance, is only included in the first word.) What if one of the twoword tokens is obtained from the other by rearranging the same letter
tokens? In that case the reply misfires, but so does the ojection: theissue becomes once again one of diachronic non-identity, with all that it
entails (Lewis 1991: 78f).

16
Case (ii)—the flowers—is not significantly different. The same, concrete flowers cannot compose a nice bunch and a scattered bundle atthe same time. Case (iii), however, is more delicate. There is a strong
intuition that a cat really is something over and above the amount of
feline tissue consisting of its tail and the rest of its body—that they havedifferent survival conditions and, hence, different properties—so it may
be thought that here we have a genuine counterexample to mereologicalextensionality. On behalf of EM , it should nonetheless be noted thatthe appeal to Leibniz’s law in this context is debatable. Let ‘Tibbles’
name our cat and ‘Tail’ its tail, and let us grant the truth of(34)

Tibbles can survive the annihilation of Tail.

There is, indeed, an intuitive sense in which the following is also true:(35)

The amount of feline tissue consisting of Tail and the rest of
Tibbles’s body cannot survive the annihilation of Tail.

However, this intuitive sense corresponds to a de dicto reading of themodality, where the description in (35) has narrow scope:(35a) Necessarily, the amount of feline tissue consisting of Tail andthe rest of Tibbles’s body has Tail as a proper part.
On this reading (35) is hardly negotiable (in fact, logically true). Yet
this is irrelevant in the present context, for (35a) does not amount to an
ascription of a modal property and cannot be used in connection with
Leibniz’s law. (Compare the following fallacious argument: The numberof planets might have been even; 9 is necessarily odd; hence the numberof planets is not 9.) On the other hand, consider a de re reading of (35):(35b) The amount of feline tissue consisting of Tail and the rest of
Tibbles’s body necessarily has Tail as a proper part.
On this reading the appeal to Leibniz’s law would be legitimate (modulo
any concerns about the status of modal properties) and one could rely on
the truth of (34) and (35), i.e., (35b), to conclude that Tibbles is distinct
from the relevant amount of feline tissue. However, there is no obvious
reason why (35) should be regarded as true on this reading. That is,there is no obvious reason to suppose that the amount of feline tissue
that in the actual world consists of Tail and the rest of Tibbles’s body—that amount of feline tissue that is now resting on the carpet—cannot
survive the annihilation of Tail. Indeed, it would appear that any reason
in favor of this claim vis-à-vis the truth of (34) would have to presupposeSpatial Reasoning and Ontology: Parts, Wholes, and Locations

17

y

•
•
•z1
Figure 1.4.

•
•x

•z2

A strongly supplemented model violating complementation

the distinctness of the entities in question, so no appeal to Leibniz’s
law would be legitimate to establish the distinctess on pain of circularity(Varzi 2000). This is not to say that the putative counterexample to (34)is wrong-headed. But it requires genuine metaphysical work to defend it
and it makes the rejection of the strong supplementation principle (P.5)
a much harder task.

1.3.3
Complementation.
There is a way of expressing the
supplementation intuition that is even stronger than (P.5). It corresponds to the following thesis, which differs from (P.5) in the consequent:(P.6)¬Pyx → ∃z∀w(Pwz ↔ (Pwy ∧ ¬Owx))

Complementation

This says that if y is not part of x, there exists something that comprises
exactly those parts of y that are disjoint from x—something that we may
call the difference or relative complement between y and x. It is easily
checked that this principle implies (P.5). On the other hand, the diagramin Figure 1.4 shows that the converse does not hold: there are two partsof y disjoint from x, namely z1 and z2 , but there is nothing that consists
exactly of such parts, so we have a model of (P.5) in which (P.6) fails.
Any misgivings about (P.5) may of course be raised against (P.6). But
what if we agree with the above arguments in support of (P.5)? Do theyalso give us reasons to accept the stronger principle (P.6)? The answer
is in the negative. Plausible as it may sound, (P.6) has consequencesthat even an extensionalist may not be willing to accept. For example,
Figure 1.5 depicts a scenario that—it may be argued—corresponds exactly to the model of Figure 1.4. It may be argued that although x andz1 jointly constitute a larger part of y (the difference between y and z2 ),and similarly for x and z2 (the difference between y and z1 ), there isnothing consisting of z1 and z2 (the difference between y and x), sincethese two pieces are disconnected. More generally, it appears that (P.6)
would force us to accept the existence of scattered entities, such as the“sum” of your left and right arms, or the “sum” of Canada and Mex-

18

z1

x

z2

y
Figure 1.5.

Does the difference between y and x exist?

ico, and since Lowe (1953) many authors have objected to this thought
regardless of how one feels about extensionality. (One philosopher who
explicitly agrees to extensionality while distrusting scattered entities is
Chisholm 1987.) As it turns out, the extra strength of (P.6) is therefore
best appreciated in terms of the sort of mereological aggregates that
this principle would entail, aggregates that are composed of two or moreparts of a given whole. This suggests that any additional misgivings
about (P.6), besides its extensional implications, are truly misgivings
about matters of composition. We shall accordingly postpone their discussion to Section 1.4, where we shall attend to these matters more fully.
For the moment, let us simply say that (P.6) is, on the face of it, not aprinciple that can be added to M without further argument.

1.3.4
Atomism and other options.
One last importantfamily of decomposition principles concerns the question of atomism.
Mereologically, an atom (or “simple”) is an entity with no proper parts:(36)

Ax =df ¬∃yPPyx

Atom

Are there any such entities? And if there are, is everything entirely
made up of atoms? Does everything comprise at least some atoms? Or
is everything made up of atomless gunk? These are deep and difficult
questions, which have been the focus of philosophical investigation since
the early days of philosophy and have been center stage also in many
recent disputes in mereology (see, for instance, van Inwagen 1990, Sider
1993, Zimmerman 1996a, and the papers collected in Hudson 2004).
Here we shall confine ourselves to a brief examination.
The two main options, to the effect that there are no atoms at all,or that everything is ultimately made up of atoms, correspond to thefollowing postulates, respectively:
(P.7)
(P.8)

∃yPPyx∃y(Ay ∧ Pyx)

Atomlessness
Atomicity

These postulates are mutually incompatible, but taken in isolation they
can consistently be added to any mereological theory X considered here.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

19

Adding (P.8) yields a corresponding Atomistic version, AX; adding (P.7)
yields an Atomless version, ĀX. Since finitude together with the antisymmetry of parthood (P.3) jointly imply that mereological decomposition must eventually come to an end, it is clear that any finite model ofM (and a fortiori of any extension of M ) must be atomistic. Accordingly,
an atomless mereology ĀX admits only models of infinite cardinality. (Aworld containing such wonders as Borges’s Aleph, where parthood is notantisymmetric, might by contrast be finite and yet atomless.) An example of such a model, establishing the consistency of the atomless versionof most mereological theories considered in the this chapter, is provided
by the regular open sets of a Euclidean space, with ‘P’ interpreted as
set-inclusion (Tarski 1935).
Now, one thing to notice is that, independently of their motivations,atomistic mereologies admit of significant simplifications in the axioms.
For instance, AEM can be simplified by replacing (P.5) and (P.8) with
(P.50 )

¬Pyx → ∃z(Az ∧ Pzy ∧ ¬Pzx)

which in turns implies the following atomistic variant of the extensionality
thesis (31):(37)

x = y ↔ ∀z(Az → (Pzx ↔ Pzy))

Thus, any atomistic extensional mereology is truly “hyperextensional”
in Goodman’s (1958) sense: things built up from exactly the same atoms
are identical. An interesting question, discussed at some length in thelate 1960’s (Yoes 1967, Eberle 1968, Schuldenfrei 1969) and taken up
more recently by Simons (1987: 44f) and Engel and Yoes (1996), iswhether there are atomless analogues of (37). Is there any predicate thatcan play the role of ‘A’ in an atomless mereology? Such a predicate would
identify the “base” of the system and would therefore enable mereology
to cash out Goodman’s hyperextensional intuitions even in the absenceof atoms. This question is particularly significant from a nominalistic
perspective, but it also bears on our present concerns. For example,
it is a relevant question to ask in connection with the Whiteheadianconception mentioned in Section 1.3.1, according to which space contains no parts of lower dimensions such as points or lines (see Forrest
1996, Roeper 1997). In special cases there is no difficulty in providing
a positive answer. For example, in the ĀEM model consisting of theopen regular subsets of the real line, the open intervals with rational end
points form a base in the relevant sense. It is unclear, however, whether
a general answer can be given that applies to any sort of domain. If not,then the only option would appear to be an account where the notion of

20
a “base” is relativized to entities of a given sort. In Simons’s terminology, we could say that the ψ-ers form a base for the φ-ers if and only if
the following variants of (P.50 ) and (P.8) and are satisfied:(P.5∗ )
(P.8∗ )

φx ∧ φy → (¬Pyx → ∃z(ψz ∧ Pzy ∧ ¬Pzx))
φx → ∃y(ψy ∧ Pyx)

An atomistic mereology would then correspond to the limit case where‘ψ’ is identified with ‘A’ for every choice of ‘φ’. In an atomless mereology,
by contrast, the choice of the base would depend each time on the levelof “granularity” set by the relevant specification of ‘φ’.
A second important consideration concerns the possibility of theories
that lie between the two extreme options afforded by Atomicity andAtomlessness. For instance, it can be held that there are atoms, though
not everything need have a complete atomic decomposition, or it canbe held that there is atomless gunk, though not everything need be
gunky (Zimmerman 1996a). Again, formally this amounts to endorsing
a restricted version of either (P.7) or (P.8) in which the variables are
suitably restricted so as to range over entities of a certain sort:(P.7φ )
(P.8φ )

φx → ∃yPPyx
φx → ∃y(Ay ∧ Pyx)

At present, no thorough formal investigation of such options has beenentertained (but see Masolo and Vieu 1999). Yet the issue is particularly
significant from the perspective of a mereological theory aimed at modeling the spatial world, especially if the theory is to countenance concrete
spatial entities along with the regions of space that such entities may
occupy. It is, after all, a plausible thought that while the question of
atomism may be left open with regard to the mereological structure ofmaterial objects (pending empirical findings from physics, for example),
it must receive a definite answer with regard to the structure of space itself. This would amount to endorsing a version of either (P.7φ ) or (P.8φ )in which ‘φ’ is understood as a condition that is satisfied exclusively byregions of space. Such a condition, of course, cannot be formulated in
the language of a purely mereological theory, but we shall see in Section 3 that a suitably enriched theory, in which the relation of location
is explicitly articulated, can do the job properly. (Actually, it is hard toconceive of a world in which an atomistic space is inhabited by entities
that can be decomposed indefinitely, so in this case it is reasonable tosuppose that any theory accepting (P.8φ ) for regions would also endorse
the stronger principle (P.8). However, (P.7φ ) would be genuinely independent of (P.7), unless it is assumed that every mereologically atomic
entity should also be spatially atomic, i.e., unextended.)Spatial Reasoning and Ontology: Parts, Wholes, and Locations

21

Similar considerations apply to other decomposition principles that
may come to mind at this point. For example, one may consider arequirement to the effect that ‘PP’ forms a dense ordering, as alreadyWhitehead (1919) had it:(P.9)

PPxy → ∃z(PPxz ∧ PPzy)

Density

As a general decomposition principle, (P.9) might be deemed too strong,
especially in an atomistic setting. However, it is plausible to supposethat (P.9) should hold at least in the domain of spatial regions, regardless of whether these are construed as atomless gunk or as aggregatesof spatial atoms. Evidently much depends on the link one establishes
between the mereology of an object and that of its spatial location andthis, again, is a question to which we attend more fully in Section 3. For
the moment, let us simply observe that the sort of philosophical issues
that lie behind these options is significantly different from those considered in the previous sections. Whether something can have a single
proper part, whether parthood is extensional, or even whether it satisfies
the complementation principle (P.6) are issues that depend greatly on
our understanding of the parthood relation. They are, in an importantsense, conceptual questions. Whether there are mereological atoms, bycontrast, or whether mereological decomposition should obey a density
principle, are substantive questions that have nothing to do with ourunderstanding of parthood as such. (For more on these questions, andon their general historical background, see Pyle 1995 and Holden 2004.)

1.4

Composition principles

Let us now consider the second way of extending M mentioned at the
beginning of Section 1.3. Just as we may want to fix the logic of P
by means of decomposition principles that take us from a whole to its
proper parts, we may look at composition principles that go in the opposite direction—from the parts to the whole. More generally, we may
consider the idea that the domain of the theory ought to be closed under mereological operations of various sorts: not only fusions, but also
products, differences, and more. Here, again, there is room for several
philosophical considerations, some of which are particularly indicativeof the tension between space-oriented and object-oriented intuitions.

1.4.1
Bounds and fusions.
Conditions on composition are
many. Beginning with the weakest, consider the claim that any two
suitably related entities have an upper bound, i.e., underlap:(P.10ψ )ψxy → ∃z(Pxz ∧ Pyz)

Boundedness

22

z

•
x •

•
Figure 1.6.

•
w
•

•y

x •

•

•

z

•
•

•y
•

A fusiona that is not a fusion, and a fusion that is not a fusionb .

Exactly how ‘ψ’ should be construed is an important question by itself—a version of what van Inwagen (1990) calls the “special composition
question”. Perhaps the most natural choice is to identify ψ with mereological overlap, the rationale being that such a relation establishes animportant tie between what may count as two distinct parts of a larger
whole. As we shall see momentarily, with ψ so construed, (P.10ψ ) is
indeed uncontroversial. However, regardless of any specific choice, it isapparent that (P.10ψ ) is pretty weak, as it holds trivially in any domain
with a universal entity of which everything is part.
A somewhat stronger condition would be to require any pair of suitably related entities to have a smallest underlapper—something composed exactly of them and nothing else. This requirement is sometimes
stated by saying that such entities must have a mereological “sum”,
or “fusion”, though it is not immediately obvious how that should beformulated in the formal language. Consider:(P.11ψa ) ψxy → ∃z(Pxz ∧ Pyz ∧ ∀w(Pxw ∧ Pyw → Pzw)) Fusiona(P.11ψb ) ψxy → ∃z(Pxz ∧ Pyz ∧ ∀w(Pwz → Owx ∨ Owy)) Fusionb(P.11ψ ) ψxy → ∃z∀w(Owz ↔ Owx ∨ Owy)
FusionIn a way, (P.11ψa ) would seem the obvious choice, corresponding to
the idea that the fusion of two objects is just their least upper boundrelative to P. (See e.g. Bostock 1979, van Benthem 1983.) However, thiscondition is too weak to capture the intended notion of a mereologicalfusion. For example, with ψ construed as overlap, (P.11ψa ) is satisfied
by the model of Figure 1.6, left: here the least upper bound of x andy is z, yet z hardly qualifies as something “made up” of x and y since
its parts also include a third, disjoint item w. In fact, it is a simple fact
about partial orderings that among finite models (P.11ψa ) is equivalentto (P.10ψ ), hence just as weak. By contrast, (P.11ψb ) corresponds to a
notion of fusion (to be found e.g. in Tarski 1929) that may seem too
strong: it rules out the model on the left of Figure 1.6; but it also rulesout the model on the right, which depicts a situation in which z may beviewed as an entity truly made up of x and y insofar as it is ultimately

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

23

composed of atoms to be found either in x or in y. Of course, such asituation violates the strong supplementation principle (P.5), but that’s
precisely the sense in which (P.11ψb ) is too strong: an anti-extensionalist
might want to have a notion of fusion that does not presuppose strong
supplementation. The formulation in (P.11ψ ) is the natural compromise:
it is strong enough to rule out the model on the left, but weak enoughto be compatible with the model on the right. This is, in fact, theformulation that best reflects the notion of fusion to be found in standard
treatments of mereology, and in the sequel we shall mostly stick to it.
Note, however, that if (P.5) holds, then (P.11ψ ) is equivalent to (P.11ψb ).
Moreover, it turns out that if the stronger complementation axiom (P.6)
holds, then all of these principles are trivially satisfied in any domain inwhich there is a universal entity: in that case, regardless of ψ, the fusionof any two entities is just the complement of the difference between thecomplement of one minus the other. (Such is the strength of (P.6), a
genuine cross between decomposition and composition principles.)We can further strengthen these principles by considering infinitarybounds and fusions. For example, (P.10ψ ) can be generalized to a principle to the effect that any non-empty set of entities satisfying a suitablecondition ξ has an upper bound. Strictly speaking there is a difficultyin expressing such a principle in a language without set variables. We
can, however, achieve a sufficient degree of generality by relying on anaxiom schema where classes are identified by open formulas. Since anordinary first-order language has a denumerable supply of formulas, at
most denumerably many sets (in any given domain) can be specified inthis way. But for most purposes this limitation is negligible, as normally
we are only interested in those sets of objects or regions that we are able
to specify. Thus, the following axiom schema will do, where ‘φ’ is any
formula in the language and ‘ξ’ expresses the condition in question:(P.12ξ )∃wφw ∧ ∀w(φw → ξw) → ∃z∀w(φw → Pwz)

Likewise, the fusion axiom (P.11ψ ) can be strengthened as follows:(P.13ξ )∃wφw ∧ ∀w(φw → ξw) → ∃z∀w(Owz ↔ ∃v(φv ∧ Owv))

and similarly for (P.11ψa ) and (P.11ψb ). (The condition ‘∃wφw’ guarantees that ‘φ’ picks out a non-empty set, so there is no danger of asserting the unconditional existence of “null entities”—a mereologicalfiction that we have already mentioned as implausible in the context
of spatial ontology.) It can be checked that these generalized formulations include the corresponding finitary principles as special cases, taking ‘φw’ to be the formula ‘(w = x ∨ w = y)’ and ‘ξw’ the condition‘(w = x → ψwy) ∧ (w = y → ψxw)’.

