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latter has a
heterogeneous
and
complex
one. We propose to rephrase this statement in
other terms. In a collective,
all member parts play the same role type w.r.t. the whole
.
For example, all trees in a forest can be said to play the role of a forest member. In
complexes, conversely, a variety of roles can be played by different components. For
example, if all ships of a fleet are conceptualized as playing solely the role of “mem-
ber of a fleet” then it can be said to be a collection of ships. Contrariwise, if this role
is further specialized in “leading ship”, “defense ship”, “storage ship” and so forth,
the fleet must be conceived as a functional complex.
Finally, we would like to call attention to the fact that collectives are not sets and,
thus, the
member-collective
and the
subcollective-collective
relations are not the same
as the
membership
(
) relations, respectively. Firstly, collectives and
sets belong to different ontological categories: the former are concrete entities that
have spatiotemporal features; the latter, in contrast, are abstract entities that are out-
side space and time and that bear no causal relation to concrete entities [1]. Secondly,
unlike sets, collectives do not necessarily obey an extensional principle of identity,
i.e., it is not the case that a collective is always completely defined by the sum of its
members. We take that some collectives can be considered extensional by certain
conceptualizations; however, we also acknowledge the existence of
intentional collec-
tives
obeying non-extensional principles of identity [21]. Thirdly, collectives are inte-
gral wholes unified by proper characterizing relations; sets can be simply postulated
by enumerating their members. This feature of the latter is named
ontological ex-
travagance
and it is a feature to be ruled out from any ontological system [19]. Fi-
nally, contrary to sets, we do not admit the existence of empty or unitary collectives.
As a consequence, we eliminate a feature of set theory named
ontological exuberance
[19]. Ontological exuberance refers to the feature of some formal systems that allows
for the creation of a multitude of entities without differentiation in content. For in-
stance, in set theory, the elements a, {a}, {{a}}, {{{a}}}, {…{{{a}}}…} are all
considered to be distinct entities. We shall return to some of these points in the next
section.
∈
) and
subset
(
⊂
4 Parthood Relations Involving Collectives
4.1 The Member-Collection Relation
According to [22], classical semantic analysis of plurals and groups distinguish be-
tween
atomic entities
, which can be
singular
or
collectives
, and
plural entities
. From a
linguistic point of view, the
member-collection
relation is considered to be one that
holds between an
atomic
entity
(e.g., John, the deck of cards) and either a
plural
(e.g.,
{John, Marcus})
or a collective term
(e.g., the children of Joseph, the collection of
ancient decks).
Before we can continue, a formal qualification of this notion of atomicity is
needed. Suppose an integral whole W unified under a relation R. By using this unify-
ing (or characterizing) relation R, we can then define a composition relation <
R
such
that (x <
R
W) iff: (i) there is a set B such that
cs
〈
R
〉
B
; (iii) (x < W) and (x
B).
Intuitively, this relation captures the idea that there is indeed a genuine connection
between a part x and the whole W (as opposed to a merely formal one). Now, one
∈
