EVENTS, T OPOLOGY AND TEMPORAL RELATIONS *We are used to regarding actions and other events, such as Brutus’ stabbingof Caesar or the sinking of the Titanic, as occupying intervals of some underlying linearly ordered temporal dimension. This attitude is so natural and compelling that one is tempted to disregard the obvious difference between time periods and actual happenings in favor of the former: events become mere “intervals
cum description”.1 On the other hand, in ordinary circumstances the point oftalking about time is to talk about what actually happens or might happen atsome time or another. We talk about ‘now’ and ‘then’ in an effort to put someorder in our description of what goes on. And since different events seem to overlap in so many different ways, a full account of their temporal relations seems to
run afoul of a reductionist strategy.
This raises two philosophical questions. The first is whether we can actually go beyond time, as it were, i.e., whether we can take events as bona fide entities and deal with them directly, just as we can deal with spatial entities such as
physical bodies or masses without confining ourselves to their spatial representations. This is a controversial issue (though probably not as controversial as it
used to be), and ties in with a number of unsettled problems concerning, e.g.,
the structure of causality or the definition of adequate identity and individuation
criteria for events. 2 The second question is whether we can perhaps do withouttime, i.e., whether we can dispense with time points or intervals as an independent ontological category and focus only on actual or potential happenings, in
opposition to the form of reductionism mentioned above—in short, whether we
can account for the temporal dimension in terms of suitable relations among
events. This is also a highly controversial issue, and relates to the classical dispute concerning relational vs. absolutist conceptions of (space and) time.3
It is this second question that we intend to focus on here. Even if the acceptance of events as part of our ontological inventory is in itself a matter of dispute, we shall assume there to be enough good arguments to justify a positive
attitude in this regard: events are part of the furniture of the world, whatever their
exact ontological make up. We shall actually assume that events are individuals,
as opposed to, for instance, states of affairs.4 The focus of our concern is whether
this assumption allows one to answer also the second question in the af-“Events, Topology, and Temporal Relations” by Fabio Pianesi and Achille C. Varzi,
The Monist, vol. 79, no. 1, pp. 89–116. Copyright © 1996, THE MONIST, La Salle, Illinois 61301.

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FABIO PIANESI AND ACHILLE C. VARZI

firmative. More specifically, our purpose is to study the possibility of giving a
positive answer in the following form: if events are assumed as bona fide individuals in our basic ontological inventory, then the basic temporal relations canbe explained away without resorting to an additional set of temporal primitives(instants and/or periods).
One classical suggestion in this direction is of course the account of Russell [1914, 1936], where time instants are construed as maximal sets of pairwise
simultaneous (or partially simultaneous) events. This treatment is echoed in various later accounts, from Whitehead [1929] and Walker [1947] to Kamp [1979]and van Benthem [1983], where events are taken as primary entities inducing periods as secondary (and instants as merely tertiary) entities. More recently,
Thomason [1989] argued that the mathematical connection between the way
events are perceived to be ordered and the underlying temporal dimension is essentially that of a free construction (in the category-theoretic sense) of linear orderings from event orderings, induced by the binary relation “wholly precedes”.
Furthermore, Forbes [1993] explored a modal account whereby the time-series ofa given world w is construed as a sequence of equivalence classes of entities existing at w, the classes being determined by facts about the possible worlds thatbranch from w.
In this paper we present an alternative account, based primarily on the basic
network of formal ontological relations—specifically, mereological and topological relations—that a domain of events must arguably satisfy. The motivations
for this approach are quite general and lie beyond the specific issue of temporalconstructions. Among other things, we also believe it may shed light on thefirst question above. In the following, however, we shall not go much beyond
the main issue that we just outlined; our only concern will be to show how
mereological and topological reasoning—which we take to be among the basic
tools for ontological analysis—provides adequate grounds for the construction of
temporal relations.
Our argument is therefore conditional: if individual events are countenanced,
then temporal relations can be construed out of them. Moreover, this should betaken quite generally, and should not carry any commitment to the view thatevents are basic entities rather than, say, secondary entities supervening on their
participants. Our construction of temporal relations is grounded on the mereotopological properties of events; it does not, however, extend to an account of
those very properties, which may therefore be taken as either primitive or derived.
1. Combining Mereology and Topology
We see mereology and topology as closely interconnected. Thus, on the one
hand, we sympathize with the view that the theory of parts—as rooted in thework of Leś niewski [1916] and Leonard and Goodman [1940]—provides a re-EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

91sourceful alternative to set theory for the analysis of the objects and events of ordinary experience.5 At the same time, we are also persuaded that a purely mereological outlook is too restrictive unless it is integrated with some concepts andprinciples of a topological nature.6 There are several reasons for this. For instance, mereologically there is no way to distinguish between a one-piece, selfconnected whole, such as a stone or a whistle, and a scattered entity made up of
several disconnected parts, such as a broken glass, a soccer tournament, or Brutus’ repeated stabbing of Caesar. Moreover, mereology alone cannot account forsome very basic spatio-temporal relations among the entities of ordinary discourse, such as the relationship between a material object and its surface, or the
relation of continuity between two successive actions or events, or the relationof something being inside, abutting, or surrounding something else. All of theseare phenomena that run afoul of plain parthood relations, and their systematic account requires a topological machinery of some sort.
There are to be sure various ways in which the two domains of mereology
and topology can be combined together.7 One can see them as two independent
provinces (following in the footsteps of inter alia Tiles [1981] and Lejewski[1982]); or one may grant priority to topology and characterize mereology derivatively, for instance defining parthood in terms of topological connection (as in
Clarke [1981]). The latter approach is nowadays rather popular, particularly inthe artificial intelligence community.8 Indeed it proves fit to account for a fairdeal of mereotopological reasoning if we confine ourselves to an ontology of
temporal intervals and/or spatial regions. If, however, we are to take an openfaced attitude towards full-fledged entities and actual happenings (without identifying them with their respective spatio-temporal co-ordinates), then the reductionof mereology to a distinguished chapter of topology seems untenable, as different
entities can be located at the very same spatio-temporal regions. An object canbe wholly located inside a hole, hence totally connected with it, without bearing
any mereological relation to the hole itself. Or two events may have exactly thesame topological connections and yet be mereologically distinct, as with the rotating and the becoming warm of a metal ball that is simultaneously rotating andbecoming warm.9
In general, therefore, we are inclined to favor the first option mentioned
above, treating mereology and topology as two conceptually independent domains. Formally this will be reflected in our use of two distinct primitives—apure mereological notion of part, and a purely topological notion of boundary.
Of course, several other sets of primitives could serve the job, but our choice
here is not entirely arbitrary, at least for the topological part. For instance, various examples of our favored strategy for combining mereology and topology canalso be found in some linguistics-oriented work on tense and aspect, such as
Kamp’s [1979], Bach’s [1986], or Link’s [1987] algebraic semantics for event