24
Finally, we get the strongest version of these composition principles
by asserting them as axiom schemas that hold for every condition ξ,i.e., effectively, by foregoing any reference to ξ altogether. Formally thisamounts in each case to dropping the second conjunct of the antecedent.
For example, the following schema is the unrestricted version of (P.13ξ ),to the effect that every specifiable non-empty set of entities has a fusion:(P.13)∃wφw → ∃z∀w(Owz ↔ ∃v(φv ∧ Owv))

Unrestricted Fusion

The extension of EM obtained by adding every instance of this schema
has a distinguished pedigree and is known as General Extensional Mereology, or GEM . It corresponds to the classic systems of Leśniewski andof Leonard and Goodman. In fact, it turns out that adding (P.13) toMM yields the same theory GEM , since (P.13) implies that every pairof overlapping things has a maximal common part (a product):(38)

Oxy → ∃z∀w(Pwz ↔ (Pwx) ∧ Pwy))

which, in turn, implies the equivalence between the weak supplementation principle (P.4) and the stronger (P.5) (Simons 1987: 31). This is byitself remarkable, for it might be thought that a composition principle
such as (P.13) should be compatible with the rejection of a decomposition principle that is committed to extensionality. On the other hand,mereological extensionality is really a double-barreled thesis: it says that
two wholes cannot be decomposed into the same proper parts but also,
by the same token, that two wholes cannot be composed out of the same
proper parts. So it is not entirely surprising that as long as PP is well
behaved, as per (P.4), extensionality might pop up like this in the presence of substantive composition statements. (It is, however, noteworthy
that it pops up as soon as (P.4) is combined with a seemingly innocent
thesis such as (38), so the anti-extensionalist should keep that in mind.)
The intuitive idea behind all these principles is in fact best appreciatedin the presence of extensionality, for in that case the relevant fusions must
be unique. Just to confine ourselves to GEM , it is natural to consider
the following fusion operator (where ‘ı’ is the definite descriptor5 ):(39)

Σxφx =df ız∀w(Owz ↔ ∃v(φv ∧ Owv))

fusionThen (P.13) and (P.5) can be simplified to a single axiom schema:(P.14) ∃xφx → ∃z(z = Σxφx)

Unique Unrestricted Fusion

5 We assume a classical logical background, with ‘ı’ defined as usual.

Much of what follows,
however, would also apply in case a free logic were used instead, with ‘ı’ assumed as part ofthe logical vocabulary proper. (See Simons 1991b for a free formulation of mereology.)Spatial Reasoning and Ontology: Parts, Wholes, and Locations

25

and the full strength of the theory can be seen by considering that its
models are all closed under the following functors, modulo the absenceof a null entity:
(40)
(41)
(42)
(43)
(44)

x + y =df Σz(Pzx ∨ Pzy)x × y =df Σz(Pzx ∧ Pzy)x − y =df Σz(Pzx ∧ Dzy)
∼ x =df ΣzDzx
U =df ΣzPzz

sum
product
differencecomplement
universe(Absent the null entity, U has no complement while products are defined
only for overlapping pairs and differences for pairs that leave a remainder). Since these functors are the natural mereological analogue of thefamiliar Boolean operators, with fusion in place of set abstraction, it
follows that the parthood relation axiomatized by GEM has the same
properties as the set-inclusion relation. More precisely, it is isomorphic
to the inclusion relation restricted to the set of all non-empty subsets ofa given set, which is to say a complete Boolean algebra with the zeroelement removed—a fact that has been known since Tarski (1935).
There are other equivalent formulations of GEM that are noteworthy. For instance, it is a theorem of every extensional mereology thatparthood amounts to inclusion of overlappers:(45)

Pxy ↔ ∀z(Ozx → Ozy)

This means that in an extensional mereology ‘O’ could be used as aprimitive and ‘P’ defined accordingly, and it can be checked that thetheory defined by postulating (45) together with the unrestricted fusion principle (P.13) and the antisymmetry axiom (P.3) is equivalent to
GEM . Another elegant axiomatization of GEM , due to an earlier workof Tarski (1929), is obtained by taking just the transitivity axiom (P.2)
and the unique unrestricted fusion axiom (P.14).

1.4.2
Composition, existence, and identity.
Arguably, thealgebraic strength of GEM speaks in favor of this theory as an accountof the structure of space, since it is rather intuitive (and common practice) to understand spatial regions in terms of sets of points mereologically related by set-inclusion. As a general theory of the mereology ofall spatial entities, however, GEM reflects substantive postulates whose
philosophical underpinnings are controversial. Indeed, all composition
principles turn out to be controversial, just as the decomposition principles examined in Section 1.3. For, on the one hand, it appears thatthe weaker, conditional formulations, from (P.10ψ ) to (P.13ξ ), are justnot doing enough work: not only do they depend on the specification of

26
the limiting conditions expressed by the predicates ‘ψ’ and ‘ξ’; they also
treat such conditions as merely sufficient for the existence of bounds
and fusions, whereas ideally we are interested in conditions that areboth sufficient and necessary. On the other hand, the stronger, unconditional formulations—most notably (P.13)—appear to go too far, notonly because they tend to obliterate any difference between weak andstrong supplementation, i.e., extensionality, but because they commit
the theory to the existence of a large variety of prima facie implausible
mereological composites. So what is the right way to go, at least insofar
as we are interested in the compositional structure of the spatial realm?
Concerning the first sort of worry, one could of course strenghten
every conditional formulation to a biconditional expressing both a necessary and sufficient condition for the existence of an upper bound, ora fusion. But then the question of how such conditions should be construed becomes crucial. For example, in connection with (P.10ψ ) wehave mentioned the idea of construing ‘ψ’ as ‘O’, the rationale beingthat mereological overlap establishes an important connection between
what may count as two distinct parts of a larger whole. However, asa necessary condition overlap is arguably too stringent. We may havemisgivings about the existence of scattered entities consisting of spatially unrelated parts, such as the top of my body and the bottom of
yours, or the collection of my umbrellas and your left shoes. But in somecases no such misgivings arise. In some cases it appears perfectly natural to countenance wholes that are composed of two or more disjoint
entities—a bikini, the solar system, a printed inscription consisting ofseparate letter tokens (Cartwright 1975). More generally, intuition andcommon sense suggest that some and only some mereological composites
exist, not all; yet it is doubtful whether the question of which composites
exist—van Inwagen’s “special composition question”—can be answered
successfully. Consider a series of almost identical mereological aggregates that begins with a case where composition appears to obtain (e.g.,
the body cells that currently make up my body) and ends in a casewhere composition would seem not to obtain (e.g., the same body cells
after their relative distance has been gradually increased to a huge extent). Where should we draw the line? It may well be that wheneversome entities compose a bigger one, it is just a brute fact that they doso (Markosian 1998b). But if we are unhappy with brute facts, if weare looking for a principled way of drawing the line so as to specify thecircumstances under which the facts obtain, then the question is truly
challenging. As Lewis (1986: 213) put it, no restriction on composition
can be vague, since existence cannot be a matter of degree; but unless
it is vague, it cannot fit the intuitive desiderata.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

27

For these reasons, although the axiom of unrestricted fusion has beena major source of complaint since the early days of mereology (see again
Lowe 1953 and Rescher 1955, with replies in Goodman 1956, 1958), it
is a fact that most formally accomplished theories accept unrestrictedcomposition principles of some sort. Apart from whatever algebraic considerations might motivate them, such principles suggest themselves as
the only non-arbitrary ways of answering the composition question. Besides, it might be observed that any complaints against such principles
rest on psychological biases that have no bearing on how the world isactually structured. In the words of van Cleve (1986: 145), little would
follow even if we did manage to come up “with a formula that jibed withall ordinary judgments” about what counts as a unit and what does not,for such judgments need not have any ontological transparency.
All of this speaks in favor of (P.13) and the like against their weaker,
conditional formulations, providing also an answer to the second worry
mentioned above: the prima facie ontological extravagance of a theory
such as GEM is not by itself a sign that the theory has gone too far.
There is, however, another worry that is worth mentioning in this connection, and this further worry concerns the ontological exuberance—if
not the extravagance—of the theory. For even granting the impossibilityof drawing a principled line between natural fusions and unnatural ones,
one could still object that positing every conceivable fusion is utterly
unjustified. Why should mereology be committed to the existence of all
such things over and above their constituent proper parts?
There are two lines of response to this question. First, it could beobserved that the ontological exuberance associated with the relevant
composition principles is not substantive. This is obvious in the case ofa modest principle in the spirit of (P.10ψ ), to the effect that entities of
the right sort have an upper bound. After all, there are small things(my fingers) and large things (my body), and it is just a fact that thelatter encompass the former. But the same could be said with respect
to those stronger principles that require the large thing to be composed
exactly of the small things—to be their mereological fusion. For one
could argue that even a fusion is, in an important sense, nothing over
and above its constituent parts. The fusion is just the parts “taken together” (Lewis 1991: 81); it is the parts “counted loosely” (Baxter 1988:580); it is, effectively, the same portion of reality, which is strictly a multitude and loosely a single thing. This thesis, known in the literature as“composition as identity”, is by no means undisputed (see e.g. van Inwagen 1994, Yi 1999, Merricks 1999). Nonetheless it should be carefully
evaluated in connection with any worry about the ontological exuberance of fusion principles. And if the thesis is accepted, then the charge

28
of ontological extravagance loses its force, too. If a fusion is nothing over
and above its constituent parts, and if the latter are all right, there canbe nothing particularly extravagant in countenancing the former: it justis them.
Secondly, one may observe that the worry in question bites at thewrong level. If, given two entities, positing their sum were to countas further ontological commitment, then, given any mereologically composite entity, positing its proper parts should also count as further commitment. After all, every entity is distinct from its proper parts. But
then the worry has nothing to do with the composition axioms; it is,rather, a question of whether there is any point in countenancing awhole along with its parts, or vice versa. And if the answer is in thenegative, then there seems to be no use for mereology tout court. Fromthe point of view of the present worry, it would seem that the only truly
parsimonious account would be one that rejects, not only some, but all
logically admissible fusions—in fact, all mereological composites whatsoever. Philosophically such an account is defensible (see Rosen and Dorr
2002) and the corresponding axiom is compatible with M :(P.15) Ax

Strong Atomicity

The following immediate corollary, however, says it all: nothing would
be part of anything else and parthood would collapse to identity.
(46)

Pxy ↔ x = y(This account is known as mereological nihilism, in contrast to the mereological universalism expressed by (P.13); see van Inwagen 1990: 72ff.)
In recent years, further worries have been raised concerning mereological theories with non-trivial composition principles—especially concerning the full strength of GEM . It has been argued that unrestrictedcomposition does not sit well with certain intuitions about persistence
through time (van Inwagen 1990: 75ff), that it requires every entity tonecessarily have the parts it has (Merricks 1999), or that it leads to paradoxes similar to the ones afflicting naive set theory (Bigelow 1996). Sucharguments are still the subject of on-going controversy and a detailed
examination is beyond the scope of this chapter. Some discussion of thefirst point, however, is already available in the literature: see especially
Rea (1998), McGrath (1998), and Hudson (2001: 93ff). Hudson (2001:95ff) also contains a discussion of the last point.

1.5

The problem of vaguenessLet us conclude this discussion of mereology by considering a questionthat is not directly related to specific mereological principles but, rather,Spatial Reasoning and Ontology: Parts, Wholes, and Locations

29

to the underlying notion of parthood that mereology seeks to systematize. All the theories examined so far, from M to GEM , assume thatparthood is a perfectly determinate relation: given any two entities x
and y, there is always a definite fact of the matter as to whether or not
x is part of y. However, it may be argued that this is a simplification.
Perhaps there is no room for vagueness in the idealized mereology ofpure space, but what about the real world? Think of a cloud, a forest, a pile of trash. What parts do they have, exactly? What are themereological boundaries of a desert, a river, a mountain? Some stuff ispositively part of Mount Everest and some stuff is positively not part,
yet there is borderline stuff whose mereological relationship to Everest
seems indeterminate. Even living organisms may, on closer look, giverise to vagueness issues. Surely John’s body comprises his heart and does
not comprise mine. But what about the candy he is presently chewing:
Is it part of John? Will it be part of John only after he swallowed it?
After he started digesting it? After he digested it completely?
In the face of such examples, it might be thought that the conceptual
apparatus on which M and its extensions are based is too rigid. It might
be thought that the world includes various sorts of vague entities, andthat relative to such entities the parthood relation need not be fully determined (van Inwagen 1990: ch. 13, Parsons and Woodruff 1995). There
are, in fact, various ways one could seek greater flexibility. One could
leave everything as is but change the underlying logic (and semantics),
for instance by allowing statements of the form ‘Pxy’ to receive no determinate truth-value (as in Tye 1990), or to receive truth-values that areintermediate between classical truth and falsity (as in Copeland 1995).
Or one could change the very basic apparatus of mereology, replacing
the ‘part of’ predicate with a new primitive ‘part of to a degree’: thisis, for example, the approach that led to the development of Polkowsky
and Skowron’s (1994) “rough mereology”, where parthood undergoes afuzzification parallel to the fuzzification of membership in Zadeh’s (1965)
fuzzy set theory. No matter how exactly one proceeds, obviously many
among the principles discussed above would have to be reconsidered, notbecause of what they say but because of their classical, bivalent presuppositions. For example, the extensionality theorem of EM , (31), saysthat composite things with the same proper parts are identical, and thiswould call for qualifications: the model in Figure 1.7, left, depicts x andy as non-identical by virtue of their having distinct determinate parts;
yet one might prefer to describe a situation of this sort as one in which
the identity between x and y is itself indeterminate, since it is indeterminate whether they really have distinct parts. Conversely, the modelon the right depicts x and y as non-identical in spite of their having the

30

x •

•

•y
•

Figure 1.7.

x •

•

•

•y
•

•

Objects with indeterminate parts (in grey)

same determinate proper parts; yet again one might prefer to suspend
judgment owing to the indeterminacy the middle element.
That there are vague objects in this sense, however, i.e., objects whose
mereological composition may to some extent be objectively indeterminate, is all but obvious. Surely a statement such as(47)

x is part of Everest

may lack a definite truth-value, if x lies somewhere in the borderline area.
But—it could be argued—this need not be due to the way the world is.
The indeterminacy of (47) may be due exclusively to semantic factors—
not to the vagueness of Everest but to the vagueness of ‘Everest’. Whenthe members of the Geodetic Office of India baptized the mountain after
the name of their British founder, they simply did not specify exactly
which parcel of land they were referring to (or which parcel of land
constituted the mountain they meant to name).6 The referent of theirterm was vaguely fixed and, as a consequence, the truth conditions of astatement such as (47) are not fully determined; yet this is not to saythat the stuff out there is mereologically vague. Each one of a large
variety of slightly distinct parcels of land has an equal claim to thevaguely introduced name ‘Everest’, and each such thing has a perfectly
precise mereological structure. To put it differently, a statement such as(48)

It is indeterminate whether x is part of Everest

admits of a de re reading, as in (48a), but also of a de dicto reading, asin (48b):(48a) Everest is a y such that: it is indeterminate whether x is partof y.
(48b) It is indeterminate whether: Everest is a y such that x is partof y.
The first reading corresponds to the initial thought, to the effect that
Everest’s parts are indeed indeterminate, with the consequence that
6 That mountains are just parcels of land is, of course, a substantive assumption:an antiextensionalist may want to deny it, as with Tibbles and the relevant amount of feline tissue(Section 1.3.2). On the ontology of topographic entities, see e.g. Smith and Mark (2003).

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

31

mereology ought to be revised as seen above. The second reading, bycontrast, corresponds to the idea that it is the semantics of ‘Everest’that is indeterminate, and there is no reason to suppose that this is due
to some objective deficiency in the parthood relation—hence no reason
to require revisions in the apparatus of mereology itself. (Ditto for theother cases mentioned above. The reason why it’s indeterminate whethera certain molecule is part of a cloud, a tree part of a forest, or the candy
part of John, is not that such things are mereologically indeterminate;rather, on a de dicto understanding the indeterminacy lies entirely in
our words, in the terms we use to pick out such things from a multitude
of slightly distinct but perfectly determinate potential referents.)
If the semantic conception is accepted, then, the problem of vaguenessdissolves. Or rather: it ceases to be an issue for mereology and it becomesa problem for semantics broadly understood—a problem that manifests
itself in many contexts besides those under consideration. (How muchmoney do you need to be rich? How slowly can you run? How late canI call you?) Again, there are many things one could do at this point. A
favored option is afforded by so-called supervaluational semantics, whose
first application to vagueness can be traced back to Fine (1975). According to such semantics, the truth-value of a sentence involving vague terms
is a function of its truth-values under the admissible precisifications of
those terms: the sentence is true if it is true under every precisification,false if false under every precisification, and indeterminate otherwise.
Thus, if x is in the borderline area, then the indeterminacy of (47) is
explained by the fact that among the many admissible ways of precisifying the term ‘Everest’, some would pick out a referent that extends far
enough to include x among its parts whereas others would not, which is
to say that (47) would be true on some but not all precisifications. By
contrast, if x were clearly part of Everest given the way the name is usedin ordinary circumstances, or if it were clearly not part of Everest, then(47) would have a definite truth-value, for every precisification would
yield the same response (always true and always false, respectively). We
need not go into the details here. But three things are worth noting.
First, none of this will have any impact on the mereological axioms
considered so far. For those axioms are expressed as (implicitly) universally quantified formulas involving no singular terms except for variables,and variables are not the sort of expression that can suffer from the phenomenon of vagueness. Variables range over all entities included in thedomain of quantification and pick out their values independently of anyvagueness that may affect the non-logical vocabulary.
Second, any model that satisfies a given axiom or theorem satisfies
also any substitution instance that can be obtained by replacing one

32
or more variables with corresponding names or descriptive terms. For
example, the following sentence is a substitution instance of (P.1):(49)

Everest is part of Everest.

and it is easily verified that a supervaluational semantics will make (49)true in every model of M . For insofar as reflexivity is meant to hold forevery entity in the domain, the truth of (49) is guaranteed no matter
which entity we elect as the referent of ‘Everest’. Likewise, the following
is a substitution instance of (31), the extensionality theorem of EM :(50)

As long as they are non-atomic, Everest and Sagarmatha are
the same if and only if they have the same proper parts.

(‘Sagarmatha’ is the Nepalese translation of ‘Everest’, though there is no
guarantee that both names admit of the same precisifications.) Again,
it is easily verified that a supervaluational semantics will make (50) truein every EM model. For no matter how we precisify the terms ‘Everest’and ‘Sagarmatha’ by tracing a precise boundary around their referents,
the extensionality of parthood will guarantee that the referents coincide
just in case their proper parts coincide too.
Finally, it is worth emphasizing that a supervaluational semantics
is perfectly adequate to classical logic (Fine 1975, McGee 1997, Varzi
2001). For example, although it does not obey to the semantic principleof bivalence, as with various instances of(51)

Either ‘x is part of Everest’ is true or ‘x is part of Everest’is false,

it certainly satisfies the logical law of excluded middle: any instance of(52)

Either x is part of Everest or x is not part of Everest

is bound to be true, for it is true on any precisification of ‘Everest’.
Things would change, however, if the language were enriched by adding
an explicit sentential operator to express indeterminacy. In that case,the following principle would give expression to the assumption thatparthood admits of no objective borderline cases:(P.16)

It is determinate whether Pxy,

Determinacy

though it is obvious that this principle may have invalid instances assoon as ‘x’ or ‘y’ is replaced by a vague singular term such as ‘Everest’,as in (48). At the moment, the logic of determinacy operators is an
open area of research, especially in the context of a supervaluational

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

33

semantics. (See Keefe 2000, §7.4 and references therein.) It is, however,
an important tool that any good theory of vagueness should countenance.
And it is bound to play a significant role in any application of the theory
to mereology and spatial reasoning broadly understood.

2.

Philosophical issues in topology

Let us now move on to the second major ingredient of a comprehensive
theory of spatial reasoning—topology. There are many reasons for thismove, but the main one is simply this: one need go beyond the boundsof a pure theory of parthood to come out with a true theory of parts andwholes. For as we have already mentioned, mereology by itself cannot
do justice to the notion of a whole (a one-piece, self-connected wholesuch as a stone or a whistle, as opposed to a scattered entity made up of
several disconnected parts, such as a bikini or a broken glass). Parthood
is a relational concept, wholeness a global property, and the latter justruns afoul of the former.
Whitehead’s early attempts to characterize his ontology of events, aspresented at length in his Enquiry (1919) and in The Concept of Nature(1920), exemplify this difficulty most clearly. The mereological system
underlying Whitehead’s ontology was not meant to admit of arbitrary
wholes, but only of wholes made up of parts that are “joined” or connected to one another—specifically, finitary sums of such parts. Thus,
Whitehead was working with a composition principle patterned after(P.11ψ ), in fact with the corresponding biconditional, with ‘ψ’ understood as a predicate expressing the relevant relation of connection. And
Whitehead’s characterization of this relation was purely mereological:7(53)ψxy =df ∃z(Ozx ∧ Ozy ∧ ∀w(Pwz → Owx ∨ Owy))

Looking at the spatial patterns in Figure 1.8, we can see how this definition is intended to work. What distinguishes the connected sum x + yon the left from the disconnected sum in the middle? Well, in the former case it is easy to find regions, such as z, that overlap both x and ywithout outgrowing the sum—regions that lie entirely within x + y. By
contrast, in the middle pattern it would seem that every z overlapping
both x and y will also overlap their complement—the entity that surrounds x+y. Thus, only the left pattern satisfies the condition expressedby ‘ψ’; the pattern in the middle violates it. However, Figure 1.8 also
shows why this account is defective. For nothing guarantees that the
7 The definition below corresponds to the formulation given in Whitehead (1920: 76). White-head’s earlier definition (1919: 102) is slightly different but essentially equivalent.

