Ontological Foundations for Conceptual Part-Whole Relations: The Case of Collectives and Their Parts - Advanced Information Systems Engineering - page 139

These axioms amount to what is referred in the literature by the name of Ground
Mereology (M) , which is the core of any theory of parts, i.e., the axioms (1-3) define
the minimal (partial ordering) constraints that every relation must fulfill to be consid-
ered a parthood relation. As previously mentioned, Mario Bunge's mereological the-
ory ( Assembly Theory ) corresponds to the axiomatization of Ground Mereology (with
the only difference of assuming the existence of a null individual which is supposed
to be part of everything else) [5]. Although necessary, these constraints are not suffi-
cient, i.e., it is not the case that any partial ordering relation qualifies as a parthood
relation. Most authors require an extra axiom termed the weak supplementation prin-
ciple (WSP) (4) as constitutive of the meaning of part and, hence, consider (1-3) plus
(4) (the so-called Minimal Mereology (MM)) as the minimal constraints that a
mereological theory should incorporate [14,19]:

(4)

∀ x,y ((y < x) → ∃ z (z < x) ∧ ¬ overlap(z,y))

Figure 1.a below illustrates this notion of weak supplementation. It shows that if y is a
part of x then there must exist another part of x which is disjoint from y (the “miss-
ing” part of x). Without this “missing” part, what differentiates y and x? Notice that x
and y are supposed to be different given that parthood is irreflexive (1). From a prac-
tical point of view, without WSP, models such as the one in figure 1.b cannot be
deemed incorrect. Now, following that model, suppose an event E which is composed
of one single subevent. Isn't this alleged part identical to the event E itself? In a sound
model, events are either atomic or are composed of at least two disjoint subevents.

Fig. 1. Invalid situation (a) and invalid conceptual model (b) according to Minimum
Mereology

There is an extension to MM that has then been created by strengthening the sup-
plementation principle represented by (4). In this system, (4) is thus replaced by
something termed the stronger supplementation principle (SSP) . The resulting theory
is named Extensional Mereology (EM) . A known consequence of the introduction of
SSP is that in EM we have that two objects are identical iff they have the same parts,
i.e., SSP entails a mereological counterpart of the extensionality principle (of identity)
in set theory. As a consequence, if an entity is identical to the (mereological) sum of
its parts, thus, changing any of its parts changes the identity of that entity. Ergo, an
entity cannot exist without each of its parts, which is the same as saying that all its
parts are essential parts .
Essential parthood can be defined as a case of existential dependence between in-
dividuals, i.e., x is an essential part of y iff y cannot possibly exist without having
that specific individual x as part [2]. This specific mode of existential dependence can
also be defined from the part x to the whole y. We say that x is an inseparable part of
y iff x cannot possibly exist without being a part of that specific individual y [2]. A