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FABIO PIANESI AND ACHILLE C. VARZI

structures, where a mereological relation defined on a field of events is typicallymatched with an independent ordering of temporal precedence. For our present
purposes, however, such a course would have a flavor of circularity. If the point
is to provide an account of the temporal dimension based on the fundamental
mereotopological features of event structures, it is essential that we take thesefeatures to be of a most general, time-independent nature. So precedence won’t do.
By contrast, reference to parts and boundaries appears to be a natural and independently motivated option, both ontologically and from a cognitive standpoint.10
1.1 Background Assumptions. The entire mereotopological machinery canbe developed within a first-order language with identity and descriptions. We
shall use ‘¬’, ‘∧’, ‘∨’, ‘→’, and ‘↔’ as connectives for negation, conjunction,
disjunction, implication, and material equivalence; ‘∀’ and ‘∃’ for the universal
and existential quantifiers; and ‘ι’ for the definite descriptor. This last operator
will not have any use per se. However, it will play a crucial role in the definition of the term-forming operator of general sum by means of which several
basic mereological and topological notions will be characterized. (Alternatively
one could use set variables along with a set fusion functor, as in Tarski [1937]
or Leonard and Goodman [1940], but this method would introduce additional
complications and would mix up mereology with set theory.) Since this sum
operator may be undefined for some arguments, the underlying logical apparatus
requires therefore some means of accounting for the possibility of non-denoting
expressions.
This can be done in a number of ways. Among the alternatives available(including Leś niewski’s [1916] original approach) we find it convenient to relyon the minimal free description theory stemming from Lambert [1962], consisting in assuming(1)∀y(y=ιxφ[x] ↔ (φ[x/y] ∧ ∀x(φ[x] → x=y))

as the only characteristic principle for descriptions. This theory is “minimal” insofar as it only captures the logic of ‘ι’ with respect to descriptions that are denotationally successful, leaving open the issue of what principles should continue to hold in the presence of referential gaps. It does, however, allow us totreat descriptive terms as genuine singular terms, and this is all we need for ourpresent purposes. For definiteness, we shall also fix the underlying free quantification theory: it is the system resulting from the classical predicate calculus byreplacing the principle of Universal Instantiation with its universal closure:(2)∀y(∀xφ[x] → φ[x/y]).

This minimalistic strategy has already been exploited in the context of
mereological theorizing by Simons [1991], and will prove sufficient also for ourEVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

93

purposes in spite of its failing to reveal the whole truth. In any case, to facilitatecomparisons we shall try to highlight those points where the choice of a different strategy would affect the argument.
1.2 Mereology. We symbolise the primitive mereological relation of parthood by ‘P’, so that ‘P(x, y)’ reads “x is (a) part of y”. Derived notions, such as
identity, overlapping, and the like, or the operations of sum, product, difference,etc. are defined as usual:
DP1
DP2
DP3
DP4
DP5
DP6
DP7
DP8
DP9
DP10
DP11
DP12

x=y
=df
O(x, y) =df
X(x, y) =df
PO(x, y) =df
PP(x, y) =df
σxφ[x] =df
πxφ[x] =df
x+y
=df
x×y
=df
x–y
=df
~x
=df
U
=df

P(x, y) ∧ P(y, x)
x is identical with y∃z (P(z, x) ∧ P(z, y))x overlaps yO(x, y) ∧ ¬P(x, y)
x crosses yX(x, y) ∧ X(y, x)
x properly overlaps yP(x, y) ∧ ¬P(y, x)
x is a proper part of yιz∀y (O(y, z) ↔ ∃x (φ[x] ∧ O(x, y)))sum of φers
σz ∀x (φ[x] → P(z, x))
product of φers
σz (P(z, x) ∨ P(z, y))sum of x and yσz (P(z, x) ∧ P(z, y))product of x and yσz (P(z, x) ∧ ¬O(z, y))difference of x and yσz (¬O(z, x))complement of xσz (z=z)universe

Note that the functors/ terms based on DP6 may be partially defined (i.e., correspond to improper descriptions) unless we go with the fiction of a null individual that is part of everything. For instance, non-overlapping entities will haveno product and the universe will have no complement. This introduces a significant deviation in the obvious correspondence between mereological operations
and the standard set-theoretic operations of union, intersection, difference, etc.
As mereological axioms we assume the following two, along with thestandard axioms for identity:
AP1
AP2

P(x, y) ↔ ∀z (O(z, x) → O(z, y))∃xφ[x] → ∃y∀z (O(z, y) ↔ ∃x (φ[x] ∧ O(x, z))).

AP1 secures that parthood is an extensional partial ordering while AP2 (the“fusion axiom”) guarantees that every satisfied condition φ picks out an entity
consisting of all φers. This yields a classical mereology as usually understood,
corresponding to a Boolean algebra with zero deleted. A sample selection of theorems relevant to the following is listed below:
TP1
TP2
TP3

P(x, x)P(x, y) ∧ P(y, z) → P(x, z)
x=y ↔ ∀z (P(z, x) ↔ P(z, y))94

FABIO PIANESI AND ACHILLE C. VARZI

TP4
TP5
TP6
TP7
TP8

x=y ↔ ∀z (P(x, z) ↔ P(y, z))x=y ↔ ∀z (O(x, z) ↔ O(y, z))PP(x, y) → ∃z(P(z, y) ∧ ¬O(z, x))¬P(x, y) → ∃z(P(z, x) ∧ ¬O(z, y))∃xφ[x] → ∃y(y = σxφ[x]).

It may be worth recalling that none of these principles is uncontroversial.
For instance, since Rescher [1955] several authors have had misgivings aboutsuch straightforward consequences of AP1 as TP2 (transitivity of ‘P’) or TP3–TP5 (extensionality). 11 In both cases, however, the objections involve reasoning
about the variety of part-whole relations that may be distinguished (e.g., between
components and complex, or quantity and mass) and may therefore be disregarded
as long as we remain at a sufficiently general level of analysis. Even such an apparently innocent consequence of AP1 as TP1 (reflexivity of ‘P’) might on someconditions be objected to, particularly insofar as the logical background that weare assuming allows for non-denoting terms. For instance, in this regard Simons[1991] suggests applying the falsehood principle of Fine [1981] to deny that
‘P(x, x)’ may be true when x does not exist (i.e., when ‘x’ does not denote).
However, such a stance would introduce a disturbing asymmetry between parthood and other basic predicates such as identity—of which parthood is a generalization—unless we are also ready to make a self-identity statement ‘x=x’ dependon the existence of x. This is in contrast with our general attitude towards thelogic of singular terms, which in fact follows the customary policy of assuming
a standard identity theory. Therefore, this type of objection also will be disregarded in the following.
The second axiom is not uncontroversial either. For one thing, in the presence of AP1 it implies the so-called “supplementation” principles expressed by
TP6-TP7, which some authors found reason to deny.12 There are indeed cases
where restrictions on the domain might involve violations of such principles (adisc with a disc removed is not a disc), but this is already a matter of regional
ontology and need not concern us for the moment. As a matter of generality, wetake it that the existence of a remainder between a whole and a proper part cannot
be denied: otherwise it would be possible for an entity to have a single properpart, and that is beyond our understanding of this notion. Likewise, AP2 hasbeen disputed for having counter-intuitive instances when φ is true of scattered orotherwise ill-assorted entities, such as the totality of red things, or Brutus’ birthand his stabbing of Caesar. 13 From a purely mereological perspective, however,this criticism also seems off target. As Lewis [1991: 81] put it, if we alreadyhave certain things, allowing for their sum is no further commitment: the sumis those things. (That the sum is always uniquely defined is guaranteed by TP8.)
In any case, one may feel uncomfortable with treating unheard-of mixtures as in-EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