34zx
Figure 1.8.

z
y

x

z
y

x

z
y

A connected sum (left) and two disconnected sums (middle, right)

item z overlapping two “joined” items x and y be itself in one piece, so
the pattern on the right depicts two entities that satisfy the condition
expressed by ‘ψ’, too. Yet this is a case where we should like to say thatx and y are not connected. Of course, Whitehead would disqualify thecounterexample because his ontology does not contain any disconnectedzs—but this is plain circularity. The account works on the assumptionthat only self-connected entities can inhabit the domain of discourse, yetthat is precisely the assumption that (53) is meant to characterize.
These considerations apply mutatis mutandis to other attempts to
subsume topological connectedness within a bare mereological framework (see e.g. Bostock 1979, Needham 1981, Ridder 2002). Nor is thisexclusively an ontological concern. These limits show up in any attempt
to account for a number of important spatial concepts besides connectedness, such as the distinction between a completely interior part anda tangential part that is connected with the exterior, of the difference
between an open entity and a closed one. All of these—and many others
indeed—are relations that any theory concerned with the spatial structure of the world should supply and which cannot, however, be defined
directly in terms of plain mereological primitives.

2.1‘Contact’ and connection

It is here that topology comes into the picture. The connection relationthat Whitehead was seeking to characterize is a topological relation.
And if it cannot be defined in mereological terms, it must be formallytreated on independent grounds.
Before looking at how this can be done, it is important again to begin with a couple of terminological caveats. As with ‘part’, the term‘connection’, and cognate terms such as ‘contact’ and ‘touching’, havedifferent meanings in ordinary language, only some of which correspondto the intended relation. Most notably, in ordinary language we do not
draw a clear distinction between a truly topological notion of connection
and a merely metric notion of contact. Consider:(54) The handle is attached to the mug.
(55) The table is touching the wall.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

35

The relation exemplified by (54) is topological: the handle and the restof the mug form a unitary whole. For practical purposes there may beroom for free rein, depending on whether the handle is glued to the rest of
the mug (what Galton 2000, §4.2, calls “adhesion”) or truly continuouswith it (“cohesion”), but either way there is an obvious sense in which
we are dealing with a single, one-piece object. By contrast, the relation
exemplified by (55) is not topological but metric: the table is so close
to the wall that we are inclined to say they are connected. If space werediscrete, this might be the right thing to say. But if space is dense, as we
may plausibly assume, then the surfaces of two bodies can never truly
be connected, short of overlapping: there will always be a narrow gap
separating them. The narrower this gap, the easier it is to disregard
it for practical purposes, but genuine topological connection can only
obtain when the gap is reduced to zero. We shall see in Section 2.5.1
that some interesting topological relations may be introduced to capture
at least some uses of the metric relation of contact. Overall, however,
metric relations cannot be squeezed into the conceptual apparatus of
topology without the help of strongly simplifying assumptions on the
structure of space. (In this sense, a diagram such as Figure 1.8, left, isambiguous, since one might think of x and y as being merely close toeach other, as when we draw a picture of a table against a wall.)A related issue concerns the distinction between connection patterns
that involve a single point of contact, as in (56), or an extended boundary
portion, as in (57), if not of an entire boundary, as in (58):(56) Colorado is connected to Arizona.
(57) France is connected to Germany.
(58) The Vatican is connected to Italy.
To some extent this is a matter of convention. We may disregard (56)
as irrelevant, or we may treat it as an acceptable case.8 Since mosttheories go for the second option, we shall follow them on this score.
As it turns out, under suitable conditions one can on such basis draw
all the relevant distinctions (Section 2.5), so the choice proves to be aconvenient one.
Finally, let us just mention the fact that topological connection is,
in a way, an idealized relation. Physically, as we know, the world consists of objects that are not continuous (or dense) in the relevant sense,and speaking of their boundaries is like speaking of the “flat top” ofa fakir’s bed of nails (Simons 1991a: 91). Physically, a mug is just a
8 This may also depend on context. In chess, for example, the choice depends on whether weare a rook or a bishop.

36
swarm of subatomic particles whose exact shape and extension involves
the same degree of arbitrariness as a mathematical graph smoothed outof scattered data. In this sense, the intuition behind the claim that(54) provides a good example of topological connection betrays a naive
conception of the mid-size world. However, this is not to say that topology is inadequate as a tool for practical spatial reasoning. For, on the
one hand, we are generally interested in describing the spatial structure of the world precisely insofar as objects are conceptualized as finite
chunks of dense matter with closed, continuous boundaries. Even if talkof boundaries and contact were deemed unsuited to the ontology of thephysical sciences, one would therefore need it when it comes to the dense
entities carved out by ordinary discourse and to the spatial regions thatthese occupy. On the other hand, the geographic examples in (56)–(58)illustrate that at least some entities countenanced by common sense measure up to the strict standards of topological connection. We do wantto say that a geopolitical unit occupies a region of space that is strictly
dense in the relevant sense, even though the underlying territory may
consist of material stuff that on closer inspection is best described as agerrymandered aggregate of zillions of disconnected subatomic particles.

2.2

Basic principles and definitions

We are now in a position to take a closer look at the idea of a topologicalextension of mereology—an extension that would take us beyond theprospects of a pure theory of parthood to deliver a genuine theory ofparts and wholes. To this end, let us expand our formal language byadding a second distinguished predicate constant, ‘C’, to be understood
intuitively as the relation of topological connection. The question of how
mereology can actually be expanded to a richer part–whole theory may
then be addressed by investigating how a P-based mereological systemof the sort outlined in Section 1 can be made to interact with a C-based
topological system.
Again, we may distinguish for this purpose “lexical” from substantive
postulates for ‘C’, regarding the former as embodying a set of minimal
prerequisites that any system purporting to explicate the meaning of
the concept of ‘connection’ must satisfy. And a natural starting point
is to assume that such lexical principles include at least the twofold
requirement that ‘C’ be reflexive and symmetric:
(C.1)
(C.2)

Cxx
Cxy → Cyx

Reflexivity
Symmetry

There is little room for controversy concerning the intuitive adequacy of(C.1)–(C.2), provided that we take ‘C’ to express, not just the relation of

37

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

external connection that may obtain between two disjoint entities thatshare a common boundary (in some intuitive sense to be made precise),but the relation of connection that may obtain between any two entities
that share at least a boundary. In this sense, mereological overlap qualifies as connection, too. We shall come back to this idea shortly. First,
let us note that ‘C’ need not, on the intended interpretation, expressa transitive relation. France is connected to Germany and Germany to
Poland, but France and Poland are not connected—they do not share any
common boundary. We can, however, consider a notion of connectionthat captures the fact that France is connected to Poland by Germany.
De Laguna (1922), a forerunner in the area of qualitative topologicalreasoning, actually based his account on a three-place primitive corresponding to this relation.9 In terms of ‘C’, De Laguna’s primitive iseasily defined:(59)

BCxyz =df Cxz ∧ Czy

By-Connection

and we can accordingly introduce the desired notion of (possibly) indirect
or mediate connection as follows:(60)

MCxy =df ∃zBCxyz

Mediate Connection

By an obvious generalization, we can also define:(61)

MCn xy =df ∃z1 ...zn (Cxz1 ∧ ... ∧ Czn y)n-Connection

It follows immediately from (C.1) and (C.2) that each MCn is reflexive
and symmetric, and the union of all such relations is transitive. In theabsence of further principles, however, e.g., principles guaranteeing theexistence of an entity connected to all the intermediate links, such atransitive union cannot be defined in the object language—unless weallow for quantification over positive integers:(62)

TCxy =df ∃nMCn xy

Transitive Connection

Now, let T be the first-order theory defined by the two basic axioms(C.1) and (C.2), in analogy with the theory M defined by the basic mereological axioms (P.1)–(P.3). T is, of course, extremely weak and a lot
will have to be added before we can say we have an interesting topology.
In particular, a model of T can be obtained simply by interpreting ‘C’
as mereological overlap, and what further principles should be added to
T so as to distinguish C from O is precisely one of the questions a good
9 Strictly speaking, De Laguna’s primitive is interpreted as “x can be connected to y by z”,

so it involves a modal ingredient. For a formal treatment, see Giritli (2003).

38
topological theory is meant to answer. For instance, should one assume
that connection is extensional, i.e., that things that are connected exactly to the same entities are identical? Should one assume that any twoconnected entities satisfy at least some form of Whitehead’s account in(53)? Or consider the binary relation defined by(63)

Exy =df ∀z(Czx → Czy)

Enclosure

It follows from (C.1)–(C.2) that this relation is reflexive and transitive,
and if C is extensional, than E is also antisymmetric—a partial ordering.
Should one assume this relation to satisfy any analogues of the axioms
for parthood? For each mereological predicate defined in Section 1 using‘P’ one could now introduce a corresponding topological predicate using‘E’ instead. Should one assume any corresponding axioms?
As it turns out, it is difficult to answer these questions in an abstract
setting (see Cohn and Varzi 2003). Obviously, much depends on how
exactly ‘C’ is interpreted, and that in turn may depend on how one
thinks ‘C’ and ‘P’ should interact. Rather than pursuing these questionsin isolation, then, let us proceed immediately to examining the main
options for combining mereology and topology.

2.3

Bridging principles

The simplest option is just to append the T -axioms to our preferred mereological theory, X, to obtain a corresponding “mereotopology” X + T ,which can then be strengthened by supplying further axioms for ‘C’.
However, this would be of little interest unless one also adds some mixed
principles to establish an explicit “bridge” between X and T .

2.3.1
Parts and wholes.
There is one sort of bridging principle that most theories, if not all, accept: it centers around the intuition
that no matter how P and C are fully characterized, they must be related
in such a way that a whole and its parts are tightly connected. Here arethree ways one can try to capture this intuition:(C.3a )
(C.3b )
(C.3)

Pxy → Cxy
Oxy → Cxy
Pxy → Exy

Integrity
Unity
Monotonicity

The first principle, (C.3a ), is perhaps the most immediate: just as
everything is connected to itself by (C.1), everything must be connected
to its constitutive parts. This is not to say that the parts must all beconnected to one another: the two main parts of a bikini are not. But
they are, in an obvious sense, connected to their sum; they are detachedfrom each other but not from the whole bikini.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

39

As it stands, however, (C.3a ) is extremely weak. It doesn’t evencapture the idea that if something is part of two things, then those things
are thereby connected. This is not to say that they are connected by that
common part, in the sense defined in (59); they are connected becauseof that common part. In other words, if sharing a common boundary is
to count as sufficient for connection, then a fortiori sharing a common
part ought to be sufficient, too. It is in this sense that overlap is to be
regarded a special (and somewhat trivial) case of connection. And thesecond principle, (C.3b ), makes this explicit.
(C.3b ) is stronger than (C.3a ), since parthood implies overlap. Moreover, since the converse need not hold (on pain of trivializing the notionof connection), (C.3b ) provides the intuitive grounds for defining a nontrivial notion of external connection, or touching, that can only holdbetween mereologically disjoint entities:(64)

ECxy =df Cxy ∧ DxyExternal Connection(Note that this relation is still symmetric, but not reflexive; it is actually
irreflexive, due to the irreflexivity of D.) This is an important notion,
which makes all the difference between mereology and mereotopology.
Yet (C.3b ) is still too weak to capture the fundamental intuition that
we are after. For while this principle guarantees that overlapping apart is sufficient for being connected to the whole, it doesn’t secure thattouching a part is also sufficient. Surely something can touch a mug(say) just by touching its handle. So it is only with the third principle,(C.3), that we get a plausible formulation of the basic idea. Connection,
if it is to behave properly, must be monotonic with respect to parthood.
It is easily checked that (C.3) implies (C.3b ), hence (C.3a ), so let us
just focus on (C.3), and let us call M T (for Minimal (mereo)Topology)
the corresponding extension of T .10 Whether (C.3) is to be classified asa “lexical” principle may be controversial and will depend, in an obvious sense, on the underlying axioms for ‘P’. Nonetheless, the principle
itself is part of virtually every mereotopological theory in the literature,
either as an axiom (Varzi 1996a, Donnelly 2004) or as a theorem. And
although M T is still far from providing an adequate characterization of
the relation of topological connection, it provides the basis for the definition of a number of important spatial relations which, like EC, cannot
be distinguished within a purely mereological setting. In particular, we
can now express the difference between a proper part that lies entirely
10 In fact, the result of adding (C.3) to T yields a slightly redundant theory; a more elegant

formulation can be obtained by dropping (C.2) and replacing (C.3) with the following variant:(C.30 ) Pxy ∧ Cxz → Czy

40
Uxn y

x3
x2

x5
x6 = y

x1
x8

x4

x7

ECxn yOxn yTPPxn y IPP xn y EQxn y TPExn y EPExn y

n=1

+

–

–

–

–

–

–

–

n=2

+

+

–

–

–

–

–

–

n=3

+

–

+

–

–

–

–

–

n=4

+

–

+

+

–

–

–

–

n=5

+

–

+

–

+

–

–

–

n=6

+

–

+

–

–

+

–

–

n=7

+

–

+

–

–

–

+

–

n=8

+

–

+

–

–

–

–

+

Figure 1.9. Basic mereotopological relations.
darker shading stands for parthood.)(Shaded cells indicate connection;

within the interior of the whole and a proper part that is connected withthe exterior:
(65)
(66)
(67)
(68)

IPPxy =df PPxy ∧ ∀z(Czx → Ozy)
TPPxy =df PPxy ∧ ¬IPPxy
EPExy =df PExy ∧ ∀z(Czy → Ozx)
TPExy =df PExy ∧ ¬IPExy

Interior PP
Tangential PP
Exterior PE
Tangential PE

Note that, given (64), interior parts satisfy the following:(69)

IPPxy ↔ (PPxy ∧ ¬∃z(ECzx ∧ ECzy))Thus, tangential parts are those parts that reach far enough to touch
something with which the whole itself is just in touch. Similarly forproper extensions. The diagram in Figure 1.9 indicates how these predicates may represent a genuine addition to the mereological vocabulary
introduced in (19)–(23) and illustrated in Figure 1.1. We shall see inSection 2.4 that this diagram may actually be misleading, owing to thedelicate role played by boundaries in the proper understanding of theconnection relation; but for the moment we take the intuitive, geometricinterpretation of the diagram to be adequate enough to serve its purpose.

2.3.2
Parthood vs. enclosure.
Things begin to be controversial as soon as we consider the possibility of stronger bridging principles.
Consider again the three principles above. Clearly the converse of Integrity, (C.3a ), is unacceptable. And unacceptable is also the converseof Unity, (C.3b ), for then connection would collapse on the relation of
mereological overlap and the definitions in (64)–(68) would lose theirintuitive appeal. On the other hand, the converse of the Monotonicity
principle (C.3) is not obviously unreasonable:(C.4)

Exy → Pxy

Converse Monotonicity

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

41

This says that a sufficient condition for one thing to be part of anotheris that whatever is connected to the former is also connected to latter.
This sounds intuitive, and several authors would actually include (C.4)
as a further bridging principle on top of M T . Indeed, a principle along
these lines may already be found in Whitehead’s latest work, Process
and Reality (1929)—a remarkable fact, since the conjunction of (C.3)
and (C.4) yields a biconditional that would allow one to define parthoodin terms of connection:(70)

Pxy ↔ ExyThus, if (53) turned out to be a defective attempt to reduce topologicalconcepts to purely mereological ones, (C.4) (together with (C.3)) reflects
a reductionist attempt in the opposite direction, to the effect that mereological concepts can be defined in terms of purely topological ones. And
although few have followed Whitehead on the first route, it is a fact thatmany authors have taken the second strategy into serious consideration.
Clarke (1981) provides the most influential example, and the so-called“Region Connection Calculus” originated with Randell et al. (1992) isthe best case in point when it comes to theories designed specifically for
applications to spatial reasoning (see also Gotts et al. 1996 and Cohn etal. 1997). So the question deserves close scrutiny: Is (C.4) a reasonableaddition to the basic postulates of M T ?
Never mind the fact that working with just one primitive may bemathematically attractive. As it turns out, it is equally possible to rely
on a single primitive even in the absence of (C.4). For instance, one canrely on the ternary relation ‘x and y are connected parts of z’ (Varzi
1994). Writing this as ‘CPxyz’, one could define ‘P’ and ‘C’ as follows:
(71)
(72)

Pxy =df ∃zPxzy
Cxy =df ∃zPxyz

and then go on to develop a theory based on the relative irreducibilityof these two predicates. (Note that (71) and (72) only carry minimal
presuppositions: that parthood be reflexive and that connected entities
have an upper bound.) So the issue is not formal economy—the useof a single primitive. It is, rather, conceptual economy. In the extension of M T obtained by adding (C.4)—henceforth RM T , for Reductive
Mereotopology—the notion of parthood is fully subsumed under that ofconnection, and the limits of mereology are overcome by turning theoriginal problem upside down: parthood cannot deliver the full story,
but connection can. However, there are at least two worries here.
The first worry concerns the material adequacy of the reduction. As
Masolo and Vieu (1999) have observed (but the point goes back to Ran-

42

y

z

x
Figure 1.10.

yzx
Two counterexamples to converse monotonicity

dell et al. 1992: §5.1), (C.4) appears to have implausible consequences
if the domain contains entities with atomic proper parts. Consider an
extended region (or object) x, and let y be x minus an atomic part z(Figure 1.10, left). On any reasonable understanding of ‘C’, everything
connected to x is connected to y, since z is connected to both. So (C.4)
would force x to be part of y. Yet, intuitively, x should count as an extension of y: it is bigger, it contains z, it contains y as a further properpart. Things get worse if we consider that (C.4) forces z itself to countas part of y, since z is connected to y and anything else is connected
to z only if it overlaps y. Yet y was defined as x minus z. Of course,such models would be ruled out if RM T were strengthened by adding
an atomlessness postulate such as (P.7). But this is precisely the point:(C.4) does not merely reinforce the bridge between P and C; it actually embodies more substantive views about the mereological structureof space. Besides, even the atomless variant ĀRM T would be open tocounterexamples. For we have the same sort of problem if we supposethat z is a non-atomic proper part of x with no interior proper partsof its own (Figure 1.10, right). To rule out this model, a stronger assumption than (P.7) would be needed, corresponding to the thesis thateverything has interior proper parts:(C.5)∃xIPPxy

Boundarylessness

In fact, it is precisely with the help of an axiom like (C.5) that we cangive expression to a Whiteheadian, boundary-free conception of space(see below, Section 2.4.3). However, this is a controversial conception.
Why should our analysis of parthood force upon us a rebuttal of theboundary concept? Why should we assume (C.5) in order to ensure acoherent implementation of a basic bridging principle like (C.4)?
The second worry is more general. Consider an object and the stuff
it is made of—for instance, a statue and the corresponding amount ofclay. As we have seen, few would regard the relationship of material constitution that holds between such entities as a case of proper parthood.
And there are many philosophers for whom constitution is not identity

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

43(improper parthood) either: see again the papers in Rea 1997. This is by
no means a pacific thesis, but never mind. The point is simply that therelationship between the clay and the statue is not obviously an instanceof parthood. Yet, on any plausible understanding of ‘C’, whatever isconnected to the clay is bound to be connected to the statue, too, so(C.4) would immediately settle the issue: the clay is part of the statue.
Indeed, since it is equally plausible to suppose that the same applies in
the opposite direction—whatever is connected to the statue is connectedto the clay—(C.4) implies that the statue and the clay are one and thesame thing. And this is a substantive tenet which, as such, ought not
to be built into the basic apparatus of mereotopology at the outset.
This worry is perhaps best appreciated by noting that the following
theorem is an immediate consequence of (C.1)–(C.4), provided P satisfies
the basic mereological axioms (P.1)–(P.3):(73)∀z(Czx ↔ Czy) → x = y

In fact, (P.1) and (P.2) (reflexivity and transitivity) are derivable from(70), and RM T turns out to be equivalent to the theory defined bytaking (73) as an axiom along with (C.1) and (C.2). (That is actually
the customary axiomatization since Clarke 1981; see also Biacino andGerla 1991.) Now, with parthood construed as enclosure, (73) is nothing
but the antisymmetry principle (P.3). Yet (73) does not merely assert
the antisymmetry of parthood; it says that connection is extensional—
that different things cannot connect to the same things. And this is justas controversial as the thesis that parthood is extensional.
It could be replied that the analogy with mereological extensionality
is in fact helpful, since the original arguments in support of (P.5) (Section 1.3.2) could now be offered on behalf of (C.4). Indeed, if a statue
and the clay are construed as things that, at some level of decomposition,
share the same constituents—e.g., the same molecules—the question ofwhether constitution is identity just is the question of whether parthoodis extensional. However, the worry does not only apply to cases of material constitution. Consider a shadow cast onto a wall. The shadow is notpart of the wall, yet anything connected to the shadow is—arguably—connected to the wall. Or consider a stone inside a hole. The stone is
not part of the hole, yet one could argue that anything connected to thestone is connected to the hole. Broadly speaking, the problem arises assoon as we allow for the possibility that distinct entities occupy the same
space (Casati and Varzi 1999). That this is a real possibility is by itself
contentious and is one of the questions to be addressed by an explicit
theory of location (Section 3). But precisely for this reason, ruling it
out on mereotopological grounds would be utterly inappropriate.

