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the parts of a whole together are also considered. To put it simply, the composite
objects in which we are interested in conceptual modeling are not mere aggregations
of arbitrary entities but complex entities suitably unified by proper binding relations.
This paper should then be seen as a companion to the publications in [2,16] and
[17]. In this research program, we have managed to show that the three classification
schemes aforementioned, namely, the linguistic-cognitive meronymic distinctions, the
mereological theories of parthood, and the so-called secondary properties are not
orthogonal. In fact, each particular meronymic distinction in the first scheme commits
to basic mereological properties, secondary properties, and even requires binding
relations of specific kinds to take place between their parts. In [17], we have managed
to show the interconnection between these classification schemes for the case of the
subquantity-quantity relation. In a complementary form, we did the same in [2,16] for
the case of the component - functional complex relation. The objective of this paper is
to follow the same program for the case of part-whole relations involving collectives,
namely, the member-collective and the subcollective-collective relations. This paper
is, thus, a substantial extension to the preliminary work reported in [18] in which only
the member-collective relation is analyzed and in some of its aspects.
The remainder of this article is organized as follows. Section 2 reviews the theories
put forth by classical mereology as well as its connection with modal secondary prop-
erties of parts and wholes. The section also discusses how these mereological theories
can be supplemented by a theory of (integral) wholes . In section 3, we discuss collec-
tives as integral wholes and present some modeling consequences of the view de-
fended there. Moreover, we elaborate on some ontological properties of collectives
that differentiate them not only from their sibling categories (quantities and functional
complexes), but also from sets (in a set-theoretical sense). The latter aspect is of rele-
vance since collectives as well as the member-collective and subcollective-collective
relations are frequently taken to be identical to sets, set membership and the subset
relation, respectively. In section 4, we promote an ontological analysis of two part-
whole relations involving collectives, clarifying on how these relations stand w.r.t. to
basic mereological properties (e.g., transitivity, weak supplementation, extensionality)
as well as regarding the modal secondary properties of essential and inseparable
parthood. As an additional result connected to this analysis, we outline a number of
metamodeling constraints that have been used to implement a UML modeling profile
for representing collectives and their subparts in conceptual modeling. Section 5 pre-
sents final considerations of this paper.
2 A Review of Formal Part-Whole Theories
In practically all philosophical theories of parts, the relation of (proper) parthood
(symbolized as <) stands for a strict partial ordering, i.e., an asymmetric (2) and tran-
sitive relation (3), from which irreflexivity follows (1):
(1)
x ¬(x < x)
(2)
x,y ((x < y)
¬(y < x))
(3)
x,y,z ((x < y)
(y < z)
(x < z))
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