95

dividual wholes; but which wholes are more “natural” than others is not a mereological issue. As noted above, the question of what constitutes an integral wholecannot even be formulated in mereological terms: it is precisely here that topology—the theory of boundaries—comes in.
1.3 Topology. The primitive topological notion of boundary is symbolizedby ‘B’, so that ‘B(x, y)’ reads “x is a boundary for y”. (We say “boundary for”
(rather than of) to avoid a reductive interpretation of boundaries as maximal
boundaries. In general, any boundary for something is a boundary of some part
thereof.) Some useful derived notions can be introduced as follows:
DB1
DB2
DB3
DB4
DB5
DB6
DB7
DB8
DB9
DB10
DB11
DB12

b(x)
=df
c (x)
=df
i(x)
=df
e(x)
=df
Cl(x)
=df
Op(x)
=df
BP(x, y) =df
IP (x, y) =df
C(x, y) =df
EC(x, y) =df
ST(x, y) =df
Cn(x)
=df

σz (B(z, x))maximal boundary of xx+ b(x)closure of x
x – b(x)interior of x~x – b(x)exterior of xx=c (x)x is closed
x=i(x)
x is openP(x, y) ∧ B(x, y)
x is a boundary part of yP(x, i(y))
x is an interior part of yO(c (x), y) ∨ O(c (y), x)
x is connected to y
C(x, y) ∧ ¬ O(x, y)
x is externally connected to y∀z(IP (x, z) → X(z, y))x straddles y∀y∀z (x=y+z → C(y, z))
x is self-connected

Again, these functors, predicates, and relations may be undefined for some arguments, as they may involve improper (e.g., denotationless) definite descriptions.
Note that nothing in these definitions implies that boundaries are always
part of the entities they bound. In fact, we accept the standard topological distinction between open and closed entities, allowing for entities with external boundaries. For example, in Pianesi and Varzi [1994a] we examine a characterizationof the standard classification of event types (Vendler [1957]) by treating processes (such as John’s climbing) as non-closed events, accomplishments (John’s
climbing of the mountain) as closed processes, and achievements (John’s reaching of the top) as parts or the corresponding boundaries. And in Casati and Varzi[1994], holes are regarded as immaterial bodies spatially bounded from the outside: the boundary of a hole is the surface of its material host.
In this regard, our basic axiomatization is therefore in line with standard
topology. More precisely, we assume the following axioms:
AB1
AB2
AB3

B(x, y) → B(x, ~y)B(x, y) ∧ B(y, z) → B(x, z)
P(z, x) ∧ P(z, y) → (P(z, b(x×y)) ↔ P(z, b(x)+b(y))),96

FABIO PIANESI AND ACHILLE C. VARZI

which are easily seen to be tantamount to (the mereologized version of) the familiar Kuratowski axioms for topological closure:
TB1
TB2
TB3

P(x, c (x))
P(c (c (x)), c (x))
c (x+y) = c (x) + c (y).

This gives us a straightforward reformulation of much of standard topology basedon mereology instead of set theory, provided only that we take care in handling
undefined operators. In particular, with AB1 we assume boundaries to be always
symmetrical, in the sense that every boundary of an entity is also a boundary of
the entity’s complement—if that complement exists.14 This ensures that boundaries are always connected to the things they bound. Here is a list of further
theorems that can be proved from AB1–AB3 and that will be required in the following:
TB4
TB5
TB6
TB7
TB8
TB9
TB10
TB11
TB12
TB13

C(x, x)C(x, y) → C(y, x)P(x, y) ∧ IP (y, z) → IP (x, z)
IP (x, y) ∧ P(y, z) → IP (x, z)
B(x, y) ↔ ∀z(P(z, x) → B(z, y))B(x, y) ↔ ∀z(P(z, x) → ST(z, y))
IP (x, y) ↔ P(x, y) ∧ ¬∃z(P(z, x) ∧ BP(z, y))BP(x, y) ↔ P(x, y) ∧ ¬∃z(P(z, x) ∧ IP (z, y))P(x, y) → ∀z (C(z, x) → C(z, y))∀x(φ[x] → B(x, y)) → B(σxφ[x], y).

The last two of these theorems are especially noteworthy. TB12 highlights
the main connection between mereological and topological notions. As alreadyhinted at above, systems in the tradition of Clarke [1981] also assume the converse of this principle, with the effect of reducing mereology to a part of topology. By contrast, the possibility that topologically connected entities bear no
mereological relationship to one another leaves room for a much richer taxonomy
of basic mereotopological relations than usually recognized.15 For instance, therelations of connection, overlapping, parthood, and interior parthood introduced
above, and common to most known systems, can be integrated by the following:
DB13
DB14
DB15
DB16

E(x, y)S(x, y)A(x, y)W(x, y)

=df
=df
=df
=df

∀z (C(z, x) → C(z, y))∃z (E(z, x) ∧ E(z, y))C(x, y) ∧ ¬S(x, y)E(x, i(y))

x is enclosed in yx is superimposed on yx abuts y
x is enclosed within y

Evidently, S is implied by O, E by P, and W by IP , but the converses do not
generally hold. (The rotation of the metal ball and its simultaneous getting warm

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

97

C
S

C
S
E

C
S
E
W

C
S O

C
S O
P
E

C O
S P
E
W IP

C

Figure 1. Some basic mereotopological relations exploiting the distinction between mereological parthood (overlapping, etc.) and mere topological enclosure (superposition, etc.).

are perfectly superimposed, but do not overlap; the stone is enclosed in the hole,
not part of it.) The resulting taxonomy of relations is depicted in Figure 1.
As for TB13, it shows that boundaries are closed under general sum andtherefore under all mereological properties. Following Smith [1993], however,
we also wish to capture some further common-sense intuitions that go beyond
the repertoire of standard topology. In particular, we need at least a rendering of
the intuitive Aristotelian-Brentanian idea that boundaries are “parasitic” entities,i.e., cannot exist independently of the larger entities they bound.16 This is incontrast with the standard set-theoretic conception of boundaries as sets of ordinary, ontologically independent points. More specifically, we assume that every
self-connected boundary is a boundary part of some larger self-connected entitywith non-empty interior:
AB4

Cn(x) ∧ ∃y B(x, y) → ∃z (BP(x, z) ∧ Cn(z) ∧ ∃w IP (w, z)).