44This last point is particularly worth stressing, for it shows once again
how the choice of a suitable set of principles may depend crucially on
whether we are interested in a theory aimed at modeling the domain of
all spatial entities or just a domain of pure spatial regions. The worry
mentioned above arises forcefully in the context of theories of the firstsort. It does, however, lose its force in relation to the second sort oftheory, since two regions cannot overlap spatially without overlapping
mereologically. Now, it is a fact that most authors committed to (C.4)
have been working on such a narrower understanding of their theory.
Whitehead’s own account was explicit in this regard: in the theory put
forward in Process and Reality (as in Clarke’s 1981 reformulation), thefield of C was meant to consist exclusively of spatial regions. On theother hand, it is also a fact that a major motivation for developing atheory of this sort has been the assumption that connection thus understood is all that matters for practical purposes. For one can always treat
the relation of connection as the “shadow” (in De Laguna’s 1922: 450 apt
terminology) of the relation of physical contact or overlap that may obtain between actual, concrete entities. In other words, such theories havetypically been developed on the assumption that the following principle
provides the necessary and sufficient link between the mereotopology ofpure space and the mereotopology of spatial entities broadly understood:(74)

x is connected to y if and only if the region occupied by x
is connected to the region occupied by y.

If so, however, then the problems mentioned above resurface even for
theories of this sort. For (74) will deliver an acceptable account if, andonly if, spatial co-location is regarded as metaphysically impossible.

2.3.3
Self-connectedness.
There are other ways of supplementing M T with bridging principles that go beyond (C.3). In particular, consider again Whitehead’s early attempts to characterize topological connection in terms of parthood, i.e. (53), with ‘ψ’ understood as‘C’. As a definition, this was found defective. However, one may certainly consider adding the corresponding biconditional as an axiom—orat least adding one of the two conditionals:(C.6) Cxy → ∃z(Ozx ∧ Ozy ∧ ∀w(Pwz → Owx ∨ Owy))(C.7) ∃z(Ozx ∧ Ozy ∧ ∀w(Pwz → Owx ∨ Owy)) → Cxy

Left Join
Right Join

Would this be a good way of tightening the conceptual link between the
mereological and the topological ingredients of M T ?
Consider (C.6). As it turns out, its status depends significantly on theunderlying axioms for ‘P’. If our mereological theory is strong enoughSpatial Reasoning and Ontology: Parts, Wholes, and Locations

45

to warrant the existence of a fusion for any pair of connected entities,i.e., if it contains the relevant instance of (P.11ψ ) as an axiom or (as inGEM ) as a theorem, then (C.6) itself is derivable as a theorem, since thefusion of x and y is sure to qualify as a z satisfying the consequent. If,
however, our theory does not warrant all the relevant fusions, then (C.6)
may still be regarded as a plausible addition to M T . Perhaps the fusionof any two connected entities turns out to be too large or gerrymandered
to be acceptable, but we may still think of connection as sufficient for
the existence of smaller fusions encompassing those portions of x and ythat are sufficiently close to their common boundary. For example, withreference to Figure 1.11, left, suppose we are only willing to acknowledge
the existence of entities that are composed of at most two disconnectedparts. Then the fusion of x and y is out, but a fusion of x2 and y2 , aswell as fusions of x and y2 and of x2 and y, would fit the bill. All of thisspeaks in favor of (C.6), though it may be argued that the existential
import of this principle goes beyond the task of establishing a necessary
conceptual link between P and C.
As for (C.7), the picture is different. Assuming this principle is virtually tantamount to excluding disconnected entities from the domain—
not all of them, to be sure, but many of them. For example, if theunderlying mereology is sufficiently weak, (C.7) is compatible with theexistence of a disconnected composite such as c in Figure 1.11, right:
on the assumption that c has no further proper parts besides a1 , a2 ,and a3 (and parts thereof), the antecedent is false so (C.7) is vacuously
satisfied. But consider a bikini, or a printed inscription consisting ofseparate letter tokens. As we have already noted, one need not buy into
unrestricted composition to appreciate the dignity of such things. Yet
their existence would be banned by (C.7). If x and y are the two mainparts of a bikini, then the consequent of (C.7) is false even though theantecedent is made true by the bikini as a whole. So, on the face of it,
this direction of Whitehead’s biconditional is definitely too strong as a
general bridging principle and there is no philosophically neutral reason
to add it to M T . One can, however, consider weaker versions, to theeffect that the consequent of the conditional must hold whenever theantecedent is made true by the right sort of entity:(C.7φ ) ∃z(φz ∧ Ozx ∧ Ozy ∧ ∀w(Pwz → (Owx ∨ Owy))) → Cxy
In particular, one can take ‘φ’ to express the property of being selfconnected. After all, this was precisely the intended import of Whitehead’s flawed definition. And we have seen that the flaw of the definition,in the if direction, was not conceptual but formal: it lied exclusively in
the impossibility of expressing the relevant restriction in mereological

46
x2

x1
x

y2

y1

a1

a2

y

Figure 1.11.

a3cLeft joining with partial fusions; right joining without connection.

terms. By making the restriction explicit, (C.7φ ) overcomes the difficulty and suggests itself as a natural bridging principle.
Surprisingly, it is not easy to express the property of self-connectedness
even in the extended language of mereotopology. If the axioms on P
are strong enough, we can follow the ordinary set-theoretic definition—
something is self-connected if it doesn’t consist of disconnected parts:(75)

SCx =df ∀yz(∀w(Owx ↔ (Owy ∨ Owz)) → Cyz)

Self-Conn.

In particular, in M T + GEM this becomes:(76)

SCx ↔ ∀yz(x = y + z → Cyz)

This is a common definition in the literature, both among theorists subscribing to the converse monotonicity principle (C.4) (see e.g. Clarke
1981, Randell et al. 1992) and among theorists rejecting it (Tiles 1981,
Varzi 1994, Smith 1996). However, if the axioms on P do not secure thenecessary composition patterns, the definition is inadequate. For example, the object c in Figure 1.11, right, is anything but self-connected,
yet it (vacuously) satisfies the definiens of (75) unless we assume theexistence of at least one sum consisting of a1 and a2 , a2 and a3 , or a1
and a3 . In the finitary case, the difficulty could be met by relying on thenotion of mediate connection: any two parts of a self-connected entity
must be at least n-connected for some n. More generally:(77)

PCx =df ∀yz(Pyx ∧ Pzx → TCyz)

Path Connectedness

This, however, involves quantification over numbers (see (62)), whichjust confirms the expressivity limits in question. Moreover, (77) does
not work in the infinitary case: the unit interval on the real line isconnected, but we cannot account for this fact in terms of the relationships between the reals themselves; reference to subintervals is necessary, specifically reference to a subinterval and its relative complement.
So, overall it appears that the notion of self-connectedness can be adequately grasped, via (75), only by theories that are at least as strong as
M T + (P.6), the complementation principle, though this is at present
an open question.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

47

•
•
•x
Figure 1.12.

•
•y

•z

An implausible model of M T + GEM .

2.3.4
Fusions.
We conclude this discussion of bridging principles by noting that even a theory as strong as M T + GEM + any of
the above axioms is incapable of capturing all the relevant links between
mereology and topology. In particular, such a theory is consistent withthe following implausible thesis (Tsai 2005: 137):(78)∃z(Cz(x + y) ∧ ¬Czx ∧ ¬Czy)

A model is given in Figure 1.12, where the curve line indicates the relevant connection relationship (besides the obvious ones imposed by (C.3)).
Clearly, this is a sign that some additional bridging principle is on
demand. The following option suggests itself:(C.8) z = Σxφx → ∀y(Cyz → ∃x(φx ∧ Cyx))

Fusion Connection

Whether this is enough to establish a good correlation between the mereological structure of composite objects and their topological behavior is aquestion that can hardly be addressed in general terms. The plausibilityof (C.8), however, seems obvious. Since we have found good reasons toalso accept (C.4) and (C.7SC ), the theory resulting by adding these threeprinciples to M T + GEM suggests itself as the natural topological extension of GEM . (Recall that (C.6) is already provable in M T +GEM .)For future reference, we shall call this theory GEM T , for General Extensional Mereotopology.

2.4

Extensions and restrictions

As it turns out, GEM T does not officially appear in the literature,mostly due to the limited study of such principles as (C.7SC ) and (C.8).
(The closest relatives are the axiomatic systems advocated by Smith
1996 and Casati and Varzi 1999.11 ) A good thing about this theory, as
11 In Casati and Varzi (1999), GEM T is identified with M T + GEM . Smith’s (1996) version

is based on a primitive ‘IP’ for (possibly improper) interior parthood. See also Pianesi and
Varzi (1996a, 1996b) for similar formulations based on the predicate ‘B’ and the operator ‘c’defined below, respectively.

48
also about the core fragment afforded by M T + GEM , is that it makes
it possible to supplement the mereotopological predicates and operators
discussed so far with a number of additional operators that mimic thestandard operators of point-set topology. For example:
(79)
(80)
(81)
(82)

ix =df Σz∀y(Czy → Oxy)ex =df i(∼ x)
cx =df ∼ (ex)bx =df ∼ (ix + ex)

interior
exterior
closureboundary

Like the mereological operators in (40)–(44), these operators are partially defined in view of the lack of a null entity that is part of everything.
For instance, if x is a boundary, it has no interior, and if x is the universal
entity U, it has no exterior. Even so, in GEM T all of these operators
are rather well-behaved. In particular, we can get closer to standard
topological theories by explicitly adding the mereologized analogues ofthe standard Kuratowski (1922) axioms for topological closure:(C.9) Px(cx)(C.10) c(cx) = cx
(C.11) c(x + y) = cx + cy

Inclusion
Idempotence
Additivity(These axioms are to be read as holding whenever c is defined for its
arguments. Here and below we omit the relevant existential conditions
to improve readability.) Indeed, (C.9) and (C.11) turn out to be provablein M T + GEM ; see Tsai (2005: 141).
The possibility of supporting such developments is of course a good
indication of the strength of GEM T . Philosophically, however, thisstrength may be regarded with suspicion, and several complaints havebeen raised in the literature.

2.4.1
The open/closed distinction.
The main sort of complaint concerns the very notion of connection that the theory is meant tocharacterize. So far we have worked mostly with an intuitive notion inmind but obviously more can and must be said—and GEM T says a lot.
In particular, the Kuratowski extension of GEM T (KGEM T for short)yields a full account of the intended interpretation of ‘C’: two things are(externally) connected if and only if they share (only) a boundary, i.e.,
if and only if the closure of one overlaps the other, or vice versa:
(83)
(84)

Cxy ↔ (Ox(cy) ∨ O(cx)y)ECxy ↔ (Cxy ∧ ¬C(ix)(iy))

Now, this shows in what sense the behavior of C in this theory closely
approximates that of standard set-theoretic topological connection; just

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

49

let ‘x’ and ‘y’ range over sets of points and interpret ‘O’ as set intersection. On the other hand, one aspect in which ordinary point-set topology appears to conflict with common sense—an aspect that has beenemphasized by authors interested in a mereotopological characterizationof qualitative spatial reasoning, such as Randell et al. (1992) and Gotts
et al. (1996)—is precisely the distinction between “open” and “closed”
entities on which it rests, and which GEM T preserves holus bolus:
(85)
(86)

OPx =df x = ix
CLx =df x = cx

Open
ClosedThis distinction goes at least as far back as Bolzano (1851: §66f). But
already Brentano (1906: 146) regarded it as “monstrous”, and we havealready seen that the sort of idealization it embodies does not sit wellwith the way we ordinarily speak. We may intuitively grasp the difference between an open and a closed interval on the real line—theobjection goes—and we may even understand how this difference applies to ideal three-dimensional manifolds such as Euclidean space. But
what does it mean to draw a similar distinction in the realm of concrete
spatial entities, where the very notion of a boundary is the result of aconceptual idealization? What does it mean to say that some objects
are closed and some are not, and that contact is only possible between
objects of one type and objects of the other?
Besides, even if common sense and ordinary language were put aside,
the open/closed distinction seems to yield genuine paradoxes as soon as
we move from the realm of pure space to its worldly population: Consider (i) what happens when a solid body splits into two halves. Before
the splitting the two halves were in contact, so we are to suppose that one
was closed and the other open, at least in the relevant contact area. But
then, after the splitting, only one of the two halves will have a completeboundary. This may not be “monstrous”, but it certainly seems implausible: the two halves—one should think—are perfectly indistinguishable.
On the other hand, consider (ii) what happens when two bodies comeinto contact. We may imagine the same experiment performed twice.
First we take an open cube and push it toward a closed cube until they
touch. Then we do the same with two closed cubes. What reason can
we offer to explain the fact that in the latter case the two cubes will
not come into contact? As Zimmerman (1996a: 12) put it, what sort of“repulsive forces” can be posited to explain such deferential behavior?
There is no quarrel that these are pressing questions. (For more examples, see Kline and Matheson 1987.) Nonetheless, there are various
things one can say in reply to such worries. Concerning (i), for example,
one could say that the paradox is grounded on a questionable model of

50

!

!
Figure 1.13.

!

!

!

!

!

!

!

Wrong (left) and right (right) topological models of splitting.

what happens when a process of “splitting” takes place (Varzi 1997).
Topologically, this is no bloodstained business. Dissecting a solid body
does not “bring to light” (Adams 1984: 400) a surface that was trapped
inside and that must by necessity belong to one of two severed halves.
Rather, the topological model is one of gradual deformation. Think ofa splitting oil drop. The drop grows longer and, as it grows, the middle
part shrinks and gets thinner and thinner. Eventually the right and left
portions come apart and we have two drops, each with its own completeboundary. Ditto with any splitting object. A long, continuous processsuddenly results in an abrupt topological change: there was one surface,
one closed body, and now there are two (Figure 1.13). Of course, one
can still raise a question about the last point of separation: Where does
this one point belong—to the left half or to the right half? However, thisis just a sign of the magic that surrounds any sort of topological change,
as when you drill a hole through an object. The instantaneous eventof a sphere turning into a torus is just as magical, and this magic has
nothing to do with the open/closed opposition. It simply reflects thefact that topological change marks one point at which common sense
reaches the limits of its theoretical competence, and a complete assessment would require a step beyond pure part-whole theorizing. It would
require a step into the territory of qualitative kinematics, for example(as in Davis 1993), if not an account in terms of the microscopic analysis
of matter. (What is—physically—the “last point of separation” involvedin the splitting?)Spatial Reasoning and Ontology: Parts, Wholes, and Locations

51

This line of reply on behalf of KGEM T applies to (ii) as well—the“merging” puzzle (Casati and Varzi 1999). Surely the positing of repulsive forces to explain the peculiar behavior of the two closed cubes
would be utterly and unagreeably ad hoc. But there are other possibilities. For instance, perhaps the two closed cubes will indeed come into
contact. From the fact that two closed entities cannot be in contact itdoes not follow that they cannot come into contact, just as from the factthat two parts are connected it does not follow that they cannot be separated. Only, the coming into contact (just as the separation) determinesa true topological catastrophe: there is a breaking through the relevantboundary parts and the two objects become one. (Think also of thetwo drops of oil merging into each other.) The two processes are dual:
merging is the reverse of splitting. And both involve a seemingly magic
moment that runs afoul of the confines of extensional mereotopology andcalls for a thorough kinematic account.
One could still press the objection here by noting that the puzzles admit of perfectly static variants, where the appeal to kinematics would beout of place. Consider the dilemma raised by Leonardo in his Notebooks:
What is it that divides the atmosphere from the water? Is it air or is it
water?. Or consider Peirce’s puzzle: What color is the line of demarcation between a black spot and its white background? (1893: 98). Moregenerally, given any object, x, does the boundary belong to x or to its
complement? Does it inherit the properties—e.g., color properties—of xor of ∼ x? There is no kinematic story to tell here. But how can one
answer without selecting one candidate at random?
Here one might reply that figure/ground considerations will help. According to Jackendoff (1987, Appendix B), for example, normally a figure owns its boundary—the background is topologically open. This may
well be the right thing to say vis-à-vis Peirce’s puzzle: the black spot
is closed, so the line is black. But what is figure and what is groundwhen it comes Leonardo’s case? We do talk about the surface of thewater, not of the air. But what goes on at the seashore? Three things
meet—water, air, soil; how can figure/ground considerations help in suchcontexts? Perhaps such dilemmas are not real. Galton (2003: 167f), forexample, argues that they arise as an artifact of the modeling process:surely properties like color or material constitution only apply to extended bodies, so it wouldn’t make sense to ask whether a boundary-like
entity is air, water, or colored. There is, however, a less dismissive way
to meet the challenge on behalf of KGEM T . For one may acknowledge
that such dilemmas are real and yet insist on a friendly attitude towards
the open/closed distinction. The actual ownership of a boundary—one
might argue—is not an issue that a mereotopological theory must be52able to settle. The theory only needs to explain what it means for two
things to be connected. Which things are open and which are closed is a
metephysical question that, plausibly enough, goes beyond the concernsof the theory. If the ocean is a closed body, then it can only touch the
air if the latter is open. If it isn’t, then it can only touch the air if thelatter is closed. And if both water and air are open, they cannot truly
touch, though they may touch a closed piece of land. That’s all thetheory says, and there is no reason to think that the theory is wrong
just because it is difficult to classify actual things into open and closed.
One last problem is worth mentioning. Consider again the cutting ofa solid object in half. We have said that this process does not bring tolight a new surface. But, of course, we can conceptualize a new, potential
surface right there where the cut would be. In fact, we can conceptualize
as many boundaries as we like, even in the absence of any corresponding discontinuity or qualitative heterogeneity among the parts. Thinkof John’s waist, the equator, the Mason-Dixon line between Maryland
and Pennsylvania. As Smith (1995, 2001) has pointed out, such “fiat”
boundaries are a major ingredient of our picture of the world. Eventhe surfaces of ordinary objects, as we have seen, may involve a certain
degree of fiat owing to the microscopic scatterdness of the underlying
stuff. Yet no fact of the matter can support the ownership of boundaries
such as these by one side rather than the other, hence there is no point
in deferring to a metaphysical theory of the extended entities at issue.
Isn’t this enough to give rise to the demarcation puzzle?
Once again the answer is in the negative. Fiat boundaries are notphysical boundaries in potentia. They are not the boundaries that would
envelop the interior parts to which they are associated in case those parts
were actually cut off. To think so would take us back to the wrong topological model illustrated in Figure 1.13, top. On a better model, fiat
boundaries are just placeholders for genuine physical boundaries and thedemarcation puzzle need not, therefore, lead to ontological anxiety. We
can say that a boundary of this sort stands for two boundaries, one foreach side. Or we can say that in drawing it we leave the question of its
belongingness (hence the open/closed distinction) indeterminate. We
do so because it is a question of no practical relevance. But precisely
for this reason the indeterminacy is innocuous: it is pragmatic, perhaps
semantic, not ontological—just like the sort of indeterminacy that afflicts the vagueness of parthood (Section 1.5). (It is in this sense thatthe diagram in Figure 1.9 is partly indeterminate: in saying that x2 is
externally connected to y, for example, we did not specify which of these
two regions owns the boundary in the relevant contact area. Ditto for
all other cases of external connection, as in Figure 1.8, left.)Spatial Reasoning and Ontology: Parts, Wholes, and Locations

