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stereotypical example of an essential part of a car is its chassis, since that specific car
cannot exist without that specific chassis (changing the chassis legally changes the
identity of the car); A stereotypical example of an inseparable part of a living cell is
its membrane, since the membrane cannot exist without being part of that particular
cell. As discussed in depth in [2], essential and inseparable parthood play a fundamen-
tal role in conceptual modeling. However, it is not the case for all types of entities that
all their parts are essential. In other words, although EM describes the basic meaning
of parthood for some types of entities (e.g., quantities [17] and events [14]), this is not
the case for entities of all ontological categories. In particular, as we have shown in
[16], for functional complexes while some of their parts are essential (inseparable),
not all of them are essential (inseparable). As discussed in section 4, EM is too strong
a theory in this sense also for the case of the member-collective and the subcollective-
collective relations.
Classical mereological theories focus solely on the relation from the parts to the
wholes. Thus, just like in set theories we can create sets by enumerating any number
of arbitrary entities, in classical mereologies one can create a new object by summing
up individuals that can even belong to different ontological categories. For example,
in these systems, the individual created by the aggregation (termed
mereological sum
)
of Noam Chomsky's left foot, the first act of Puccini's Turandot and the number 3, is
an entity considered as legitimate as any other. However, as argued by [10], humans
only accept the aggregation of entities if the resulting mereological sum plays some
role in their conceptual schemes. To use an example: the sum of a frame, a piece of
electrical equipment and a bulb constitutes a whole that is considered meaningful to
our conceptual classification system. For this reason, this sum deserves a specific
concept in cognition and a name in human language. The same does not hold for the
sum of bulb and the lamp's base.
According to Simons [14], the difference between purely formal mereological
sums and, what he terms,
integral wholes
is an ontological one, which can be under-
stood by comparing their existence conditions. For sums, these conditions are mini-
mal: the sum exists just when the constituent parts exist. By contrast, for an integral
whole (composed of the same parts of the corresponding sum) to exist, a further
unify-
ing condition
among the constituent parts must be fulfilled. A unifying condition or
relation can be used to define a closure system in the following manner. A set B is a
closure system under the relation R, or simply, R-closure system iff
cs
〈
R
〉
B =
def
(cl
〈
R
〉
B)
∧
(con
〈
R
〉
B)
(5)
where
(cl
〈
R
〉
B)
means that the set B is closed under R (R-Closed) and
(con
〈
R
〉
B)
means that the set B is connected under R (R-Connected). R-Closed and R-Connected
are then defined as:
cl
〈
R
〉
B =
def
∀
x ((x
∈
B)
→
((
∀
y R(x,y)
∨
R(y,x)
→
(y
∈
B)))
(6)
(7)
con
〈
R
〉
B =
def
∀
x ((x
∈
B)
→
(
∀
y (y
∈
B)
→
(R(x,y)
∨
R(y,x)))
An integral whole is then defined as an object whose parts form a closure system
induced by what Simons terms a
unifying
(or
characterizing
)
relation
R.