Thus, to continue with our earlier examples, an achievement such as John’sreaching of the top cannot occur except as a culmination of a process (John’s
climbing), i.e., except as boundary part of an accomplishment (John’s climbingof the mountain).
It is understood that further principles would be needed in order to obtain atleast a rough approximation of the folk theory of spatio-temporal continua.
Here, however, we shall content ourselves with AP1–AP2 and AB1–AB4, regarding this as a minimal theory.
2. Event Structures
Let us now see how this general mereotopological framework can be specialized to a domain of events. As already pointed out, on a rather popular conception events are regarded as intrinsically dependent on the temporal dimension.

98

FABIO PIANESI AND ACHILLE C. VARZIThis conception ranges from the strong reductionist view that events are nothing
but temporal intervals 17 to weaker forms that take events as primitive entities
endowed with some primitive temporal relation 18 defined on them. Work in the
tradition of Kim’s [1969] theory of events as property-exemplifications (that is,
exemplifications of specific properties by specific individuals at specific times)
also treats time periods as primitive entities upon which events supervene.19 By
contrast, in the following we shall argue that events can be given an independent
characterization that accords with our intuitions about their mutual relations andderives the temporal ordering by imposing suitable restrictions on the underlying
mereotopological structure. More precisely, we will show that the formal connection between the way events are supposed to be ordered and the underlying
temporal dimension is essentially that of a construction of a linear ordering fromthe mereotopological properties of an oriented domain structure in which events
are included as bona fide individuals. In other words, temporal relations are a byproduct of what we call an event structure.
2.1 Divisors. Our characterization of event structures relies on the auxiliary
concept of a divisor. Intuitively, a divisor is an event that separates the entire
domain of events into two disconnected parts, thus making it possible to choose
one part as corresponding to the sum of all events that temporally precede the divisor, and the other as the sum of all events that follow it. This is illustrated in
Figure 2. (Of course, the choice of precedence vs. following—past vs. future—
will be arbitrary, as long as successive choices for different divisors be done in aconsistent way. If we think of an event domain as comprising the totality of all
happenings, there is no a priori way to fix the temporal orientation. But ourconcern here is the temporal ordering, not the direction of time.)
x
x1

x2

x3

past

futureFigure 2. A divisor, x , separating the domain of events into two disconnectedparts. (Here and below we use dots to represent events; divisors are representedby (groups of) solid dots and are highlighted by shaded dashed lines. For instance, here x is a divisor consisting of three parts, x 1, x 2, and x 3, none ofwhich is itself a divisor. Thin lines are, intuitively, courses of events as determined by the relation of topological connection.)

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

99

As a preliminary characterization, this intuition can be expressed with thehelp of both mereological and topological notions by requiring every divisor to
split all of its neighborhoods into disconnected parts:
DE1

D(x) = df ∀y (IP (x, y) → ¬Cn(y–x)).

Within certain limits that will soon become clear, this is sufficient to capture
the intuitive distinction between “global” and “local” events. We may think of adivisor as an all-encompassing event made up of all that happens during a certain“period”. By contrast, an action such as Brutus’ stabbing of Caesar, or an incident such as the sinking of the Titanic, are “local” events: many other actions
and events occur at the same time, but in different places. This is of course a
crucial distinction arising from our acceptance of events as distinct from the intervals they occupy. Events are full-fledged entities.20
Note that if we refer to divisors for the purpose of characterizing the notions of past and future, these latter will be deprived of any absolute meaning andbecome relative notions: given any divisor x, the suggestion is to interpret theevents on one side of x as past events, and the ones on the other side as future
events, relative to x. There is no past or future except with respect to some division of the whole of history. It will follow that we will need to guarantee thatsuch a relativistic account be nevertheless suited to the task; that is, we must
make sure that all the relevant divisors partition past from future events in a coherent may. (For instance, an event that lies in the past relative to a given divisor x must also lie in the past relative to any divisor y that lies in the future ofx.) This means that ‘D’ must be refined, since nothing so far prevents two divisors from generating orthogonal divisions.
We also want our construction to allow for a certain degree of control on its
grain. Intuitively, any two events that are part of the same divisor should countas simultaneous. However, for that purpose the general notion defined by DF1 is
far too unconstrained. For one thing, it allows for divisors with all sorts of
topologies, including undesired ones (for instance, multiply-connected divisors,i.e., divisors with “holes”, such as the sum of a line and a tangential circle, or aplane and a tangential torus). Second, we need some means for associating each
event in the domain with the “right” divisor, viz. the minimal one containing it.
(Otherwise, for any two events x and y we could pick out a sufficiently large divisor z containing both, thereby making them simultaneous.) Moreover, thepredicate ‘D’ picks out the distinguishing property of a divisor relative to the full
mereotopological structure of the given domain of events, but of course there are
domains admitting of an indefinite, potentially infinite number of divisors, only
some of which can (or need) be collected in a temporally coherent structure.
In short, not all divisors can or need be considered together. Rather, every
domain can be associated with various dividing devices δ (including divisors that

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FABIO PIANESI AND ACHILLE C. VARZI

ignore some mereological distinctions, treating as punctual events, for instance,
entities that do in fact have a part structure), giving rise to a variety of event
structures. Furthermore, the possibility of varying the grain itself may be a welcome one. This can be helpful, for instance, in accounting for the various degrees
of precision that natural language permits when talking about events and time. 21
2.2 Structures. Given all of this, we define an event structure quite generally as an ordered pair 〈E, δ〉 made up of a non-empty domain of events, E, alongwith a distinguished predicate, δ, to be thought of as singling out a coherent setof divisors. The specific conditions are as follows.
First, we assume E to be mereotopologically connected:
AE1∀z(O(z, x) ∨ O(z, y)) → C(x, y).

We are here taking E as a domain closed under all mereological and topologicalprinciples set forth in the previous section, and we understand individual variables to range over this domain. (Thus, for instance, not only do we assume thatthe sum of any number of events exists, by AP2; we also assume that it is an
event.22 If one feels more comfortable with a less liberal use of the word “event”,
one can think of E as the mereotopological closure of the domain of bona fide
events, whatever sorts of entity that might involve.) Accordingly, condition AE1
could also be written as
AE1'

Cn(U),

which effectively corresponds to the statement that the whole universe (the universal event) is self-connected. This is stipulative and reflects the idea that there
are no gaps in history: something is always happening, whether remarkable ornot. 23
Second, the divisor-specific predicate δ is to pick out an exhaustive set ofmutually coherent divisors incorporating the above-mentioned requirements ofminimality and granularity. Generally speaking, it can be characterized as a maximal predicate satisfying at least the following conditions:
AE2
AE3
AE4
AE5

δ(x) → D(x)
δ(x) ∧ δ(y) →. δ(x+y) ↔ C(x, y)δ(x) ∧ δ(y) →. δ(x ×y) ↔ O(x, y)δ(x) ∧ δ(y) →. δ(x–y) ↔ X(x, y).