53

2.4.2
Connection by coincidence.
All of this, of course, issubject to controversy. If the foregoing remarks are found compelling,
then the strength of KGEM T is vindicated and such theorems as (85)and (86) deliver a full and correct understanding of ‘C’. If not, however,then KGEM T will be deemed inadequate and the intended interpretation of ‘C’ remains unsettled. Are there any other options? We may
distinguish two main alternatives, depending on whether or not a rejection of the open/closed distinction is taken to be compatible with arealist attitude towards the ontological status of boundaries.
The realist option finds its best expression in those theories that attempt to provide a detailed reconstruction of the view Brentano put
forward in reaction to Bolzano’s “monstrous doctrine”, as in Chisholm(1984, 1993) and Smith (1997). According to this view, boundaries are
genuine denizens of the world of spatial entities, but their lack of properinterior parts makes them peculiar in two important respects (Brentano
1976, part I). First, they can never exist except as belonging to entities
of higher dimension. There are, in other words, no isolated points, lines,or surfaces, for boundaries are, in Chisholm’s terms, dependent entities.
Second, and more to the point, insofar as boundaries are not possessedof divisible bulk, they do not occupy any space and can therefore sharethe same location with other boundaries. They can coincide, and the
topological relation of external connection is to be explained, not via the
open/closed opposition, but in terms of genuine boundary coincidence.
Thus, we can speak of the surface of an object. But this single surface is
to be recognized as being made up of two parts, two perfectly coinciding
boundaries bounding the object and its complement, respectively.
As is clear, a rigorous formulation of such theories is no straightforward business. For one thing, it is not immediately obvious how toformulate the dependence thesis, both because of the modal ramifications that a good theory of ontological dependence would require (seee.g. Correia 2005) and because the relevant notion of dimension is by itself hard to characterize mereotopologically (Chisholm 1984). Ignoringsuch complications, and assuming GEM , one can capture the gist of thethesis as follows (Smith 1996):(C.12) SCx ∧ Bxy → ∃z(SCz ∧ BPxz ∧ ¬∃wBzw)

Dependence

where
(87)
(88)

Bxy =df Px(by)BPxy =df Bxy ∧ Pxy

Boundary
Boundary Part

In other words, every self-connected boundary is part of some selfconnected entity which it bounds and which is not itself a boundary.

54(The restriction to self-connected entities is to avoid that (C.12) be trivially satisfied by a scattered z containing x as an isolated proper part.)
Without the full mereological support of GEM , however, things are significantly more complex, among other reasons because of the apparent
elusiveness of the self-connectedness predicate ‘SC’ (Section 2.3.3.)
Secondly, and more to the point, a lot depends on how exactly one
understands the relation of spatial coincidence invoked by such theories to explain the phenomenon of (external) connection. Chisholm andSmith treat it as an undefined primitive, suitably axiomatized so as
to guarantee that coinciding entities have coinciding parts. (See also
Smith and Varzi 2000 for a similar treatment of the relation of coincidence between fiat boundaries.) Alternatively, one can embed the theory of coincidence into a theory of spatial location broadly construed:
to say that things coincide is to say that they literally share the same
location. Clearly, the choice between these two options is not just a
matter of taste. Treating coincidence as a primitive is in principle compatible with different metaphysical conceptions of the nature of space,
whereas the second option is best understood within the framework of a
substantivalist (Newtonian) conception, i.e., a conception according towhich space is an entity in its own right. In any event, it is apparent
that both options yield theories that are not strictly mereotopological,
since a third primitive—coincidence or location, respectively—needs tobe brought into the picture to provide a full account of the connection
relation. We shall not go into the details of the first option here, but
we shall have more to say about the second option in Section 3. For
the moment, let us just observe that construing coincidence explicitly interms of spatial co-location amounts to a partial reduction of topology
to mereology: connection between entities of a kind (space occupiers)
is reduced to overlap between entities of a different kind (their spatial
receptacles), as per the following principle:(89)

x is connected to y if and only if the region occupied by x
overlaps the region occupied by y.

This is by itself interesting, though we are obviously left with the taskof providing an account of the topology of space as such. And if theaccount is to match the strength of a theory such as KGEM T , thenthe open/closed distinction will at least be partially preserved. It will
be obliterated from the Brentanian realm of space occupiers, but space
itself would be Bolzanian. (Compare the initial worry: we can grasp
how the distinction applies to ideal manifolds such as the real line or
Euclidean space; it is when it comes to the realm of ordinary objects
that their classification into “open” and “closed” is problematic.)

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

Figure 1.14.

55

Open, closed, and semi-closed discs with the same area.

2.4.3
Omitting boundaries.
The alternative route is to avoid
the puzzles raised by the open/closed distinction by dismissing boundary
talk altogether. This is the anti-realist option.
Philosophically, this route is often motivated on its own grounds, forinstance because of the dubious ontological status of boundaries vis-à-vis
the microscopic analysis of the physical world (Stroll 1988), or because of
their suspect nature qua lower-dimensional entities (Zimmerman 1996b).
In the context of formal theories, however, the main motivation for doing away with boundaries is precisely the rejection of the open/closed
distinction vis-à-vis common sense. To use an example from Gotts etal. (1996: 57), Figure 1.14 depicts a disc with and without its boundary,
and with just part of its boundary. Of course, the depiction has to show
the boundary as having some finite thickness, which strictly speaking
it does not possess. But this is the very point that appears counter tocommon sense: all three discs, if superimposed, would cover exactly thesame area; yet the second disc includes unextended parts that the others do not, while the third includes some that the first does not. Suchdiscriminations—it is argued—are not warranted.
There are radical as well as moderate variants of this view. The
radical variants are represented by those theories that follow Whitehead(1929) in doing away with all boundaries. This amounts to assuming theboundarylessness axiom (C.5) in its full strength: everything has interior
proper parts. The moderate variants, by contrast, only assume someweaker version of the axiom in which the variable is suitably restricted
so as to range over entities of a certain sort:(C.5φ ) φx → ∃zIPPyx

Restricted Boundarylessness

In particular, relative to our present concerns it is natural to construe‘φ’ as a distinguished property of all concrete spatial bodies. This would
allow for the possibility that space as such include points and otherboundary-like elements, which means that the open/closed distinction
would be partially preserved. But as we have just seen, restricting thedistinction in this way may be enough to bypass the intuitive puzzles
that it raises, so this may well be a good compromise. For example,
Cartwright (1975) holds that concrete spatial bodies are the material

56
content of (regular) open regions of space, connection relations between
the former being explained in terms of overlap relations between theclosures of the latter:(90)

x is connected to y if and only if the closure of the region
occupied by x overlaps the closure of the region occupied by y.

As a mereotopological theory, this is of course another hybrid—just as
the theory behind (89)—for it requires an explicit treatment of locative
relations. Still, there is no question that (90) allows for a systematicboundary-free account of the mereotopology of concrete spatial bodies.
(The real challenge, rather, is to justify the claim that only some regions
are receptacles, e.g., only open regular regions; see Hudson 2002).
Let us focus on the radical variants. We have seen that positing (C.5)
is a necessary move for any reductive mereotopology based on the converse monotonicity axiom (C.4), and it is a fact that most theories that
accept one axiom accept the other as well. But let us put that aside for a
moment and let us just focus on (C.5). Where X is any theory including
M T , let B̄X be the corresponding boundaryless extension obtained byadding this axiom. What sort of mereotopology do we get?
As it turns out, the number of options is significantly constrained,
both mereologically and topologically. For example, surely X cannot
be atomistic, since (C.5) implies the atomlessness axiom (P.7). So any
model of B̄X is perforce infinitary. And surely the interaction between
compositional and decompositional principles will have to be carefully
re-examined. In particular, it is easy to verify that in B̄M T the weak
supplementation principle (P.4) is incompatible with the unrestrictedfusion axiom (P.13) and, more generally, with any version of the strong
fusion axiom (P.13ξ ) in which the condition ‘ξ’ is satisfied by all interior
and tangential proper parts of any given thing. For suppose we allow
for such fusions. Then every entity would have an interior as well as aclosure, and the following would hold:(91)

PP(ix)(cx).

By (P.4), this would imply(92)∃z(Pz(cx) ∧ ¬Oz(ix)),

which in turn would imply(93)∃zPz(bx),

contradicting (C.5). Thus, B̄M T + (P.4) + (P.13) is inconsistent, as is
any theory B̄X including (P.4) along with (P.13ξ ) with ‘ξ’ as indicated.

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

57

This is not surprising, since the whole point of going boundary-free is,in the present context, to avoid the open/closed distinction, hence thedistinction between interiors and closures reflected in (91). However,this means that (C.5) prevents the formulation of any reasonably strong
theory unless we are willing to give up weak supplementation, and thismay certainly be regarded as a major drawback of the approach.
In fact, one may consider both options here. One may (i) regard thecompositional weakness of the theory as a necessary price to pay topreserve mereological supplementation and avoid the topological conundrums surrounding the open/closed distinction. But one may also (ii)
go for a stronger theory with generalized or even unrestricted fusions,dismissing the conundrums precisely by forgoing supplementation. Afterall, it could be argued that the open/closed distinction is problematic
only insofar as it is cashed out in terms of a discrimination between entities that do and entities that do not possess their boundaries, and in
the absence of boundaries the discrimination dissolves. In the literature
the first option is more widespread, its closest representative being thegreatly influential Region Connection Calculus (RCC) originated with
Randell et al. (1992)—a reductive extension of B̄M T + (P.5) with binary sums and complements. But the second option, which is closer
to Whitehead’s original approach, is also well represented, as evidenced
by Clarke (1981), Biacino and Gerla (1991), and Asher and Vieu (1995)
inter alia (the first three admitting unrestricted fusion, the latter admitting fusions of interior parts). Indeed, we have seen in Section 1.4.2 thatthe idea of restricting mereological fusion in order to avoid undesired
entities is by itself suspicious. If we agree with the thought that a fusion
is nothing over and above the things that compose it, then the intuitive
problems raised by the open/closed distinction are hardly solved by eschewing formal commitment to such things as interiors and closures.
i.e., fusions of interior and of tangential parts. For such parts are all
there already (and, of course, one needs IPP and TPP to be distinct inorder to state (C.5) in the first place). In this sense, option (ii) might
be regarded as preferable on philosophical grounds, though the failureof supplementation would remain a hindrance.
Unfortunately, all of these theories include the monotonicity axiom
as well as its converse, (C.4), i.e., they are all of the reductive sort,which makes it difficult to assess their relative pros and cons vis-à-vis
the two options in question. In fact, not only do such theories include(C.4) and (C.5); they also rest on a sui generis characterization of thefusion operator in which C takes over the role of O, which makes it
difficult to compare them to KGEM T . For example, in Clarke’s theory,
which goes as far as to include analogues of the Kuratowski axioms,

58yFigure 1.15.

xz

The interior of a closed disc x is a fusion, but not a fusion∗ , of its parts.

the unrestricted fusion principle does not equal (P.13) but, rather, thefollowing schema:(C.13) ∃wφw → ∃z∀w(Cwz ↔ ∃v(φv ∧ Cwv))

Topological Fusion

This leads to a correspondingly sui generis fusion operator:(94)

Σ∗ xφx =df ız∀w(Cwz ↔ ∃v(φv ∧ Cwv))

fusion∗(Recall that RM T treats C as extensional: see (73).) And it is easily
checked that Σ∗ does not coincide with the operator Σ defined in (39):
if parthood reduces to enclosure, then the interior of a closed entity x
qualifies as the fusion of x’s proper parts, but not as their fusion∗ . (Thisis because in the absence of boundaries the interior is sure to overlap
any y that x overlaps, and vice versa, though it will to be disconnectedfrom any z to which x is externally connected; see Figure 1.15). This
amendment is plausible enough. But it means that the mereotopologicaloperators defined in (40)–(44) and in (79)–(82) must be revised accordingly, and at the moment there is no systematic comparison between the
behavior of such revised operators in a boundary-free theory and the
behavior of the original operators in a boundary-tolerant theory. Just
to give an example, note that re-defining ‘∼’ in terms of Σ∗ :(95)∼∗ x =df Σ∗ zDzx

complement∗

implies that nothing is connected to its own complement and, therefore,that the universe is bound to be disconnected. Of course we can rely ona different notion of complement (as suggested in Randell et al. 1992:168), or one can change the definition of self-connectedness in such a
way as to avoid at least the latter consequence (as suggested in Clarke
1985: 69). But this is playing with definitions. The “old” notions, whenrevisited in terms of (94), continue to make good sense no matter how
we change the official nomenclature, so we can hardly leave it at that.
Perhaps the best way to assess the strengths and weaknesses of these
theories is to note that the departure from the ordinary conception offusion affects the very distinction between open and closed entities. Consider the following variants of (79) and (85):

59

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

(96)
(97)

i∗ x =df Σ∗ z∀y(Czy → Oxy)OP∗ x =df x = i∗ xinterior∗
Open∗

It is easily checked that any theory at least as strong as B̄RM T + (C.13)
has the following theorem:(98)OP∗ x → ¬ECxy.

Thus, open∗ entities never touch anything. It is only closed∗ and semiclosed∗ entities (defined similarly), that can touch something without
sharing any parts.12 And if the open/closed distinction is replaced by
the open∗ /closed∗ distinction, then the intuitive import of the relevant
misgivings is up for grabs, and the choice between a conservative attitude towards mereological supplementation (option (i)) and a liberal
attitude towards mereological composition (option (ii)) calls for independent thinking. We are no longer dealing with a partitioning of thedomain into entities that do and entities that do not possess a boundary;
we are dealing with a partitioning into entities that do and entities that
do not connect externally.
Be that as it may, all of this suggests that a thorough comparison
between these two strategies for construing boundaryless mereotopologies is a challenging task (Cohn and Varzi 2003). What is clear is thatthe strategies are mutually incompatible in spite of their common motivation and this, in all fairness to KGEM T , is disturbing. No matter
how one feels about subtracting or adding elements to the domain, there
is something puzzling in the thought that a topological “monstrosity”
should by cured through mereological surgery. Indeed, philosophicallythis puzzling feature is especially striking when it comes to explaining
the intended interpretation of these theories. As it turns out, both canbe modeled on a domain with a standard point-set topology, interpreting‘C’ as in (99) for type-(i) theories, and as in (100) for type-(ii) theories:(99)

x is connected to y if and only if the closure of x and theclosure of y have a point in common.
(100) x is connected to y if and only if x and y have a point incommon.
(See e.g. Randell et al. 1992: 167, Gotts 1996b and Pratt and Schoop
2000 vs. Clarke 1981: 205, Biacino and Gerla 1991, and Asher and Vieu
12 Actually, Clarke’s definition of the closure operator does not exactly parallel (81).

notation it reads as follows:c∗ x =df Σ∗ z¬Cz(i∗ x).
However, this peculiarity does not affect the main point made in the text.

In our60
1995, respectively). There is, of course, nothing wrong with this sort ofmodels when it comes to proving the consistency or even the completeness of such theories. And there would be nothing wrong with (99) and(100) as genuine models if we were dealing with boundaryless theoriesof the moderate sort, as seen above. It is disturbing, however, that one
can hardly do any better when it is comes to theories that are meant
to be radically eliminativist—when it comes to explaining how contact
relations may obtain in a world that is truly lacking the topological glue
provided by points, lines, and surfaces even in the realm of pure space.
(Whether one can do better is an open question. For example, withreference to type-(i) theories, Bennett 1996a suggests that RCC can beinterpreted by encoding it into the bimodal propositional modal logic
S4u , though the encoding is imperfect, as shown in Aiello 2000, andits natural canonical model is itself topological, as evidenced in Renz
1998 and Nutt 1999. Likewise, Stell and Worboys 1997, Stell 2000, andDüntsch et al. 2001 provide algebraic interpretations of RCC that dispense with any reference to point-based topologies, but the ontologicaltransparency of such interpretations is itself a delicate matter.)

2.5

Expressivity and ontology

Let us conclude this philosophical excursus on topology with some general considerations concerning the delicate interplay between the expressive power of a theory and its ontological presuppositions. Regardless ofwhether we rely on the full strength of KGEM T or on theories of weaker
import, we have seen that the move from mereology to mereotopology
represents an important step towards the formulation of an adequate
model of our spatial competence. The mereological distinction between
a whole and its parts is crucial, but so is the distinction between interior and tangential proper parts, or the distinction between a connected
whole and a scattered one, and such distinctions are intrinsically topological. This is not to say that mereotopological concepts exhaust the
picture; geometric and morphological considerations also play a significant role when it comes to practical matters. But there is no questionthat a great deal of our spatial competence is grounded on our capacity
to “parse” the world in terms of parthood and connection relationships.
The interesting question, rather, is whether such relationships can be
fully captured by the formal behavior of the binary predicates ‘P’ and‘C’ when characterized by means of formal principles of the sort that wehave been discussing up to now, and to what extent the answer depends
on one’s specific views when it comes to matters of ontology. Here aresome indicative examples.

61

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

x

x
x
y

Figure 1.16.

y

x
y

y

Four patterns of external connection, from weakest to strongest.

