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for instance, John as a guide, is responsible for the group formed by {Paul, Marc,
Lisa} and for the other group formed by {Richard, Tom}. The intention is not to ex-
press that John is responsible for the groups {Paul, Marc, Lisa}, {Paul, Marc}, {Marc,
Lisa}, {Paul, Lisa}, and {Richard, Tom}, i.e., that being responsible for the group
{Paul, Marc, Lisa}, John should be responsible for all its subgroups. A simple solu-
tion to this problem is to consider groups of visitors as maximal sums, i.e., groups that
are not parts of any other groups. In this case, depicted in figure 3, the cardinality
constraints acquire a different meaning and it is no longer possible to say that a
group
of visitors
is composed of other
groups of visitors
in this technical sense.
responsible for
Group of Visitors
Guide
1..*
1
Fig. 3.
Representation of Collections as Maximal Sums
The solution above is similar to taking the meaning of a quantity K to be that of a
maximally-self-connected-portion of K [17]. However, in the case of collections,
topological connection cannot be used as a
unifying or characterizing
relation
to form
an integral whole, since collections can easily be spatially scattered. Nonetheless,
another type of connection (e.g., social) should always be found. A question begging
issue at this point is: why does it seem to be conceptually relevant to find
unifying
relations
leading to (maximal) collections? As discussed in the previous section, col-
lections taken as arbitrary sums of entities make little cognitive sense: we are not
interested in the sum of a light bulb, the North Sea, the number 3 and Aida's second
act. Instead, we are interested in aggregations of individuals that have a purpose for
some cognitive task. So, we require all collectives in our system to form closure sys-
tems unified under a proper characterizing relation. For example, a group of visitors
of interest can be composed by all those people that are attending a certain museum
exhibition at a certain time. Now, by definition, a closure system is maximal (see
formula (5)), thus, there can be no group of visitors in this same sense that is part of
another group of visitors (i.e., another integral whole unified by the same relation).
Nonetheless, it can be the case that, among the parts of a group of visitors, further
structure is obtained by the presence of other collections unified by different relations.
For example, it can be the case that among the parts of a group of visitors A, there are
collections B and C composed of the English and Dutch speaking people in that
group, respectively. Now, neither the English speaking segment nor the Dutch speak-
ing segment are
groups of visitors
in the technical sense just defined, since the latter
has properties lacking in both of them (e.g., the property of having both English and
Dutch segments). Moreover, the unifying relations of B and C are both specializations
of A's unifying relation. For example, A is the collection of all parties attending an
exhibition and the B is the collection of all English speakers among the parties attend-
ing that same exhibition. We return to this point in section 4.2.
By not being homeomeros and infinitely divisible, collectives actually bear a
stronger similarity to functional complexes than to quantities in the classifications of
[10,11]. In [11], for instance, the authors propose that the difference between a collec-
tive and a functional complex is that whilst the former has a
uniform
structure, the