In other words, the events fulfilling δ form a maximal class of divisors closed
under the mereological operations of sum, product, and difference (within certain
obvious limits). These latter conditions do not generally hold for ‘D’, but areeasily motivated by the intended meaning of “divisor” that we are considering(Figure 3). Generally speaking, if every event must be included in some divisor

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

101(with some limit exceptions to be discussed shortly), then any two connected divisors must make up a (“thicker”) divisor; and if divisors are to divide the domain into two main parts (intuitively, past and future), then they must be “minimal”, in the sense of not consisting of two or more disconnected divisors. (Theymay, of course, consist of two or more disconnected parts, as in Figure 3(b);likewise, since the assumption of connectedness of the universe is not preserved
locally, there is no guarantee that a divisor always splits its complement into
two self-connected parts: see Figure 3(a)). This explains AE3. Moreover, if all
x

x

(a)

(b)

x

x

(c)

(d)

x

y

(e)

x

(f)

y

x

x

Figure 3. Some effects of AE3–AE6: (a) a divisor, x , separating its complement
into two self-disconnected parts; (b) a self-disconnected and yet legitimate(minimal) divisor; (c) an illegitimate self-disconnected divisor by AE3 (sum of
two separated divisors); (d) a non-minimal self-connected divisor by AE3 (sumof two minimal adjacent divisors); (e) two illegitimate divisors by AE4 (productis not a divisor); (f) an illegitimate divisor by AE5 (subtracting it from any
divisor overlapping the event in the middle does not yield a divisor).

102

FABIO PIANESI AND ACHILLE C. VARZI

divisors must operate in a “parallel” fashion, i.e., have a uniform temporal orientation, then the common part of any two overlapping divisors (AE4) and the difference between any two crossing divisors (AE5) must themselves be divisors.
Conditions AE3 and AE4 can be strengthened in an obvious way by requiring δ to be closed under infinitary sums and infinitary products of connected/
overlapping divisors:
AE3*
AE4*

∀x(φ[x] → δ(x)) →. δ(σx φ[x]) ↔ Cn(σx φ[x])∀x(φ[x] → δ(x)) →. δ(πxφ[x]) ↔ ∃z∀x(φ[x] → P(z,x)).

We are not sure that AE3* is immune from unpalatable consequences, so we refrain from including it in our basic characterization of event structures. (It would
imply that the entire universe is a divisor by AE1, so some restriction would be
required.) On the other hand, given the intuitive notion of a divisor that δ ismeant to capture, AE4* provides a natural extension of AE4, and we shall assume it unrestrictedly. In set-theoretic terms, it has the effect of making the setof divisors into a closure system, and this is really what we mean when we saythat the common part of any number of overlapping divisors picked up by δmust itself be a divisor. The corresponding closure operator associates each eventx with the smallest divisor containing x. We shall call it the divisor of x:
DE2d(x) =df πz(δ(z) ∧ P(x, z)).

That d is indeed a closure operator is essentially guaranteed by the following
theorems, which assert the analogues of the usual increasing, idempotency, andmonotonicity properties:
TE1
TE2
TE3

P(x, d(x))
d(d(x)) = d(x)P(x, y) → P(d(x), d(y)).

However, d is only a partial operator, as it is not defined for arguments that are
not contained in any divisor in δ. This is as it should be in case we are talkingof, say, the universal event U, for the whole of history has no past and no future.
On the other hand, if we consider events that are indefinitely extended in the past
but not in the future, or vice versa, then we do want to say that such events
precede every future event, or vice versa that they follow every past event. Likewise, if U is bounded, then the problem arises of explaining the temporal location of its boundaries. There is no such thing as the divisor of the Big Bang; yet
we want to say that nothing happened before, and everything followed. To coverthese and similar cases, we extend d to a total operator d* :DE3

d* (x) = df x + σz ∃y (P(y, x) ∧ z=d(y)).

This new operator satisfies TE1–TE3 and includes d as a special case.

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

103

2.3 Oriented structures. The notion of an event structure provides a characterization of the notion of an event (or a family of events) separating in some intuitive sense what has already happened from what is still to come. It does not,however, say anything concerning the way in which the separation is performed.
That is, it guarantees—as we shall see—that the relative notions of past and future behave coherently throughout, but it remains neutral with respect to thequestion of whether a given event actually lies in the past or in the future of another event. In short, event structures are not temporally oriented.
We can make this more precise with the help of some additional terminology. Let us say that an event x is flanked by two events z1 and z2 if these lie on
two opposite sides, as it were, of the divisor of x; and let us say that x is complemented by z1 and z2 (or that x separates z1 from z2 ) if these are maximal
among the events by which x is flanked:
DE4
DE5

F(z1 , x, z2 ) =df ¬O(z1 +z2 , d(x)) ∧ ¬C(d* (z1 ), d* (z2 ))S(z1 , x, z2 ) =df F(z1 , x, z2 ) ∧ z1 +z2 = ~d(x).

(It is easily verified that DE5 implies that neither z1 nor z2 be a divisor, or a partof a divisor. In fact, both events must extend all the way in one of the two
directions induced by the double-sidedness of the divisor predicate δ.) The pointis then that whenever x is a “dividing” event, i.e., an event whose minimal divisor d(x) is defined, we can find a unique pair of events z1 and z2 such thatS(z1 , x, z2 ). But since such a pair also satisfies S(z2 , x, z1 ) (i.e., S is a relationthat is symmetric with respect to its middle argument), there is no way of tellingwhich is the past and which the future of x.
We can, however, do the following (Figure 4). Given an initial triple z1 , x,z2 such that S(z1 , x, z2 ), we can define a function g associating every dividing
event y with the totality of events that lie—as it were—on the same relative side
as z1 and z2 , respectively:DE6g(y, zi) =df ιz∃w(S(z,y,w) ∧ (O(y,zi)→P(z, zi)) ∧ (¬O(y, zi)→P(zi, z))).

Clearly, g(y, z1 ) and g(y, z2 ) are perfectly symmetric and interdefinable:
TE41
TE42
TE51
TE52

S(g(y, z1 ), y, g(y, z2 ))S(g(y, z2 ), y, g(y, z1 ))g(y, z1 ) = σz (P(y, g(z, z2 ))g(y, z2 ) = σz (P(y, g(z, z1 )).

And, clearly, there is still no way to say which is the past and which the future.
However, we can now say that if z1 occurs, say, before x (and z2 after), theng(y, z1 ) must always occur before y (and g(y, z2 ) after) no matter how y is chosen.
In other words, the choice of some initial point of view relative to some arbitrary
event is all we need to fix a coherent orientation for the whole event structure.