2.5.1
Modes of connection.
So far we have followed thefamiliar course of explaining connection in general terms, i.e., irrespective of the size (dimension) of the relevant contact area. In introducingthat notion, however, we have mentioned the possibility of distinguishingconnection ties of different strength—e.g., ties involving a single pointof contact (as between Colorado and Arizona), an extended portion of acommon boundary (France and Germany), or an entire boundary (Vatican and Italy). Even without bringing in boundaries, one may want todraw such distinctions to fully grasp, for example, the difference between
a whole consisting of two spheres that barely touch from the whole consisting of two halves of a single sphere: both wholes are self-connected,
but the second is surely more firmly connected than the first. And these
distinctions have ramifications. For example, since tangential parthoodis defined in terms of external connection, we may want to distinguish
those proper parts that barely touch the exterior from those that firmly
touch it—and so on. Moreover, the number of distinctions grows withthe dimensionality of the entities we consider. Figure 1.16 illustrates thefour main patterns of external connection (no overlap) that can be distinguished in 2D space. But in 3D we might want to further distinguish,
for example, two cubes barely touching at a vertex, two cubes barely
touching along an edge, two cubes touching along a face, and so on.
Now, can all such distinctions be expressed in terms of the mereotopological primitives ‘P’ and ‘C’ (or just ‘C’, if one goes reductive)?
As it turns out, within a sufficiently rich mereotopological theory suchas KGEM T the answer is in the affirmative. To illustrate, with reference
to Figure 1.16 the difference between the first two cases can be explained
as follows. In both cases, x and y are connected; but whereas in the firstcase this simply means that one can go from x to y without ever going
through the exterior of the connected sum x + y, in the second case it isalso possible to go from x to y without ever leaving the interior of x + y.
More precisely, let us distinguish between the connectedness of a whole(‘SC’) and the connectedness of its interior, and let us say that an object
is firmly self-connected just in case the latter condition holds:

62
(101)

FSCx =df SCx ∧ SCix

Firm Self-Connectedness

Then we can say that two entities are firmly connected when they haveparts that add up to a firmly self-connected sum:
(102)

FCxy =df ∃wz(Pwx ∧ Pzy ∧ FSC(w + z))

Firm Connection

This captures the difference between the second case of Figure 1.16,
where the relevant connection relationship is firm, and the first, where it
isn’t. The stronger connection patterns corresponding to the third andfourth cases can then be defined by reference to the complement of x+y:
(103)
(104)

CCxy =df FCxy ∧ ¬FCx(∼ (x + y))
PCxy =df FCxy ∧ ¬Cx(∼ (x + y))Complete Connection
Perfect Connection(Strictly speaking, (103) and (104) would call for refinements, owing to
the possibility that x and y have internal holes; see Cohn and Varzi 2003
for a more general picture.) On this basis, the generalization to spaces ofhigher dimensionality is not difficult. For instance, in 3D space the difference between two entities touching at a point and two entities touchingalong an edge can be described by further distinguishing two patternsof non-firm connection, depending on whether the common boundary is
atomic or a self-connected composite (a line segment, or curve).
Now, all of this is easy in KGEM T . When it comes to weaker theories, however, things get more difficult. For the definitions above involvethe self-connectedness predicate along with the interior, complement,
and fusion operators, all of which may be absent in a theory deprived of
the necessary compositional strength. For instance, in a boundarylesstheory with no open/closed (or open∗ /closed∗ ) distinction, we can supply
for the lack of the interior operator by redefining firm self-connectedness
as follows (Bennett 1996b: 345):(1010 ) FSCx =df ∀y(IPPyx → ∃z(IPPzx ∧ Pyz ∧ SCz))
This makes it possible to go ahead with definitions (102)–(104) and capture the relevant distinctions in the 2D case. (See also Borgo et al. 1996
for a boundaryless theory in which FC is treated as primitive.) Yet it
is not clear how one can capture the further distinctions available in
3D and higher-dimensional spaces without appealing explicitly to thedimensionality of the relevant boundaries or to special assumptions concerning the structure of space (see Gotts 1994a). And of course things
get worse in a theory that lacks the complementation principle (P.6), forin that case, as we have seen, the notion of self-connectedness is alreadyproblematic. In short, the expressive power of a theory depends cruciallySpatial Reasoning and Ontology: Parts, Wholes, and Locations

63

on the underlying ontology, which in turn is reflected in the strength of
the relevant compositional and decompositional principles.
Similar considerations apply to further conceptual distinctions that
may be deemed relevant in the context of spatial reasoning. Consider
again the common-sense notion of contact exemplified by such statements as (55): the table is touching the wall. We have said that thisnotion is not topological but metric. This is actually true in KGEM T ,assuming that both entities—table and wall—are treated alike, i.e., asboth closed or both open. For KGEM T has the following two theorems:
(105)
(106)

ECxy → (CLx → ¬CLy)
ECxy → (OPx → ¬OPy)

By contrast, in a boundaryless theory the picture is different. For example, insofar as RCC admits of models satisfying (99), it can treat thetable and the wall as genuinely connected as long as their closures overlap. This may be at odds with physics, but it captures the common-sense
intuition. (Ditto for a moderate variant such as Cartwright’s—see again(90).) So which of these accounts is better depends on the seriousness
with which we handle the spatial ontology of common sense. On theother hand, none of this should be taken to imply that KGEM T lacks
the resources to account for the loose notion of connection countenanced
by common sense. Suppose we understand this notion in the following
sense: the table is touching the wall insofar as nothing can be squeezed
between them. The metric flavor of this notion lies in its modal ingredient: to say that nothing can be squeezed between two objects is to
say that they are “vanishingly close” to each other—that their relative
distance is arbitrarily small. We can, however, define a predicate that
captures this ingredient in mereotopological terms: we can say that two
objects are at least loosely connected in the relevant sense when one isconnected to the closure of every open neighborhood of the other:(107) LCxy =df ∀z(OPz ∧ Pyz → Cx(cz))

Loose Connection(See Asher and Vieu 1995 for a similar definition.) This relation captures the intuition that nothing can lie between two entities that touch,even when those entities are closed. And surely enough the definition
is consistent with KGEM T . So it is not that KGEM T lacks the conceptual resources to do justice to common sense. It is, rather, that therelation of loose connection is bound to be empty in those models of
KGEM T where the open entities form a dense ordering, i.e., where thefollowing holds:(C.14) OPx ∧ OPy ∧ PPxy → ∃z(OPz ∧ PPyz ∧ PPzy) Open Density

64

z
x
Figure 1.17.

yAn interior tangential part?

And whether all models should satisfy this axiom is, on the face of it,a question about the ontological make-up of the world. (One can argue
that as long as the open/closed distinction holds, common sense only
requires a denial of (C.14) when the variables are restricted to the rangeof ordinary entities, as opposed to their spatial receptacles. For a fullaccount of the mereotopology of discrete space, see Galton 1999 and
2000, §2.6, and the generalizations in Li and Ying 2004.)

2.5.2
Dimensionality.
Consider a second example. We haveseen that in a boundaryless theory everything must have interior properparts, as per (C.5). However, the notion of an interior part is itself,
in a way, a relative one, depending on the dimensionality of the space
we are considering (Galton 2004). In 1D space, the middle portion of aline segment y, or any portion x that does not extend to y’s extremities,
would qualify as an interior part of y, so y itself would satisfy the axiom.
As soon as y is embedded in 2D (or higher) space, however, all of its parts
would be tangential, as they can all be connected to things to which yitself would be externally connected—e.g. a disc z (Figure 1.17). Thus,in such higher spaces y would not satisfy (C.5).
Now, this is not by itself a disturbing fact if we take boundaryless theories to reflect a general intuition to the effect that all entities in the domain are of equal dimensionality. On the other hand, this very intuitionis philosophically problematic. It is puzzling that whether something
exists—a line segment, for instance—should depend on the dimensionof space, for one may want to declare one’s ontological commitments
while remaining neutral with respect to the difficult question of thedimensionality of space. Indeed, this becomes a necessity if one does
not have the resources to address such a question in the first place. In
ordinary point-set topology, one can say that a domain is of dimension< n if and only if every open cover O1 , ..., Ok can be refined to a closed
cover C1 , ..., Ck such that every point occurs in at most n + 1 of the Ci s.
This characterization is not available in mereotopology unless one goessecond-order, and surely it cannot be mimicked in a boundaryless theory that eschews the open/closed distinction. Gotts (1994b) and Bennett

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

65(1996b) suggest a way to bypass such difficulties by means of a different
characterization, which only requires P (or rather: E) to be closed under
the operations of complementation and binary sum, but the adequacyof such proposals is an open question. (By contrast, Galton 1996 showsthat an adequate characterization is available in a boundary-based theory such as KGEM T , or weaker variants thereof; see also the layered
mereotopologies of Donnelly and Smith 2003 and Donnelly 2004). So,again, we see here how the expressiveness of the theory depends cruciallyon the ontology it countenances, which in turn may be hard to specify
within the theory itself.

2.5.3
Counting the holes.
Finally, consider an example thatlies somewhere between the previous ones—the notion of a simply connected whole, i.e., intuitively, a whole with no holes. Topologically,as also from the perspective of common sense, this notion is just as
significant as the notion of a self-connected whole. Just as there is a
big difference between an apple and a bikini, there is a big difference
between an apple and a donut. Indeed, topology is often defined, intuitively, as a sort of rubber-sheet geometry that focuses precisely onthese two differences, ignoring shape, size, and all sorts of other spatial
properties that concern geometry broadly understood. Now, we haveseen that self-connectedness is easily defined in mereotopological terms,
though the adequacy of the definition presupposes the complementation
principle (P.6) and is not, therefore, entirely neutral from an ontological standpoint. What about simple connectedness? More generally,
the definition of self-connectedness can be extended so as to classify every object in terms of the (maximum) number of self-connected partsof which it consists—what is sometime called its “separation number”.
This can be done by defining a corresponding sequence of predicates,
one for each positive integer, as follows:13(108) SN1 x =df SCx
Separation1(109) SNn+1 x =df ∃yz(x = y + z ∧ ¬Cyz ∧ SCy ∧ SNn z)
Sep.n+1
Can we also provide a mereotopological characterization of the genus ofan object, so as to classify it in terms of the number of holes it has?
As it turns out, the answer to this question is in the affirmative, again
provided that we assume the complementation principle (P.6), but thisaffirmative answer has interesting ontological ramifications.
Here is how the basic account goes (Gotts 1994a). Let us say thatsomething has dissectivity n (n a positive integer) just in case it is self13 Here ‘n + 1’ indicates arithmetical addition, as opposed to mereological summation.

66
z

z

z

y

y

y

Figure 1.18.

Simple connectedness and genus classification in 2D.

connected and can be decomposed into n+2 self-connected, disjoint parts
with the following property: two of them, y and z, are not connected,
whereas all the others are connected to both of them but disconnectedfrom one another. Formally, this amounts to the requirement that there
be two disconnected parts y and z that are connected by a remainder
whose separation number is n (‘connected by’ in the sense of (59)):(110) DSn x =df SCx ∧ ∃yzw(x − (y + z) = w∧ BCyzw ∧ ¬Cyz ∧ SNn w)

Dissectivityn

Then we can say that something is simply connected just in case its
maximum dissectivity equals 1:(111) SSCx =df DS1 x ∧ ¬DS2 x

Simple Connectedness

With reference to Figure 1.18, for example, only the left pattern is simply
connected, for the others have higher dissectivity numbers. This is how it
should be, and it can be checked that the definition would yield the rightclassification also with reference to objects of different dimensionality.
A donut has dissectivity 2, so it is not simply connected; a solid ballis. Indeed, we can now use (110) also to provide a mereotopologicalcharacterization of the genus of an object. Something is of genus n, i.e.,
has n holes (n ≥ 0), if and only if its maximum dissectivity is n + 1:
(112) Gn x =df DSn+1 x ∧ ¬DSn+2 x

GenusnAgain, Figure 1.18 illustrates the definition in the 2D case, but it canbe checked that (112) yields the correct classification also for objects ofhigher dimensionality: a solid ball has genus 0, a donut has genus 1, apretzel has genus 2, and so on.
So this is the basic account, which fully answers the questions above:
provided we work with a mereotopology that is closed under complementation and binary sums, we can define simple connectedness and, moregenerally, classify any object in terms of the number of holes it has.
There are, however, two additional questions one may ask at this point.
The first is whether this basic account can be refined so as to do justice
to further distinctions that could be drawn in view of the dimensionality

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

Figure 1.19.

67

A donut, a punctured block, and other deviant solids of the same genus.

issues discussed above. We may, for example, want to tell a genuine
donut from the deviant cases in Figure 1.19, all of which have the same
genus. And here, as one might expect, the answer depends more heavily on the strength of the theory. In KGEM T we can go quite far; inweaker theories we may not, as we may not be able to distinguish thevarious kinds of non-firm connection that are needed to operate the relevant discriminations. For example, Gotts (1994b, 1996a) has shown thatthe issue can eventually be settled within the boundaryless framework of
RCC, but this result rests on various assumptions on the topology anddimensionality of the entities in the domain that cannot themselves be
expressed in the language of the theory. More importantly, it rests on
the interpretation of ‘C’ given in (99): two bodies are connected if andonly if their closures have a point in common. We have seen that suchan interpretation is of dubious legitimacy in a boundaryless ontology (asGotts himself laments in 1996b). So, once again, we reach a point where
the expressive power of a theory depends crucially on the ontologicalcommitments that one is willing to make.
The second question is whether the basic account can be refined so
as to do justice to further distinctions that could be drawn in view ofthe various ways—and there are many—in which an object can be perforated. And here it appears that even a strong, boundary-based theory
such as KGEM T may show its limits. In fact, there is a sense in which
the limits in question are not just the limits of the mereotopological approach of which the theory is expression; they are the limits of topology
as a general theory of space. Let us focus on the 3D case.
For one thing, we have been speaking of ‘holes’ in the sense of perforations, but we may also want to classify a self-connected object interms of the number of its internal ‘cavities’. To some extent this iseasy: on the assumption that the universe U is self-connected, it is sufficient to identify the number of internal cavities with n − 1, where nis the separation number of the object’s complement. In 2D space, thiscoincides with the genus of the object—there is no difference between a
2D perforation and a 2D cavity. In 3D the numbers may diverge: a solid
donut has genus 1 but 0 cavities, since its complement is self-connected.
Dropping the assumption on U, we can express this as follows:(113) ICn x =df SCx ∧ ∀y(SCy ∧ IPPxy → SNn+1 (y − x))Cavityn68

Figure 1.20.

Cavities, donut-cavities, and donuts with donut-cavities.

This definition works for every object in any dimension (except, ofcourse, for U). But this is just the beginning. A cavity may comein different forms. It may be a solid cavity, so to say, but it may itself be donut-shaped. It may also have the shape of an irregular donutof the sort illustrated in Figure 1.19. Or it may be “knotted” in various fashions—as a trefoil knot, for instance, or a granny knot. Clearlysuch distinctions are not covered by (113). Moreover, consider an object
with two donut-shaped cavities. The cavities may lie next to each other,so to say, or they may be interlocked like the rings of a necklace. Or
consider an object with a perforation—a donut—which also has an internal, donut-shaped cavity. The perforation may go through the “hole”
in the cavity or it may lie next to it. All of these and many others are
distinctions that are easily described in words just as they can easily
be depicted (Figure 1.20), and reflect significant differences in the spatial structure of the objects in question. It is far from clear, however,whether one can capture them in mereotopological terms.
Secondly, a perforation may come in different forms, too. It can bestraight or it can be knotted, and the knot may or may not wrap aroundanother perforation, just as it may or may not go through the “hole” ofan internal, donut-shaped cavity. It can also branch in the middle, so as
to have more than two openings. Indeed, it can branch in many different
fashions, as it can “merge” in various ways with internal cavities of various kinds. Again, all of these possibilities reflect significant distinctions
that are easily described in words and can easily be depicted, but it is far
from clear whether one can give a proper characterization in mereotopological terms—even with the full strength of KGEM T . In some cases
it is not even clear to what extent such a characterization should justparallel the standard topological account of such patterns. Standardly,
for example, a block with two parallel, straight perforations is equivalent
to a block with a single, Y-shaped perforation—both have genus 2 (andcan be transformed into each other by mere elastic deformation). This
much can be said in mereotopological terms, using (112) above. But
here is where standard topological considerations might be regarded as
inadequate for a good description of the spatial structure of ordinary
objects, and of the intuitions underlying our spatial reasoning broadlySpatial Reasoning and Ontology: Parts, Wholes, and Locations

Figure 1.21.

69

Same genus, different holes.

understood. The topological equivalence between such patterns—and
between such patterns and many others; see Figure 1.21—appears todeliver a partial account of the relevant spatial structures, for the genusof an object only captures the intrinsic topology of the object, not theway it relates to the environment. To get a better picture it seems necessary to keep an eye on the holes, not just on the object. And this is
obvious from the fact that in describing such patterns we tend to do soby describing the mereotopology of the holes and the way they relate toeach other; we do not describe the objects themselves. We tend to treat
holes as objects in their own right, as “negative objects” about which we
can say exactly the same sorts of thing we say about ordinary, “positive”
objects. And we count both sorts of objects in the same way: we counttwo straight perforations and one Y-shaped perforation.
If this is correct, then there are two things one can say. One can say(and accept) that the limits of mereotopology vis-à-vis such fine-grained
distinctions are just the limits of topology, mutatis mutandis. Or one
can say that the limits in question reflect precise ontological assumptions concerning the domain of application of the theory, specifically adismissive attitude towards the ontological status of holes. This is not
to say that holes are left out of the picture. Surely any theory with unrestricted fusions has room for such things, for mereologically speaking
a hole is nothing but part of the object’s complement. Rather, the pointis that mereotopology by itself says nothing specific about which partsof the complement qualify as holes. The boundaries of a hole simply
cannot be determined by purely mereological or topological considerations. Of course, we have seen that mereotopology says nothing about
the boundaries of material objects either. But draw such boundaries as
you like, chose the objects you like, unless you also draw the boundariesof their holes (if any) you cannot get a full picture of the mereotopological structure of the objects themselves. And to draw the boundaries ofsomething is to confer ontological dignity to it.
In Casati and Varzi (1994, 1999) it is argued that this alternative wayof construing the limits of mereotopology has far reaching consequences.
Suppose we take holes seriously: a hole in an object is something withwell-defined boundaries. Then the fine-grained distinctions mentioned

70

Figure 1.22.

The objects of Figure 1.21 have topologically different internal skins.

above can be recovered by looking at the mereotopological interplay
between matter and void, at the properties of the boundary where an
object comes into contact with its holes. More precisely, let the interface
between two entities x and y be the product of their boundaries:(114) x|y =df bx × by

Interface

And let the internal skin of an object x be the interface between x andthe fusion of its holes. Using ‘H’ for the binary relation ‘is a hole in’,this can be defined as follows:(115) sx =df Σz∃y(Hyx ∧ z = y|x)skin

Then the distinctions in question are distinctions that reflect the mereotopology of an object’s skin. With reference to Figure 1.21, for example,
it can be checked that the skin of the doubly perforated block on the left
is the disconnected sum of two cylinders, i.e., topologically, two spherical surfaces with two punctures each. By contrast, the block with a
Y-shaped hole has a connected skin that is equivalent to a spherical surface with three punctures, while the other blocks have skins equivalent
to a torus with two punctures and to a bitorus with one puncture, respectively (Figure 1.22). (Note that a puncture is not a hole but a mere
boundary. The surfaces of the objects in Figure 1.21 do not have boundaries, yet their internal skins do—and that makes all the difference.)
Now, in a boundaryless theory all of this is beyond reach. But in a sufficiently strong boundary-based theory such as KGEM T the notion of
an internal skin is perfectly meaningful and well defined for every objectof positive genus, and its mereotopological classification does not present
any special challenge. This confirms once again the greater expressive
power that comes with an ontological commitment to boundaries. It
also shows, however, that such a commitment is not enough: the existential quantifier in definition (115) shows that an explicit commitment
to holes is also needed. To the extent that the binary predicate ‘H’ isto be treated as a primitive, it is clear that this requires a step beyond
KGEM T and its pure mereotopological extensions. (For an axiomatic
treatment of ‘H’ and of its interplay with ‘P’ and ‘C’, see Casati and
Varzi 1994, Appendix, and Varzi 1996b.)

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

3.

71

Location Theories

Let us finally turn to the relation of spatial location. Intuitively, thisis the relation that holds between an entity and the spatial region thatit occupies, and we have already seen that this relation can hardly be
reduced to a chapter of mereology and/or topology. Even if it were—assomeone inclined to favor a Leibnizian, relationist conception of spaceagainst its Newtonian, substantivalist foes would urge—methodologicalprudence would suggest that we regard the reduction as a theorem, not
as a starting point, hence that the relation of location be treated as anindependent primitive next to parthood and connection. Exactly how
this relation should be characterizeed, and how it should interact withthe principles governing those other primitives, is precisely the sort ofquestion that a good theory of location should aim to answer.

3.1

Varieties of Location

Before looking at the main options, the usual terminological caveats are
in order. As with ‘part of’ and ‘connected to’, locative predicates havevarious meanings in ordinary language and it is important to be explicit.
For one thing, we often speak so as to specify the location of an object
by reference to another object, as opposed to a spatial region. Consider:
(116)
(117)
(118)

The biceps muscle is located in the arm.
The parking area is located next to the stadium.
The elevator is located inside the main building.

