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In-Depth Information
In the previous sections, we have discussed that collectives are not necessarily ex-
tensional and that, if a member of a collective is essential to the collective, then the
collective is an extensional entity, i.e., all its members are essential. How do subcol-
lectives stand w.r.t. to this secondary property? Suppose that we have a collective W
composed of the subcollective W' and W'' such that the former is an essential part of
W but not the latter. Now, as we have discussed, the structure of collectives is defined
via specialization of the collective's unifying relation, i.e., the members of W' and
W'' are also members of W. This implies that all subcollectives of W are
inseparable
parts
of it, i.e., W' and W'' come to existence by refining the structure of W and by
grouping the specific members of W. As a consequence, they cannot exist without
that whole. A second observation we can make is that if there is an x which is a mem-
ber of W', and x is an essential member of W then x must also be an essential member
of W'. The argument can be made as follows. If x is an essential part of W then W
cannot exist without x; If W' is an inseparable part of W then W' cannot exist without
W; due to the transitivity of existential dependence [14], we have that W' cannot exist
without x, ergo, x is an essential part of W'. Finally, since we are admitting that col-
lectives are not necessarily extensional entities, it is conceivable that a whole W has
an essential part W' composed of members which are not essential for either W or
W'. For instance, suppose that by law, all juries must have at least two members
which are older than sixty years old. Although this subcollective would be essential to
the whole, it is conceivable that its individual members are exchangeable. By the
same reasoning, one could admit a particular subcollective to be essential to a whole,
without requiring the other collectives of that whole to be likewise essential.
4.3 Towards a UML Profile for Modeling Collectives and Their Parts
We summarize the results of these sections in a proposal that has been incorporated in
a UML profile for representing the member-collective and the subcollective-collective
relations (table 1). Since a profile is constituted by syntactical constraints and, since
UML conceptual models are always defined at the type level, the meta-properties of
irreflexivity, anti-symmetry and transitivity (at instance level) cannot be captured by
profile constraints. We have included a constraint to guarantee weak supplementation
for these relations taking into consideration the type-level nature of a UML class
diagram, i.e., taking into consideration the minimum cardinality constraints of all
parthood relations connected to the same type representing a whole.
Metaclass
Description
A «collective» represents a type whose instances are collectives, i.e., they are
collections of entities that have a uniform structure. Examples include a deck
of cards, a forest, a group of visitors, a pile of bricks.
«collective»
A
subcollective-
collective
This parthood relation holds between two collectives. Examples include: (a)
the north part of the Black Forest is part of the Black Forest; (b) The collec-
tion of Jokers in a deck of cards is part of that deck; (c) the collection of
for
ks in a cu
tlery set i
s part of
that cutlery set. We use the symbols
C
to represent the shareable and non-
shareable (see http://www.uml.org/) versions of this relation, respectively.
and
C
