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important thing to highlight is that if (x <
R
W) then there is no y such (y <
R
x). In other
words, the closure set defined by relation R are the R-atoms of W. This is because, the
whole W unified under R is maximal under this relation (by the definition of an R-
closure system). The fact that no R-part of W can be unified under the same relation R
of course does not imply that these R-parts need to be atomic in an absolute sense. In
fact, given an element x such that (x <
R
W), x itself can be an integral wholes unified
by a different relation R'. However, it should be clear by now that the sets of R'-
atoms of x and the set of R-atoms of W (of which x is a member) are disjoint.
An example of a relation that takes place between an atom under relation R and an
integral whole unified under that relation is the
member-collective
relation (symbol-
ized as
M(part,whole)
). Following the above discussion, we have that these relations
are never transitive, i.e., they are intransitive. Thus, if M(x,W) then x is atomic for W,
and if we have M(y,x), we also have necessarily that
M(y,W). In other words, for
the case of the member-collective relation, to say that a member must be a
singular
entity coincides with this entity being an
atom
in the sense just discussed, i.e., an atom
w.r.t. to a characterizing relation unifying that specific whole.
The following example illustrates the intransitivity of the member-collection rela-
tion: “
I am member of a club C (collective) and my club is a member of an Interna-
tional Association of clubs C' (collective). However, it does not follow that I am a
member of this International Association of Clubs C' since this only has clubs as
members, not individuals”
. However, an even more general statement about the in-
transitivity of this relation can be made. Since members of a collective are considered
to be atomic w.r.t. the context in which the collective is defined, if an individual x is a
part of (member of) a collection y, then for every z which is part of (member of, func-
tional part of, sub-collection of) x, then z is not a part of (member of) y. In other
words, the member-collective relation causes the part to necessarily be seen as atomic
in the context of the whole, hence, “blocking” a possible transitive chain of part-
whole relations. Thus, for instance, although an individual John can be part of (mem-
ber of) a Club, none of John's parts (e.g., his heart) is part of that Club.
Regarding the weak supplementation axiom, some authors claim that this axiom is
too hard a constraint to be imposed to the member-collective relation [4]. From a
formal point of view, this view implies that we accept
reflexive characterizing rela-
tions
for collectives as integral wholes. Such an approach seems at first to be some-
how afforded by common sense. For instance, we can conceive a book of poems
composed of a single poem, a CD composed of a single track, a purchase order com-
posed of single order item, or a journal issue composed by a single article. Now, are
there disadvantages to such an approach? We can foresee two of them.
Firstly, abandoning weak supplementation would set this relation apart from all the
other parthood relations that we have considered, since this axiom (considered to be
constitutive of the very meaning of part) is assumed by the relations of
component-
functional complex
[16],
subquantity-quantity
[17], and
subcollective-collective
(sec-
tion 4.2). Secondly, this choice opens the possibility for the creation of collectives
with one single member. But what then would be the difference between John,
{John}, {{John}}, {…{{John}}…}, etc? If entities such as these are generally
adopted, then our system can face the objection of
ontological extravagance
, and we
should be reminded that avoiding this feature was one of the motivations of mereol-
ogy in the first place [19]. Given these two reasons, we adopt in this paper the view
¬