Pretty clearly, such cases are of no special interest, as they reflect different ways of asserting mereotopological relations of the familiar sort:in (116) the locative predicate is just a variant of ‘part of’, in (117)
it expresses the relation of external connection, and in (118) it stands
for a relation of containment that can be cashed out in terms of interior proper parthood. Of course, establishing mereotopological relations
may be an indirect way of specifying a genuine location: insofar as the
biceps muscle is part of the arm, for instance, the muscle is bound to belocated within the region occupied by the arm, though this is by itself an
intuition that needs to be spelled out carefully (Section 3.3). Moreover,
not every case of relative location can be explained in this fashion (Section 3.4). For the moment, let us just emphasize that the main concernof a theory of spatial location as we understand it here is with thosecases in which an object’s location is specified directly, as in
(119)
(120)

The peak of Mount Everest is located at 27◦ 590 N 86◦ 560 E.
The new library will be located at this site.

72
It is in this sense that the theory presupposes an ontology that includes
spatial regions as bona fide entities in their own right. Indeed, we shallassume that the location primitive is a relation whose second argument
can only be a region of space—a “place”. (Never mind the question ofwhat sort of linguistic expressions can serve the purpose of referring toplaces, as opposed to things that have a place. Statements such as
(121)
(122)

The bookcase is located in the living room.
The United Nations are located in Manhattan.

are somewhat ambiguous in this respect, but we may suppose that thecontext will always suffice to determine the intended meaning.)
Secondly, there are various ways in which an object may be said tobe located at a region. In a very loose sense, I am located at any region
that is not completely free of me (this room, or even the adjacent dining
room if I am reaching a foot out of the doorway); in a stricter sense, I
am only located at those regions that host me entirely (this room, if I
am not reaching out of the doorway); and in a stricter sense still, I am
only located at one region, namely the region that corresponds exactly
to the volume of my body. In the following we shall use ‘located at’ as
designating the last, strictest relation; the weaker relations can be introduced by definition. More precisely, suppose we use ‘L’ for the predicate
of exact location. Then three additional predicates can immediately be
defined as follows (from Parsons 200614 ):
(123)
(124)
(125)

GLxy =df ∃z(Ozy ∧ Lxz)
ELxy =df ∃z(Pzy ∧ Lxz)
ULxy =df ∃z(Pyz ∧ Lxz)

Generic Location
Entire Location
Ubiquitous Location

Thus, I am generically, in fact entirely located in Manhattan, but not
ubiquitously (or exactly) located there; I am generically, in fact ubiquitously located at the region occupied by my left arm, but not entirely (orexactly) located there; and if I reach an arm in my neighbor’s window,
then I am generically, but neither entirely nor ubiquitously (let alone
exactly) located at the region corresponding to her living room. As weshall see, under suitable conditions these predicates are interdefinable,
so the choice of ‘L’ as a primitive is ultimately immaterial.
Finally, it goes without saying that the location of an object may
change over time. This would suggest treating ‘located at’, not as a binary predicate, but as a three-place predicate involving a spatial as well
14 Parsons’s term for generic location is ‘weak location’, and his term for ubiquitous location is‘pervasive location’. Our different terminology is dictated merely by notational convenience,in view of the predicates ‘whole location’ (WL) and ‘proper location’ (PL) introduced below.

73

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

GLxny ELxny ULxny

x2
x1

x5

x3

y

x4
Figure 1.23.

L xny

n=1

–

–

–

–

n=2

+

–

–

–

n=3

+

+

–

–

n=4

+

–

+

–

n=5

+

+

+

+

Basic locative relations.

as a temporal argument (or as a temporally indexed binary predicate).
However, we have seen that the same goes for parthood and connection:
unless one accepts a radical form of mereotopological essentialism, an
object may in principle change its parts or its topological relations without ceasing to exist. In the preceding sections we have tried to keep
things simple by treating ‘P’ and ‘C’ as binary predicates, and we shalldo the same with our location primitive ‘L’. In a way, this means that
we are assuming the relevant time to be fixed throughout. But one may
also consider an alternative reading, to the effect that the variables of
the theory range over four-dimensional entities extended in space-time.
(See again the brief discussion following (32), Section 1.3.2.) Not muchof what follows depends on the strategy one favors, but for simplicity
we shall continue to speak of the location of an object as a 3D region of
space rather than—possibly—a 4D region of space-time.

3.2

Basic Principles

With these conventions in place, let us officially expand our formal language by adding the new binary predicate ‘L’, intuitively understood as
the relation of exact location holding between an object and a region of
space. To make this interpretation explicit, we may begin by assuming
the following axiom:(L.1)

Lxy ∧ Lxz → y = z

Functionality

This guarantees that nothing can have more than one exact location,
which is all that is needed to justify the definitions in (123)–(125) (Figure 1.23.). Indeed, it is easy to see that in the presence of an extensional
mereology, (L.1) has the following corollaries:
(126)
(127)
(128)

Lxy ↔ (ELxy ∧ ULxy)
ULxy ↔ (GLxy ∧ ∀z(Ozy → GLxz))
ELxy ↔ (GLxy ∧ ∀z(GLxz → Ozy))74
Thus, although we have settled on the strictest possible primitive, thepredicate ‘L’, one could equally well settle on the weakest predicate, ‘GL’,and define the rest via (126)–(128). More precisely, if the mereologicaltheory in the background is at least as strong as EM , it turns out that
the L-based system defined by (L.1) plus (123)–(125) is equivalent to theGL-based system defined by (126)–(128) plus the following:(L.2)

GLxy → ∃zLxz

Exactness(See Parsons 2006; for a different choice of primitives, compare also
Perzanowski 1993.) Note that the conjunct ‘GLxy’ is redundant in (127)as long as O is reflexive. However, this extra conjunct is needed in (128)unless one assumes that everything is located somewhere:
(L.3)

∃yLxy

Spatiality

This is clearly an assumption that reflects a substantive thesis (a central
tenet of most nominalistic ontologies), so it is fair to keep it separate.
To be sure, there is a sense in which (L.1) may also be read as a
substantive thesis: functionality is reasonable only to the extent that weare thinking of so-called “particular” entities, entities such as material
bodies or events, as opposed to “universal” entities such as properties.
That a material body cannot be in two places at once was already a central thesis of Aristotle’s theory of location (Morison 2002). But Aristotle
also held the view that universals, too, exist in space and time: they exist wherever and whenever they are exemplified. Wisdom, for example,
exists whenever and whevever there are wise people—and whenever awise person exists, wisdom exists in its entirety wherever that person is
located. Wisdom can therefore be multi-located, and the same goes for
all universals. Since this view is still very popular (Armstrong 1989), thefunctionality principle (L.1) would be objectionable. However, we canbypass this issue by taking ‘L’ to represent the location relation that ischaracteristic of particulars. In that sense, (L.1) is, if not a conceptual
truth, a perfectly reasonable starting point.
What else is needed in order to fix the intended meaning of ‘L’ inthis sense? Since the idea is that every object must be located at aregion, some restriction must be imposed on the second argument of
the relation. This would be a trivial task if the language contained
an explicit predicate for regions. However, we can make do without
such a predicate and try to characterize regionhood directly in terms ofsuitable axioms on L. There are two options here, depending on whether
we think that spatial regions are themselves entities located somewhere.
If we think so, then the obvious thing to say is that such entities canonly be located at themselves (Casati and Varzi 1999: 121):

Spatial Reasoning and Ontology: Parts, Wholes, and Locations(L.4)

Lxy → Lyy

75

Conditional Reflexivity

This would immediately imply that no distinct regions can be exactly
co-located, i.e., effectively, located at each other:(129)

Lxy ∧ Lzw ∧ Lyw → y = w

Moreover, given (L.1), conditional reflexivity would ensure that L is bothantisymmetric and transitive:
(130)
(131)

Lxy ∧ Lyx → x = y
Lxy ∧ Lyz → Lxz

It follows that relative to the sub-domain of regions L would behave as
a partial ordering. By contrast, if we think that regions do not have alocation—they are locations—then the obvious option is given by:(L.5)

Lxy → ¬Lyz

Conditional EmptinessIn this case, the restriction of L to the class of regions would again
qualify as a partial ordering—a strict ordering—but only in a trivial
sense: effectively, it would just collapse to the empty relation.
There is, arguably, no deep metaphysical issue behind these two options: both (L.4) and (L.5) are equally good stipulations, and the difference would disappear as soon as we focus on cases of proper location:(132)

PLxy =df Lxy ∧ ¬Lyx

Proper Location

However, there are some differences that are worth mentioning. For one
thing, given (L.1), the first option makes it possible to define regionhoodin a perfectly straightforward way:(133)

Ry =df ∃xLxy

Region(We are speaking of regions in a broad sense, including boundaries aslimit cases.) By contrast, (L.5) would support this definition—and variants thereof—only on the assumption that there are no unoccupied regions, i.e., regions that fail to correspond to the location of some object
or event. To put it differently, if all location is proper location, it is notpossible to define regionhood unless the following principle is accepted:(L.6)

Ry → ∃xPLxy

Fullness

And philosophically this principle is just as controversial as the spatiality principle (L.3). (Among other reasons, one might want to allow for76boundary-like regions while rejecting the existence of boundary-like objects, as seen in Section 2.4.3.) Secondly, it is also apparent that the two
options differ with regard to (L.3) itself: (L.4) is compatible with thisprinciple, (L.5) isn’t. Thus, the second option makes it impossible toassert the thesis that everything is located somewhere—a thesis which,albeit controversial, is certainly not inconsistent. Again, this is a limitation that would dissolve if ‘R’ were available in the language, in whichcase the thesis in question could be reformulated as follows:(L.30 )

¬Rx → ∃yLxy

But precisely because ‘R’ cannot be defined absent (L.6), the limitation
is not immaterial.
For these reasons, in the following we shall favor the first option andassume the conditional reflexivity axiom (L.4). Together with the functionality postulate (L.1), this yields a minimal theory of (exact) spatial
location, which we shall label S: this theory is incompatible with (L.5),
but it includes the exactness principle (L.2) as a theorem and can bestrengthened by adding the spatiality principle (L.3), the fullness principle (L.6), or both.
At this point, we could consider various ways of strengthening S (orits extensions) by imposing suitable axioms on the predicate ‘R’ definedin (133). For example, it seems reasonable to assume that regionhood isboth dissective and cumulative, i.e., that any part of a region is itself aregion, and that the sum of any regions (if it exists) is a region, too:
(L.7)
(L.8)

Ry ∧ Pxy → Rx
z = Σxφx ∧ ∀x(φx → Rx) → Rz

Dissectiveness
Cumulativity

It may also be reasonable to consider additional postulates concerningwhether the class of all regions forms a dense domain, or whether it
forms an atomless, possibly a boundaryless domain as opposed to an
atomistic domain every element of which consists (intuitively) of points:(L.9) Rx ∧ Ry ∧ PPxy → ∃z(Rz ∧ PPxz ∧ PPzy)R-Density(L.10) Rx → ∃y(Ry ∧ PPyx)
R-Atomlessness(L.11) Rx → ∃y(Ry ∧ IPPyx)
R-Boundarylessness(L.12) Rx → ∃y(Ry ∧ Ay ∧ Pyx)
R-Atomicity
More generally, it may be reasonable at this point to consider whether
the domain of regions should be closed under various mereotopological principles, regardless of whether such principles hold of the entities
that may occupy those regions. For example, even an anti-extensionalist
about material objects will presumably deny that different regions maySpatial Reasoning and Ontology: Parts, Wholes, and Locations

77

consist of the same proper parts, and even those who have misgivings
about strong composition principles for arbitrary obects might be happy
to endorse unrestricted composition of spatial regions. All such extensions of S are obviously worth examining, and they are crucial if wewant to fix the intended range of the relational predicate ‘L’, but there
is no need here to review all the options: suffice it to say that the availability of ‘R’ makes it possible to examine them in a systematic fashion. (An interesting question, for instance, is whether one can provide
a purely mereotopological characterization of Euclidean space; see Tsai
2005: §7.4, for a negative answer). Rather than focusing on the structureof space per se, let us see how S can be further extended by considering more closely the relationship between the two terms of the location
relation—the structure of regions and the structure of their tenants.

3.3

Mirroring Principles

To this end, let us begin by noting that the four relations L, GL, UL, andEL do not exhaust all the options. Additional locative relations can be
specified by replacing the plain mereological predicates in (123)–(125)with finer-grained mereotopological predicates—for example:
(134)
(135)
(136)
(137)

TELxy =df ∃z(TPPzy ∧ Lxz)
IELxy =df ∃z(IPPzy ∧ Lxz)
TULxy =df ∃z(TPPyz ∧ Lxz)
IULxy =df ∃z(IPPyz ∧ Lxz)

Tangential EL
Interior EL
Tangential UL
Interior UL

More generally, given any mereotopological relation ψ, there is a corresponding locative relation Lψ defined by:(138)

Lψ xy =df ∃z(ψzy ∧ Lxz)

ψ-Location(Thus, GL = LO , EL = LP , UL = LP̆ , etc.) Such generalizations are
straightforward, but they bear out that the language of location can beas rich as the underlying mereotopological vocabulary. Indeed, this is
only half of the story. According to (138), to be ψ-located at a region yis to be exactly located at some region, z, that is ψ-related to y. Thus,
the resulting variety of locative relations is defined with reference to themereotopological structure of the range of L—the structure of space.
But one could also consider the obvious alternative, and characterize
locative relations by reference to the mereotopological structure of thedomain of L—the structure of space’s tenants. In this alternative sense,
to be ψ-located at a region y is to be ψ-related to some object, z, thatis exactly located at y:(139)

Lψ xy =df ∃z(ψxz ∧ Lzy)

ψ-Location (2)78Now, the interesting question is whether these two ways of characterizing
locative relations should coincide—whether ‘Lψ ’ and ‘Lψ ’ should always
stand and fall together. This is trivially true when the relata are of thesame kind, i.e., regions, for in that case locative relations collapse tomereotopological relations in view of the following S-theorems:
(140)
(141)

Rx → (Lψ xy ↔ ψxy)
Rx → (Lψ xy ↔ ψxy)

But what about the general case? This is not just an interesting question
to ask if we want to get the map straight; it is also a question that calls
for interesting philosophical decisions.
To address the matter properly, let us consider the two directions of
the equivalence separately, corresponding to the following theses:(L.13)(L.14)

Lψ xy → Lψ xy
Lψ xy → Lψ xy

Bottom Mirroring
Top Mirroring

3.3.1
Bottom mirroring and co-location.
Informally, thefirst of these theses says that the structure of space should be at leastas rich as the structure of its tenants For example, suppose I am exactly
located at region r. Then my right foot, which is part of me, is LP -locatedat r. But if my foot is part of me, then it is reasonable to suppose thatits exact location is part of my exact location, which is precisely what
(L.13) would imply: my foot is LP -located (i.e., entirely located) at r.
For another example, since my foot is connected to the rest of my body,which is located at a certain region r0 , then it is reasonable to supposethat the location of my foot is connected to r0 , too: my foot being LC located at r0 implies its being LC -located at r0 . In general, adding (L.13)
to S would secure that if two objects are ψ-related (where ψ is P, C, or
any other mereotopological relation), then so are their locations:(142)

Lxy ∧ Lzw ∧ ψxz → ψyw.

Plausible as all this might sound, it is however easy to see that (L.13)
does not generally hold. A simple counterexample is depicted in Figure 1.24, left. In this model there is just one (atomic) region, r, and acomplex object, a, consisting of two (atomic) parts, b and c. Object a
is located at r, hence its parts b and c are LP -located at r. Yet they
are not LP -located at r because there is no part of r at which they areexactly located. Note that the mereotopological relations in this model
can easily be extended so as to satisfy all the axioms of KGEM T , so
the counterexample does not depend on the strength of the underlying
mereotopological theory. It depends esclusively on the behavior of L.

79

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

a

a

•

b•

•

•c

b•

•r

•c

a •

•b

c1 •

• c2

•r

•r
r1 •

• r2

Figure 1.24. Three violations of Bottom Mirroring. (Location relationships are represented by gray arrows; reflexivities omitted.)

Now, it might be observed that this model (or its KGEM T closure)would be ruled out if we assumed that every part of a spatially locatedentity had a spatial location—a restricted form of the spatiality principle(L.3). More generally, it might be thought that in order to justify (L.13)
one must assume the following:(L.15)

ψxz ∧ Lzy → ∃wLxw

Conditional Spatiality

Bottom mirroring should not hold holus bolus, but only when x and its
ψ-relata are genuine spatial entities (in which case (L.13) turns out to beeqivalent to (142)). However, it is easy to see that even S + (L.15) would
fail to warrant this claim. A counterexample is depicted in Figure 1.24,
center. Here b and c are LPP -located at r, since each is a proper part ofa, which is located at r. Yet there is no proper part of r at which a orb is located, which is to say that neither is LPP -located at r.
We begin here to see the hidden force of (L.13). By requiring thatthe structure of space mirror the structure of its tenants, this principle
rules out the possibility that mereotopologically distinct entities be colocated, i.e., located at the same region. Of course, it may be difficult
to imagine a concrete scenario corresponding to the model in question,and certainly very difficult to provide a less abstract representation of it.
But it is not difficult to provide a good example if, for instance, we giveup mereological extensionality. Consider again Tibbles, the cat, and the“mere” mereological sum of his tail with the rest of his body, Tib + Tail
(Section 1.3.2). Obviously, even if one treated these entities as distinct,
one would still like to say that they share one and the same location.
Yet, if ‘ψ’ expresses the relevant mereotopological relationship of distinctness cum sameness of proper parts, it is obvious that the region towhich Tibbles (or Tib + Tail) bears the relation Lψ is not the region towhich it bears the relation Lψ (Figure 1.24, right). Moreover, even in
the presence of mereological extensionality one may conceive of situa-

80
tions where spatial co-location seems possible. We have already seen, forinstance, that according to Chisholm’s (1984) Brentanian theory, topological connection is explained precisely in terms of spatial coincidenceof boundaries: boundaries are located in space but do not occupy space,hence they can coincide while being distinct (Section 2.4.2). For a scenario compatible with the full strength of KGEM T , consider Davidson’s(1969) example concerning event identity: arguably the rotation and thegetting warm of a metal ball that is spinning fast are two events, yetthey occur exactly in the same region and they share that location withthe ball itself. Finally, if our ontology is rich enough to include immaterial or otherwise ethereal creatures for which genuine interpenetrationis possible, then again co-location seems conceivable. Already Leibniz
mentioned shadows as a case in point (New Essays, II-xxvii-1); othercandidates include clouds (Shorter 1977), holes (Casati and Varzi 1994),
ghosts (van Inwagen 1990: 81), and even angels (Lewis 1991: 75).
These are just some examples. But they are indicative of the many
philosophical motives that may lie behind a rejection of the principle
according to which proper spatial co-location is impossible—a principlethat can be put thus:(L.16)

PLxy ∧ PLzy → x = z

Exclusiveness

Giving up this principle involves giving up (L.13), at least in its generalform. And the question of what special instances one should posit, i.e.,
what values of ‘ψ’ and ‘x’ satisfy bottom mirroring, calls for a detailed
case-by-case investigation—and for explicit ontological decisions.

