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that weak supplementation should be part of the axiomatization of the member-
collective relation. This obviously does not imply that we cannot have single-track
CD's or single-article journal issues. Following [1], in these cases, we consider the
relation between, for instance, the tracks and the CDs to be a relation of
constitution
as opposed to one of parthood. Relations of constitution abound in ontology. An ex-
ample is the relation between a marble statue and the (single portion of marble) that
constitutes it [1]. In fact, a more detailed analysis of WCH initial proposal showed
that some of their original meronymic relations are in fact cases of constitution [1,11].
Finally, let us take the case of the secondary (modal) property of essentiality. As
we have previously discussed, unlike sets and mereological sums, collectives do not
necessarily have an extensional criterion of identity. That is, whereas for some collec-
tives the addition or subtraction of a member renders a different individual, it is not
the case that this holds for all of them. However, when this is the case, all member-
collective relations that the extensional collective participates as a whole are relations
of essential parthood. This is because, since a collective (by definition) has a uniform
structure, then all members of a collective must be indistinguishable w.r.t. the whole.
As a consequence, it cannot be the case that some members of a collection are essen-
tial while others are not. In summary, member-collective relations are only relations
of essential parthood if the collective in the association end connected to the whole is
an extensional individual. In the converse reading, if a collective is extensional then
all its parts (members) are essential.
4.2 The Subcollective-Collective Relation
In contrast with the member-collective relation, from a linguistic point of view, the
subcollective-collective
is a relation that holds between two plural entities, or collec-
tives constituted by such plural entities, such that all atoms of the first are also atoms
of the second [22].
Let us start with an example. Figure 4 depicts an integral whole termed the DSRG
(Distributed Systems Research Group) unified by the relation of
carrying out re-
search in the area of distributed systems at University X (UtX)
. The R-atoms of
DSRG are them {John, Mary, Peter, Mark}. The fact that no R-part of W can be uni-
fied under the same relation R also does not imply that these R-parts cannot be further
structured to form new wholes. In other words, for example, we can take two different
relations R'and R'' which are specializations of R, such that they can be used to form
new closure systems among the R-parts of DSRG. Let R' be the relation of
carrying
out research in the same sub-area of modeling of distributed systems at UtX,
and R'
be
the relation of
carrying out research in the same sub-area of performance of dis-
tributed systems at UtX
. This situation is depicted in figure 5.
Indeed, the fact that R' is a specialization of the condition R implies that the possi-
ble relata of R' are the R-atoms of R, i.e., R'
R. When this is the case, we name the
integral whole W' unified under R' a
subcollective
of the whole W unified under R.
Let us name the relation between W' and W the relation of
subcollective-collective
,
symbolized as C(W',W). We then have that C(W',W) iff: (i) all formal parts of W'
are formal parts of W; (ii) the characterizing relation R' of W' is a specialization of
the characterizing relation R of W. Now, suppose an integral whole W'' unified by
relation R'' and C(W'',W'). By the above definition of the C-parthood relation, we
⊆