104

FABIO PIANESI AND ACHILLE C. VARZId (y)u1 = g(y, z1)

u 2= g(y, z 2)















...

...

w1
x

y












...

z1 = g(x, z 1) d(x)

...






w2

z 2 = g(x, z 2)

Figure 4. Pictorial illustration of DE4–DE6. Both x and y are flanked by w1 andw2 ; x separates z 1 from z 2 , while y separates u1 from u2 ; these latter in turncoincide with g(y , z 1 ) and g(y , z 2 ), respectively, while g(x , z 1 ) and g(x , z 2 )reduce to z 1 and z 2 .

This leads to the following general definition. We say that a triple 〈E, δ, e〉
is an oriented event structure iff 〈E, δ〉 is an event structure as defined in the previous section, and e is a distinguished element of E such thatAE6

∃x ∃y (S(e,x,y)).

If 〈E, δ, e〉 is such a structure, we can define a pair of (possibly partial) maps ƒand ƒ' in the obvious way using TE51 (or TE52 ):DE7DE8

ƒ(x) = df g(x,e)ƒ'(x) = df σz (P(x, ƒ(z))).

We then just stipulate that e represents the past. That is, we simply treat ƒ as afunction of temporal orientation associating each event in the domain with thetotality of events that temporally precede it; and, correspondingly, we treat ƒ' as
a function associating each event with the totality of what follows it. (We take itthat if ƒ(x) is not defined, then x has no past, and likewise for ƒ'(x).) This isadmittedly a conventional choice—the alternative stipulation would do just as
well. But the following facts (for instance) show that it is a coherent choice:
TE6
TE7
TE8
TE9

ƒ(x) = ƒ(d(x))ƒ(x)+ƒ'(x) = ~d(x)P(x, ƒ(y)) → P(ƒ(x), ƒ(y))δ(x) ∧ δ(y) ∧ O(y, ƒ(x)) ∧ O(y, ƒ'(x)) → P(x, y).

TE6–TE7 relate to the intuitive interpretation of the divisor operator δ; TE8
relates to the underlying mereology; and TE9 relates to both, reflecting the provi-

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

d(y) ƒ'(y) =ƒ'(d(y))ƒ'(x) =ƒ'(d(x))

ƒ(x)





























...

d(x)

y2

y

... ...

ƒ(x) =ƒ(d(x))





















x

y1

...

...

ƒ(y) =ƒ(d(y))

...

105

x

ƒ'(x)

Figure 5. Left: pictorial illustration of the mereotopological relations expressedby TE6–TE8 (TE6'–TE8'). Right: x and y 1 satisfy, whereas x and y 2 violate, thegeneral requirement on divisors expressed by TE9 (compare also Figure 3(e)).

sion that all divisors be uniformly oriented (Figure 5). Moreover, DE8 guarantees perfect duality between ƒ and ƒ'. Thus, just as TE7 and TE9 are symmetricwith regard to these functions, the analogues of TE6 and TE8 for ƒ' hold too:
TE6'
TE8'

ƒ'(x) = ƒ'(d(x))
P(x, ƒ'(y)) → P(ƒ'(x), ƒ'(y)).

Note how the stipulation of the “anchor” element e is grounded on theemerging directionality of event structures. For one can easily verify that if〈E, δ, e1 〉 and 〈E, δ, e2 〉 are two oriented event structures, with orientation functions ƒ1 , ƒ 1 ' and ƒ2 , ƒ 2 ' (respectively), then the following holds:
TE10

P(e1 , e2 ) ∨ P(e2 , e1 ) → ƒ 1 =ƒ2 ∧ ƒ 1 '=ƒ2 '.

That is, oriented structures whose anchor elements are related by parthood induce
the same ordering. Since complementation reverses parthood, there are therefore
only two ways of orienting an event structure 〈E, δ〉, and these are given by any
pair of oriented structures 〈E, δ, e1 〉 and 〈E, δ, e2 〉 whose anchor elements e1 ande2 do not overlap. Furthermore it is easy to prove that such structures satisfy
TE11

ƒ1 =ƒ2 ' ∧ ƒ1 '=ƒ2i.e., they induce two opposite orientations.
2.4 Special Structures. Further constraints on E, δ, or e can of course be
added to select (oriented) structures with special characteristic properties. For example, one may want to rule out the possibility that there be atomic events (i.e.,
events with no proper parts) that are open. Or one may want to impose stronger

106

FABIO PIANESI AND ACHILLE C. VARZI

conditions on the granularity of δ: for instance, one may want to require that an
event’s divisor be never “thicker” (i.e., intuitively, of longer duration) than theevent itself; or one may ask for maximum granularity, requiring δ to be “closed
downwards” (i.e., to include every D-divisor that is included in some element).
Likewise, it is possible to impose stronger conditions on the topology of E thanjust self-connectedness, such as density or, alternatively, discreteness: the former
asserts that between any two successive closed events (or, equivalently, between
any two successive open events) there is always a third one; the latter amounts
to the opposite requirement that every closed (open) event has some closed(open) immediate successor and some closed (open) immediate predecessor. In
our formalism these conditions would correspond to the following postulates respectively, the last two corresponding to the two sides of discreteness:
AE7
AE8
AE9
AE10
AE11
AE11'

Op(x) → ∃y PP(y, x)O(x, d* (y)) ↔ O(d* (x), d* (y))δ(x) ∧ P(y, x) ∧ D(y) → δ(y)P(c (x), ƒ( c (y))) → ∃z (P(c (x), ƒ(z)) ∧ P(z, ƒ( c (y))))∃y P(c (x), ƒ( c (y))) →∃z (P(c (x), ƒ( c (z))) ∧ ¬∃w (P(c (x), ƒ(w)) ∧ P(w, ƒ( c (z)))))∃y P(c (x), ƒ'(c (y))) →∃z (P(c (x), ƒ'(c (z))) ∧ ¬∃w (P(c (x), ƒ'(w)) ∧ P(w, ƒ'(c (z))))).

We shall not pursue these lines of development here. For our present purposes, let us only stress once again that the operators and mappings introduced
above may possibly be undefined for some arguments. Different models of the
proposed conditions may therefore be obtained according to the underlying logical apparatus.
3. The Construction of TimeWe are now ready to see how the temporal dimension can be retrieved fromthe mereotopological properties of an oriented event structure. More precisely,
let 〈E, δ, e〉 be such a structure, and let ƒ and ƒ' be the corresponding functionsof orientation (we shall keep these parameters fixed throughout this section).
Then we shall show that the members of E bear relations to one another comparable to the temporal relations commonly assumed in connection with time ontology and temporal reasoning.
To this end, note first of all that along with T6–T9, the following facts
hold for all events x and y in the given domain E :
TE12
TE12'
TE13

P(x, ƒ(y)) → ¬O(x, ƒ'(y))
P(x, ƒ'(y)) → ¬O(x, ƒ(y))
P(x, ƒ(y)) → P(y, ƒ'(x))

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

TE13'
TE14
TE14'

107

P(x, ƒ'(y)) → P(y, ƒ(x))
P(x, y) → P(ƒ(y), ƒ(x))
P(x, y) → P(ƒ'(y), ƒ'(x)).