3.3.2
Top Mirroring.
Consider now the converse of (L.13),
namely the principle of top mirroring, (L.14). Informally, this says thatthe structure of space should be mirrored in the structure of those entities that inhabit it. For example, if the location of my body properly
includes a region r, then it is reasonable to suppose that my body properly includes something located at r: LPE -location, hence LPE -location.
Pretty clearly, however, there are numerous relations ψ for which thissort of implication appears problematic. The location of my body is aproper part of any region that includes this room, but there is no obvious reason to think that every such region is the location of some existing object—no reason to think that LPP -location implies LPP -location.
(Compare Figure 1.25, left.) Similarly, my body is LEC -located at many
regions, viz. regions externally connected to my body’s current location;
yet there is no obvious reason to think that my body is LEC -located atthose regions, since many of them may be (partly) empty (Figure 1.25,
center.) Just as bottom mirroring appears to presuppose the spatiality

81

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

a

a

•

a

•

•

•r
r1 •

•r
• r2

r1 •

• r2

r1 •

Figure 1.25.

Three violations of Top Mirroring.

• r2

principle, or at least its restricted variant (L.15), top mirroring appears
to presuppose fullness, or at least the following restricted version:(L.17)Ry ∧ ψzy ∧ Lxz → ∃wPLwy

Conditional Fullness

And this is a substantive presupposition that few might grant, regardlessof their views concerning purely mereotopological matters.
Indeed, even when ψ is the seemingly innocent relation of proper
extension, as in our first example, (L.17) appears problematic. Misgivings about this principle—hence about the corresponding instance of
top mirroring—come in various forms. First of all, there are argumentsthat purport to show that the principle is empirically false (Parsons
2000). Second, the principle rules out a priori the possibility of spatiallyextended mereological atoms, as in Figure 1.25, right. To the extent
that one can conceive of such things, it is argued, it should not be aconceptual truth that every region ubiquitously occupied by an object isexactly occupied by a part of that object. (See Markosian 1998a; othercontemporary philosophers who endorse the possibility of extended simples include Parsons 2004 and Simons 2004, but the view goes back to
Democritus’s claim that atoms come in an infinite variety of shapes andsizes.) Third, there are arguments to the effect that (L.17) sits ill withthe thought that ordinary material bodies can gain or lose some parts(van Inwagen 1981). To illustrate, consider again Tibbles, the cat whose
tail gets annihilated at t, and suppose we agree that it survives the accident. Prior to t, (L.17) would suggest that in addition to the whole
cat there exist also two externally connected proper parts: Tail and Tib(the remainder). Now consider the following statements:
(143)
(144)
(145)
(146)

Tib (before t) = Tib (after t)
Tib (after t) = Tibbles without Tail (after t)
Tibbles without Tail (after t) = Tibbles with Tail (before t)
Tibbles with Tail (before t) 6= Tib (before t)

These four statements are jointly inconsistent, unless one is willing togive up the transitivity of identity, so something must give. (146), how-82
ever, is trivially true: there is no way one could identify a whole catwith its tailless portion. And (145) is true by assumption: to give it
up is to deny that Tibbles survives the accident, unless one is willing toconstrue cats as four-dimensional entities whose temporal parts are numerically distinct (Heller 1984). As for (144), its denial would obviously
incur in a commitment to properly co-located entities, let alone a violation to mereological extensionality (Wiggins 1968). Thus—the argument
goes—unless one is ready to accept such unpalatable consequences, theonly option is to give up (143): that identity is false for the simple reason that prior to the accident Tib does not even exists; it only exists
after the accident, and it exists as tailless Tibbles.15 (One might also
say that before the accident Tib does not actually exist. The view that
undetached parts are mere “potential entities” has been the focus of an
intense debate in early modern philosophy; see Holden 2004. Brentano
1933 endorsed a similar view, too, and some authors have applied it
explicitly to the puzzle in question—e.g., Smith 1994, §3.5.)
Whether any such arguments are found compelling is, of course, an
open issue. Nonetheless, the obvious moral is that (L.14) can hardly
be regarded as a conceptual truth about location. Even if we confine
ourselves to its single, prima facie plausible instance in which ψ is therelation of (proper) extension, i.e., the converse of (proper) parthood,
top mirroring is a substantive metaphysical thesis whose addition to S
must be independently motivated.

3.3.3
Further locative relations.
In discussing such matters,
it is useful to keep in mind that the lack of a full correspondence between
the mereotopology of space and the mereotopology of its tenants may
find expression in the failure of other principles or equivalences thatmight otherwise suggest themselves. Consider, for instance, the following
relation (from Parsons 2006, §4):
(147)

WLxy =df ∀z(Pzx → GLzy)

Whole Location

Intuitively, this says that an object is located in this room (for instance)
if every part of the object is generically located in this room, i.e., if noneof it is missing from the room. This might sound like a different way
of saying that the object is entirely located in this room, in the original
sense of (124) (corresponding to LP -location) or in the alternative senseof (139) (LP -location). In fact, however, all these notions are distinct.
15 The puzzle raised by (143)–(146) has been introduced to contemporary discussion by Wig-

gins (1968), but it goes back at least to the Stoics; see e.g. Sorabji (1988: §1.6). For a detailed
overview, see Simons (1987, §3.3) and the introduction to Rea (1997).

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

83

Not only do LP -location and LP -location come apart, as seen above. They
also differ from whole location: the diagram in Figure 1.24, left, corresponds to a model in which an object, a, is both LP - and LP -located at aregion, r, in spite of not being wholly located there; the diagram in Figure 1.25, right, depicts a model in which a (an extended atom) is wolly
located at region r1 in spite of being neither LP - nor LP -located there.16(Note that both of these models are extensional, and would continue toexhibit these features even if closed under every KGEM T -axiom.)
The notion of whole location is just one example. In general, for any
mereotopological relation ψ, one can define two additional locative relations besides Lψ and Lψ , obtained by switching to a universal quantifier
and replacing ‘L’ by ‘GL’:
(148)
(149)

L∀ψ xy =df ∀z(ψzy → GLxz)
L∀ψ xy =df ∀z(ψxz → GLzy)

Here WL = L∀P̆ , and it should be obvious from our single example thatthe equivalence between the new predicates in (148)–(149) and the old
predicates in (138)–(139) is generally an open question. Indeed, at thispoint we get the full picture by further generalizing these four basic
patterns in the obvious (recursive) way: if λ is any locative relation,then so are the following:
(150)
(151)
(152)
(153)

L∃ψλ xy =df ∃z(ψzy ∧ λxz)
L∃ψλ xy =df ∃z(ψxz ∧ λzy)L∀ψλ xy =df ∀z(ψzy → λxz)
L∀ψλ xy =df ∀z(ψxz → λzy)

There are lots of redundancies and empty relations in this picture, whose
complexity depends significantly on the mereotopological axioms governing ψ. Nonetheless, it is only through a careful study of such intricacies—
and of the corresponding mirroring principles, at the moment vastly
unexplored—that a reasonably complete theory of location can emerge.

3.4

Relative locations

The axiomatization of the region predicate ‘R’ and the positing of suitable mirroring principles constitute the two main directions in which
theory S can be extended. Let us briefly mention a third direction,
whose ramifications span philosophical and methodological issues alike.
We have said that in ordinary language location is often understood as
16 In Casati and Varzi (1999, §7.2), entire location is called ‘whole location’ and labelled ‘WL’.

84
a relation between two objects, as opposed to an object and a spatial
region, and we have said that such understandings need not be taken to
express any fundamental relationships: often, such “relative” locations
are mereotopological relations in disguise, as in examples (118)–(120).
There are, however, cases that resist this sort of explanation. Consider:
(154)
(155)
(156)

The brain is located inside the cranial cavity.
The swimming pool is located behind the house.
The bus stop is located right across the old oak tree.

Surely the truth conditions of these statements can hardly be explainedin terms of mereotopological relations, so one can hardly leave it at that.
Can S be strengthened so as to account for such cases as well?
In a way, the answer is straightforward. Consider (154). Although
there is no direct mereotopological relationship between the brain andthe cranial cavity (unless one thinks the latter literally surrounds theformer), one can still explain the relevant truth conditions by reference
to the mereotopology of the corresponding spatial locations: statement(154) is true if, and only if, the location of the brain is an interior properpart of the location of the cranial cavity. Equivalently, (154) is true if
and only if the brain is located entirely in the interior of the spatial
region occupied by the cranial cavity. This suggests that cases such as
this can easily be accommodated within the present framework. When
we say that a “target” object, x, bears a certain locative relation to a“reference” object, y, we mean to say that x bears that relation to y’splace. To make this clear, let us introduce an explicit location functor,
whose uniqueness follows directly from the functionality axiom (L.1):(157)

px =df ıyLxy

place(See Donnelly 2004 for a location theory with p treated as a primitive.)
Then, for any locative relation λ and any spatial objects x and y, we
can define a corresponding predicate of relative location ‘Rλ’ as follows:(158)

Rλxy =df λx(py)Relative Location

In the brain-cavity case, λ is the relation IEL defined in (135), i.e., LIPP ,but the same pattern would apply to a large variety of other cases,
provided the reference object has a location somewhere. In fact, thesame pattern can be applied to account for the initial examples in (116)–(118), too: the biceps muscle is RLP -located at the arm; the parking area
is RLEC -located at the stadium; the elevator is RLIPP -located at the mainbuilding. In the presence of suitable bottom mirroring conditions, thesethree claims are equivalent to the pure mereotopological claims obtainedby replacing the relation RLψ with ψ itself.

85

Spatial Reasoning and Ontology: Parts, Wholes, and Locations
above

above

behindbehind
right

left

left

in front

in front
below

Figure 1.26.

right

below

Object-centered (left) and observer-centered (right) frames of reference.

Cases such as (155) and (156) are different. Here the difficulty does
not just lie in the fact that the target object and the reference object
do not stand in any mereotopological relation to each other. The difficulty is that the spatial relationships reported by such statements—
corresponding to such prepositions as ‘behind’ and ‘across’, but also‘above’, ‘underneath’, ‘left of’, etc.—have little or nothing to do withmereotopology. There is in fact a large literature devoted to the semantics of spatial prepositions (beginning with the classic work of Herskovitz
1986) and it is fair to say that their treatment requires a degree of sophistication that goes far beyond the conceptual apparatus developed
above. Among other things, there are well-known complications owing
to the fact that their treatment calls for a systematic distinction between
object-centered frames of reference, as in (155), and observer-centered
frames, as in (156) (see Figure 1.26), whereas mereotopological relations
are completely independent of subjective or perspectival considerations.
Still, this is not a limitation that speaks against the employment of aprimitive such as ‘L’. It is, rather, an indication that the ensuing theory, S, may have to be matched with a more sophisticated backgroundthan a mereotopological theory can afford. Thus, suppose we allow the
relational predicate ‘ψ’ in the definitions of Section 3.3 to stand for relations that are not purely mereotopological: relations such as ‘behind’,‘across’, etc. (in each of their multiple uses). Then the idea illustrated
with reference to the brain-cavity example can in principle be appliedalso to (155), (156), and the like. To say that the swimming pool is located behind the house is to say that it is RLbehind -located at the house.
To say that the bus stop is located across the tree is to say that it
is RLacross -located at the tree. And so on. Exactly how these relations
should be formalized, i.e., what principles should be posited to fix thelogical behavior of the relevant ψ, is where things get difficult and may
require a lot of detailed work. (We have, after all, seen how difficult it
is to do this when ψ is a mereotopological relation.) But that is not to
say that relative locations require a sui generis treatment that a suitable
extension of S could not accommodate.

86
On the other hand, here is where the main metaphysical assumption
underlying S may be questioned. The accommodation comes with definition (158), which allows one to handle relative locations in terms of aprimitive relation, L, whose range consists exclusively of spatial regions.
In other words, it allows one to express a spatial relation between atarget and a reference object as a relation between the target and thereference’s place. But one might object that this has things the wrong
way round. Relative locations are ontologically neutral with respect tothe status of space, whereas the proposed treatment depends cruciallyon the assumption that places, and spatial regions generally, are entities
of a kind. A relationist about space might therefore reject the accountand take the opportunity to reverse the order of the explanation: to be
RLψ -related to a place is to be ψ-related to the place’s tenant, and talkabout places is shorthand for talk about spatial objects. Also a substantivalist about space might think that when it comes to spatial reasoning,objects are conceptually prior to their locations, since we cannot identify the latter independently of the former (Strawson 1959, ch. 1.) Evena forerunning substantivalist such as Newton emphasized that absolute
places, defined as in (157), are scarcely useful for locating things in theworld: we do not locate an object on a moving ship with reference toan immobile environment but, rather, with reference to the ship itself(Principia, Definitions, Sch. 4). In short, there may be philosophicalas well as methodological reasons for resisting the treatment of locative
relations indicated above, and if these reasons are taken seriously, thencases such as (154)–(156) run afoul of the basic framework outlined here
and call for independent treatment. An articulated proposal in this spirit
may be found in Donnelly (2005), but much recent literature devoted
to the formal representation of direction and other qualitative spatial
relations (from the works of Mukerjee and Joe 1990, Frank 1992, andHernández 1994 to more interdisciplinary works such as those collectedin van der Zee and Slack 2003) may be viewed in this perspective.

3.5

Location, connection, and parthood

Let us conclude with a few remarks about the whole conceptual packagethat we have been putting together. We have at this point three main
primitive notions: location, connection, and parthood. Some structural
relationships among these notions have been examined, but the generalquestion of their ontological intertwining is still open. Generally speaking, parthood and connection are independent from each other, unless
one accepts the converse monotonicity principle (C.4) (Section 2.3.2).
But what about location? Let us keep with the assumption that location

Spatial Reasoning and Ontology: Parts, Wholes, and Locations

87

is a relation between a thing and its place. Is that relation completely
autonomous or does it entail a mereotopological linkage of some sort?
A purely mereological linkage seems out of the question. There is no
reason to think that I share any parts with the space I occupy, just as
there is no reason to think that movement—change of location —is aform of mereological change. This is not to say that location implies
disjointness, since the first argument of L may itself be a region, or a
hybrid fusion including regions among other things. In general, however,it seems perfectly reasonable to assume that the implication holds forordinary cases of location—a thesis that can be put as follows:(L.18)

PLxy → ∃z(Pzx ∧ Lzy ∧ Dzy)

Spatial Disjointness

What about a topological linkage? In this case the picture seems
different. If an object is located at some place, then it might be plausible
to suppose that the object and its place are connected in some way:
(L.19)

Lxy → CxySpatial Connection

In what way would I be connected to my place? Since overlap is excluded, the relevant linkage would have to be one of external connection.
However, this means that the plausibility of (L.19) depends crucially on
the interpretation of ‘C’.
If we go along with the standard interpretation, corresponding to aboundary-based theory such as KGEM T , the prospects are slim. On
that interpretation, two things can be externally connected only if one
is open and the other closed, at least in the relevant contact area: twoclosed entities, or two open entities, can only connect through mereological overlap (Section 2.4.1). Now, suppose I am a closed body. Then
(L.19), together with (L.18), imply that the region at which I am located—
my place—must be open. That region, however, has a closure, and one
should think that the closure of a spatial region is itself a region: its
boundary is pure space, too. But my boundary is not pure space: whatever it is, it is part of me, and none of me is made of space. Thus, theclosure of my place and I are disjoint—which is to say that we are notconnected after all. A different way of putting this involves the thoughtthat my place and I must have the same topology: if I am closed (for instance), then my place must be closed too, hence we cannot be externallyconnected. Strictly speaking, this involves an appeal to the principle ofbottom mirroring, specifically the following instance of corollary (142):(159)

Lxy ∧ Lzw ∧ IPPxz → IPPyw

My interior (x) is a proper interior part of me (z), hence its place (y)must be an interior proper part of my place (w). We have seen that bottom mirroring may not generally hold, but this specific instance seems

88fair. Yet (L.19) would require the opposite: it would require my place
to be an interior proper part of the place of my interior—absurd.
By contrast, suppose we go along with a different interpretation. For
instance, consider those interpretations that explain connection explicitly in terms of spatial location. We have seen three such interpretations,corresponding to the following necessary and sufficient conditions for two
entities x and y to be connected: (i) the place of x is connected to theplace of y (Section 2.3.2, thesis (74)); (ii) the place of x overlaps theplace of y (Section 2.4.2, thesis (89)); or (iii) the closure of the place ofx overlaps the closure of the place of y (Section 2.4.3, thesis (90)). With
the help of ‘L’, and assuming the spatiality axiom (L.3), these threeinterpretations can now be formally stated as follows:
(160)
(161)
(162)

Cxy ↔ C(px)(py)Cxy ↔ O(px)(py)Cxy ↔ O(cpx)(cpy)And, plainly, each of these statements is perfectly compatible with thespatial connection principle (L.19). In fact, since C and O are both reflexive, each statement entails (L.19) in view of the following S-theorem:(163)

Lxy → px = y = py

This is not surprising. After all, these theories establish an intimate
relationship between topology and spatiality, and the claim that location
implies connection is but one way of making that explicit. One mighteven go as far as to say that on these theories location is connection ofa kind: it is the relation of connection that always holds in the special
case where one of the relata equals the place of the other. On the other
hand, it is fair to note that all of this depends on the assumption that L isconditionally reflexive (L.4). Should one decide to go for the alternative
option (L.5), treating L as a relation whose domain does not contain any
spatial regions, (163) would not hold and (L.19) would not follow from(160)–(162). Indeed, on that way of interpreting ‘L’, the picture would
be perfectly reversed: nothing would be connected to its place because
places would lack a place of their own, hence (L.19) would be just as
unacceptable in these theories as it is in KGEM T .
These are just some examples, but they suffice to show that the ontological intertwining between locative and mereotopological concepts cannot be assessed without a clear stand on some very basic semantic issues
concerning those concepts. Of course, this is also true of the interplay
between mereological and topological concepts, but in the present casethe stakes are higher. Some of the options leave room for no intertwining whatsoever; other options trivialize the intertwining by explainingSpatial Reasoning and Ontology: Parts, Wholes, and Locations89

one sort of concept directly in terms of the other. There is, to be sure,some room for compromise. After all, KGEM T is a pretty strong theory and its weaker children are compatible with (L.19). Similarly, there
is strictly speaking no constraint to supplement its main alternatives
with the explicit equivalences in (160)–(162), so the status of (L.19) isin principle open for those theories, too. One final remark, however, isin order. For suppose we do accept (L.19) together with (L.8), that is,
suppose we do establish an overt mereotopological linkage of external
connection between a spatial object and its spatial location. Then it
follows that no spatial entity will ever qualify as a interior proper partof anything, except for empty regions of space:(164)

PLxy → ¬IPPxz ∧ ¬IPPyz

This is an immediate consequence of the definition of ‘IPP’ (Section 2.3.1):
my “interior” proper parts, for example, would immediately turn into
tangential proper parts by virtue of being connected to something withwhich I have no parts in common—their places; and the “interior” properparts of my place, at least those proper parts that are not empty, would
immediately turn into tangential proper parts by virtue of being connected to something with which they have nothing in common—theirmaterial guests. This is bad news, for it means that our mereotopology
would collapse altogether. Or rather, it means that it would have to belargely re-written by replacing throughout our topological primitive ‘C’with the following impure connection predicate:(165)

RCxy =df Cxy ∧ (Rx ↔ Ry)

Restricted Connection

It is, of course, this notion of connection that we had in mind in setting
up KGEM T and its variants. RC never cuts across levels: in order for
two entities to be RC-related, both of them or neither of them must
be regions of space. But then we have come around the circle, for theRC-variant of (L.19) is clearly false.
We face, here, the fundamental limit of topology—and mereotopology—
as a general theory of space. One way or the other, the analysis situs
cannot do proper justice to the fact that objects are situated, which
is why the theory of location is independently needed. One way or theother, spatial reasoning must come to terms with the fundamental metaphysical mystery on which it depends—embedding in space.

Acknowledgments
Parts of this chapter draw on previous material. In particular, Section 1
has some overlap with Varzi (2003) while Section 2 has some overlap
with Chapters 1, 4 and 5 of Casati and Varzi (1999).

90

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