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Distributed Systems Research Group
R
R
John
Mary
Peter
Mark
R
R
R
R
R
Fig. 4.
Examples of an integral whole (collective) and its members
Distributed Systems Research Group
Modeling of
Distributed Systems
Mary
Performance Analysis of
Distributed Systems
Mark
John
Peter
R'
R''
Fig. 5.
Examples of a collective and its subcollectives
have that R''
R', and that all parts of W'' are also parts of W'. Due to the transitiv-
ity of both formal parthood (<) and the subset relation (
⊆
R, and
also that all formal parts of W'' are also formal parts of W. Again, by definition, we
conclude that C(W'',W) holds. In other words, the
subcollective-collective
relation is
always transitive.
A second property we would like to demonstrate is the following. Suppose we have
that M(y,x) and C(x,W), and that R' and R are the characterizing relations of x and
W, respectively. Since M(y,x), we have both that (y < x) and that y is an R'-atom of x.
From C(x,W), we have that all formal parts of x are formal parts of W, but also that
R'
⊆
), we have that R''
⊆
, we have both that (y < W) and
that y is a R-atom of W. From this, we conclude that M(y,W). In other words, transi-
tivity always holds across a member-collective relation combined with a subcollec-
tive-collective relation.
Now, how does the subcollective-collective relation stand w.r.t. weak supplemen-
tation? Suppose that we have two collectives x and y such that C(y,x). As we have
previously discussed, x is closure system unified by relation R, and y must be a clo-
sure system unified by a specialization of this condition R'. Since the subcollective-
collective relation is irreflexive, we have that R' is necessarily a proper subset of R,
i.e., there are R-atoms of x which are not R'-atoms of y. This, at first, seems to imply
that we can always have an integral whole z which is unified by another specialization
R'' of R (the complement of R w.r.t. R'). However, the fact that there are members of
x which are not part of y does not mean that these members can define a genuine
integral whole. In other words, it can be the case that the only relation in common
between these entities is that they obey the condition implied by R and the negation
the condition implied by R'. As discussed in [1,5], characterizing entities based on
negative properties is a poor ontological choice. For this reason, if we have that
C(y,x) we do not require that there is a z different from y such that C(z,x), but we do
require that there is a z such that
⊆
R. Again, due to the transitivity of both < and
⊆
¬
overlap(z,y) and M(z,x).