The first pair assert that past and future (relative to a given event x) do not overlap; the second pair assert that whatever happens before a certain event y must be
such as to include y among its future events and, vice versa, whatever happens
after an event x must include x among its past events; the last pair assert thatwhenever an event is included in another, the past and the future of the lattermust be included in the past and in the future of the former, respectively.
As we have it, of course, only events that have a (minimal) divisor can be
fully matched for precedence, since ƒ or ƒ' may otherwise be undefined. This is
an important consequence of TE7:TE15∃y (y=ƒ(x)) ∧ ∃y (y=ƒ'(x)) → ∃y (y=d(x)).

In particular, it follows that ƒ(ƒ(x)) and ƒ'(ƒ'(x)) are always undefined. (For what
would it mean to say that “the past” and “the future” of an event have a past andfuture of their own?) Furthermore, note that if x is a non-dividing event, for instance an open-ended event reaching infinitely backwards or forwards, then ƒ(x) is
defined only if ƒ'(x) is not defined, and vice versa. Accordingly, when considering
principles such as TE12–TE14', we should always keep in mind that the underlying semantics may be only partially defined. Alternatively, we can think of suchnon-dividing events as having a degenerate future or a degenerate past, respectively. This is captured in the following equivalences:
TE16
TE16'

ƒ(x) = σz (P(x, ƒ'(z))) = πz (∃y PP(y, x) ∧ z=ƒ(y))ƒ'(x) = σz (P(x, ƒ(z))) = πz (∃y PP(y, x) ∧ z=ƒ'(y)).

Thus, ƒ yields the “limit” past of an event, which becomes undefined in the caseof events infinitely extending backwards—likewise for ƒ'.
This provides grounds for the intended interpretation of the relative properties expressed by TE12–TE14'. Two further interesting facts are the following:
TE17
TE17'

ƒ'(ƒ(x)) = ~ (ƒ(x))ƒ(ƒ'(x)) = ~ (ƒ'(x)).

These theorems, in a sense, generalize TE7 yielding the corollaries:
TE18
TE18'

P(x, ƒ'(ƒ(x)))
P(x, ƒ(ƒ'(x))).

Together with TE14 and TE14', these ensure that ƒ and ƒ' behave as a pair of Galois connections in E. Moreover, it is worth observing that composing ƒ and ƒ'
induces corresponding topological structures:

108

FABIO PIANESI AND ACHILLE C. VARZI

TE19
TE20
TE21
TE22

P(x, ƒƒ'(x))ƒƒ'(ƒƒ'(x)) = ƒƒ'(x)ƒƒ'(x+y) = ƒƒ'(x) + ƒƒ'(y)P(x, y) → P(ƒƒ'(x), ƒƒ'(y))(likewise for ƒ'ƒ). Thus, the interaction of ƒ and ƒ' yields a pair of well-behaved
topological closure operators. In particular, TE20 shows that their composition
reaches a fixed point immediately after two applications.
All these facts, then, support our intended interpretation of E as a domainof events and they allow us to think intuitively of ƒ and ƒ' as functions of temporal orientation, as desired. In particular, we can say that an event x temporally
precedes (wholly) an event y just in case x is part of ƒ(y), while x and y temporally overlap just in case they overlap or have some parts whose minimal divisors overlap:
DE9
DE10

TP(x, y) =df P(x, ƒ(y)))
TO(x, y) =df O(d* (x), d* (y)).

(In DE9, the same definiens can of course be used to introduce the converse relation of y temporally following x; by TE13, this will be tantamount to:
DE11

TF(y, x) = df P(y, ƒ'(x)).

As for DE10, reference to d* covers the case when x and y have no proper divisor.) Note that parthood excludes temporal precedence (following) whereas overlap implies temporal overlap:
TE23
TE24

P(x, y) → ¬TP(x, y)O(x, y) → TO(x, y).

Note also that these relations make it possible to introduce an entire family of
additional notions, in analogy with the basic mereological setting. For instance,
one can define relations of temporal inclusion, temporal coincidence, temporalspanning, etc. in an obvious way:
DE12
DE13
DE14

TI (x, y) =df ∀z (TO(x, z) → TO(y, z))TC(x, y) =df TI (x, y) ∧ TI (y, x)TS(x, y) =df TO(x, y) ∧ ¬TI (x, y).

These can easily be seen to behave in analogy to the corresponding mereologicalrelations of parthood, identity, crossing, etc. For instance, TI is a partial orderingand TC an equivalence relation. In fact, it is apparent that in the case of divisors,these temporal relations reduce to their basic mereological correlates:
TE25
TE26

δ(x) ∧ δ(y) →. TO(x, y) ↔ O(x, y)δ(x) ∧ δ(y) →. TC(x, y) ↔ x=y

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

TE27
TE28

109

δ(x) ∧ δ(y) →. TI (x, y) ↔ P(x, y)δ(x) ∧ δ(y) →. TS(x, y) ↔ X(x, y).

All of this sheds light on the temporal relations implicit in our oriented event
structures, where non-dividing events can be simultaneous or partially simultaneous without necessarily bearing mereological relations to one another.
At this point we need not go any further. We do not, to be sure, have a fullcharacterization yet. (This will ultimately depend on the specific event structure
under consideration.) Nevertheless the above suffices to establish a substantial
point. For we can now prove that regardless of the specific choice of E, δ, and e,
the following general conditions always hold:
TE29
TE30
TE31
TE32
TE33
TE34
TE35

TO(x, x)TO(x, y) → TO(y, x)TP(x, y) → ¬TO(x, y)TP(x, y) → ¬TP(y, x)TP(x, y) ∧ TP(y, z) → TP(x, z)
TP(x, y) ∧ TO(y, z) ∧ TP(z, t) → TP(x, t)TP(x, y) ∨ TP(y, x) ∨ TO(x, y).

These are the mereological counterparts of the seven axioms for strict linear orders employed by Kamp [1979] in his construction of time instants out of ordered events, axioms which are common to most recent theories of temporal reasoning. (Counterparts of the axioms for interval structures of van Benthem[1983], for instance, can be established by taking divisors as the counterparts ofintervals and then reasoning in terms of the relations of temporal precedence andoverlapping as introduced above.) In other words, TE29–TE35 correspond to theseven principles usually assumed to axiomatize the fundamental temporal relations. Since they can now be seen to follow from our basic axioms AE1–AE6(the proof is routine), it follows that the temporal characterization that we wereafter is actualy complete.
4. Concluding Remarks
We must stress that deriving relational properties such as those expressedby TE29–TE35 does not amount to construing time out of events. What wehave is a construction of temporal relations—strictly speaking, relations that canbe interpreted as temporal relations—out of mereotopological relations among
events. Even so, we believe the above supports the claim that the notion of anoriented event structure, albeit silent about time, permits at last a full retrieval of
the temporal dimension.
This eventually answers our initial question. At least for the purpose oftalking about what actually happens or might happen, time need not be posited

110

FABIO PIANESI AND ACHILLE C. VARZIas an independent notion—be it as a primary ontological category (intervals or
instants) or in the form of some primitive, irreducible relation of temporal precedence. Rather, time can simply be viewed as a by-product of the possibility oforienting the domain of all happenings. If one will, this can then be read as evidence in favor of a (somewhat moderate) form of relationalism. More precisely,
our construction allows one to see the dispensability of time as a function of
three main factors: (i) the assumption of events as bona fine individual entities(this is a matter of ontology), (ii) the notion of a divisor (which is intrinsicallymereotopological), and (iii) the choice of an anchor event relative to which the“arrow of time” can be oriented (and this is formally a stipulative matter). A full
assessment of these factors in relation to the absolutist/relativist dispute goesbeyond the aims of this paper. But it will be interesting to see how much of (i),(ii), and (iii) constitutes a necessary provision for any relational undertaking.
Particularly, it will be interesting to look further into the bearing of the relevant
mereological and topological underpinnings. And it will be interesting to see
whether, or to what extent, a similar machinery can be applied to other domains—for instance, to provide an analogous construction of spatial relations
out of the fundamental mereotopological properties of common-sense spatial entities such as physical objects and chunks of matter.
Fabio Pianesi
Istituto per la Ricerca Scientifica e Tecnologica
Povo (Trento), Italy
Achille C. Varzi
Columbia University

NOTES
*

This is a much revised and extended version of a short conference paper
which originally appeared as Pianesi and Varzi [1994b]. Section 1 also draws
from a conference paper which appeared as Pianesi and Varzi [1994a]. We arethankful to participants in both conferences for helpful discussion. We are alsoindebted to Nicholas Asher, Roberto Casati, Gennaro Chierchia, Carola Eschenbach, Enrico Franconi, Alessandra Giorgi, Nicola Guarino, Jean Petitot, Chris
Pinon, Graham Priest, Barry Smith, Laure Vieu, and Graham White for illuminating exchanges and comments on various drafts. (The final version is of course
the result of joint work. For institutional requirements, however, F. Pianesi
takes responsability for sections 1 and 3, and A. C. Varzi for the remainder.)1. The expression is from van Benthem [1983: 113]. Much AI work inplanning and temporal reasoning can be viewed as embodying this reductionist

EVENTS, TOPOLOGY, AND TEMPORAL RELATIONS

111

approach, particularly under the impact of Allen [1984]; see the papers collected
in Allen et al., eds. [1990] and Ford and Anger, eds. [1991] for some indicative
examples.
2. A selection of classic papers on these issues is collected in Casati and
Varzi, eds. [1996].
3. We refer here to the classic works of Sklar [1976], Newton-Smith[1980], Earman [1989].
4. This aligns us with Davidson [1967] as opposed to, e.g., Chisholm[1970].
5. Specifically with respect to events, this view goes back to Whitehead[1919] and is by now rather popular. Among other things, it played a role in thedebate on event identity (see, e.g., Davis [1970], Thalberg [1971], Thomson[1977]) and is rooted in much recent work on natural language semantics (compare Landman [1991] and Moltmann [1996], and references therein).
6. This was clearly pointed out by Cartwright [1975] and Tiles [1981] and,more recently, by Simons [1987], Bochman [1990], and Smith [1993] inter alia.
7. See Varzi [1994, 1996b] for a first assessment.
8. Applications include spatiotemporal reasoning (Randell and Cohn
[1989], Randell et al. [1992a]), naive physics (Randell et al. [1992b]), and thesemantic analysis of spatial prepositions in natural language (Aurnague and Vieu
[1993]).
9. The hole example is from Casati and Varzi [1994], ch. 7; the ball example from Davidson [1969: 232]. Both cases are admittedly controversial, andwill obviously fail if one treats holes or events as material entities (as in Lewis
and Lewis [1970] and Quine [1960], respectively). The issue is taken up in
Casati and Varzi [1995, 1996].
10. We refer here to the material reviewed in Smith [1994].
11. Against transitivity see, e.g., Cruse [1979] and Winston et al. [1987];
on extensionality see Wiggins [1979] and Simons [1987].
12. See, e.g., Chisholm [1978: 201] on Brentano’s views.
13. The view that scattered entities may not have individual sums may be
traced back to Whitehead’s restricted mereology of events [1919] and underlies
much later literature on the topic; see, e.g., Lowe’s [1953] and Goodman’s
response in [1956].
14. On non-symmetric boundaries see Chisholm [1984, 1992/93] andSmith [1996]; for discussion we refer to Varzi [1996a].
15. Compare Varzi [1993, 1996b].
16. Brentano [1976]. See Chisholm [1984], Bochman [1990], and especially
Smith [1993, 1996] for discussion and elaborations.
17. See, e.g., Allen [1984] and the other works cited in note 1.
18. For instance, the strict ordering of Kamp [1979] or Thomason [1989].

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FABIO PIANESI AND ACHILLE C. VARZI19. A noteworthy exception is Chisholm [1990], who puts forward a timefree variant of the property exemplification view.
20. Some authors deal with this distinction by treating events as endowedwith a “multidimensional” part structure, distinguishing for instance between
spatial and temporal mereological relations. See Moltmann [1991] for a proposal
in this spirit.
21. See Pianesi and Varzi [1996] for some work in this direction.
22. In this strong form, AP2 is endorsed, e.g., by Thomson [1977];
misgivings in Taylor [1985: 25] and Lombard [1986: 25]. (Lewis is not soliberal either: see his [1986].) Compare also the discussion in Bennett [1988],
ch. 10.
23. It may be objected that dismissing the possibility of temporal vacua,i.e., time intervals in which nothing happens (a possibility that some authors
have argued to be logically coherent: see Shoemaker [1969] for an influential
argument) is question begging. This is true if we assume that every event must
involve some change (as advocated, e.g., by Quinton [1979], Taylor [1985], orLombard [1986]). If we do not make such an assumption, however, changelesstime periods may in principle be viewed as involving “static” events. Alternatively, the possibility of changeless time could be accommodated byreasoning in terms of possible events, rather than actual happenings. This would
mean adding a modal dimension to the construction presented below, and would
correspond to a rather common account of the relationalist view as opposed to apure reductionist account (see, e.g., Butterfield [1984], Teller [1991], and Forbes[1993]). On the other hand, if the universe consisted of two or more separate
parts, one could apply the reasoning below to each of them, ending up with afamily of disconnected worlds each having its own temporal ordering. Here weshall not pursue this possibility for reasons of simplicity.

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