1. Introduction
1.1. General background
The part-whole relation is indispensable for ontology development. There are two major
approaches to parthood (despite many other possible lines of inquiry such as Galton’s
[
1
]). On the first approach, mereology [
2,3
] explores a single kind of parthood relation
that is standardly taken to be a partial order (reflexive, anti-symmetric, and transitive).
Such mereological parthood relation is included in virtually all upper ontologies [
4
]. In
contrast, the second approach [
5
] studies multiple kinds of part-whole relations that
appear in natural language discourse and that are sometimes utilized in domain
ontologies. For instance, the part-of relation found in the statement “Each soccer player
is part of a soccer team” is a member-of relation. Characteristically, some domain-level
relations may be transitive while others are non-transitive. As Keet [
4
] says, it is an open
problem how to articulate upper and domain ontologies with respect to parthood: for
example, which kind of parthood relation should be covered in upper ontologies.
This paper focuses on one of the most intensively studied mid- or domain-level
relations: functional parthood (aka “component-integral object relation” [
6
]). Broadly
speaking, it is such that, especially when the part and the whole are both so-called
ordinary (material) objects (e.g. molecules, tables, and planets), the part plays some sort
of “functional role” (in a non-technical sense of the term) in the whole, so that the whole
can possess “functional integrity” [
2
]. Functional parthood is typically illustrated with
biological objects and (technical) artifacts [
7
]: e.g. Vieu’s [
8
] example of this cell being
a functional part of the heart and Winston et al.’s [
6
] example of wheels being functional
parts of cars.2
Functional parthood is intimately connected with fundamental issues about parthood.
For instance, there is a debate between monism and pluralism concerning parthood [
9
].
Monism insists that there is a single privileged kind of parthood, whereas pluralism
counters that there are multiple kinds of parthood relations that are equally basic from an
ontological viewpoint (see, for details, McDaniel’s [
9
] more precise formulation). In
particular, monism agrees (but pluralism disagrees) that various domain-level relations,
including functional parthood, can be reduced to mereological parthood and other
relations. (We will illustrate later this point with a monist approach to functional
parthood.) Although monism may be a prevailing view of parthood, there is nowadays
an increasing usage of a pluralistic approach in ontology. Examples of topics investigated
along pluralistic lines include part-whole relations relating to physical objects [
10,11
],
locations [
12
], and a specific language [
13,14
].
1.2. Purpose and methodology
In this paper we will outline a “dispositional perspective” on functional parthood. This
proposal is motivated by further development of biological and biomedical ontologies.
Biological reality is so complex that we can observe many other kinds of material entities
than typically conceived ones such as cells and organs. To illustrate this point, it has been
suggested [
15,16
] that a fuller representation of the biological domain should require a
more refined classification of material entities than the three subtypes of the category of
“materiel entity” of the upper ontology Basic Formal Ontology (BFO) [
17
]: objects (e.g.
cells and organs), object aggregates (e.g. a group of bacteria in blood), and fiat object
parts (e.g. an upper torso). There is also a growing demand for a granularity-sensitive
representation of the life sciences [
18
] because different kinds of material entities are
closely related to different granular levels of reality [
19
].
The part-whole relation plays a vital role in modeling the correlation between
variegated material entities in biomedicine as well as in many other domains. Indeed,
(time-indexed) mereological parthood has been extensively utilized in biological and
biomedical ontologies [
20
] in support of data-driven life sciences [
21
]. However, more
fine-grained part-whole relations remain largely unspecified from a foundational
perspective, although e.g. Relation Ontology (RO) [
20
] includes some specific subtypes
of the mereological part-whole (or whole-part) relation such as “member of” and “has
component”. It will be thus worthwhile to consider carefully functional parthood in order
to have a more accurate representation of multifarious biological objects.
As for related work, functional parthood tends to be conceptualized and formalized
in terms of the notion of (generic) functional dependence: roughly, a functional version
of ontological dependence that is based on the primitive functioning-as predicate [
8,22
].
To borrow Vieu’s [8, p. 150] explanation: “ ‘Functioning as a X’ refers to the fact that,
at a given time, the entity is actually displaying a X function, the function generally
expected for it to be described as a X.” For instance, this heart “functions as” a heart
2 Winston et al. [
6
] originally discuss a part-whole relation between the terms “wheel” and “car”. In this
paper we treat this example ontologically as a part-whole relation between individuals, i.e. between this
particular (steering) wheel and this particular car.
when it is pumping blood through the body. However, the idea of functional dependence
may be too weak to capture fully the alleged “functional role” involved in functional
parthood.3 Therefore, it would be more desirable to investigate functional parthood using
some more specific view of functions (refer to e.g. Mizoguchi & Borgo [
23
] for a
previous example of this line of research).
In this direction, we will propose a dispositional approach to functional parthood
with a focus on its usefulness in biological and biomedical ontologies. In the remainder
of this paper, Section 2 presents a dispositional interpretation of functions [
24
] and a
theory of parthood between dispositions [
25
]. Section 3 sketches out how functional
parthood can be considered in terms of “disposition-parthood”. Section 4 concludes the
paper with a brief discussion and some remarks on future work. As for formalization (in
Courier) in first-order predicate logic, we will restrict the domain of discourse to
particular entities (e.g. this heart). We will also use unary predicates to refer to classes
(i.e. without reifying classes) and omit universal quantifiers for the sake of readability.
2. Preliminaries
2.1. A dispositional account of functions
To examine functional parthood vis-à-vis the biological domain, we will presuppose an
ontology of dispositions [
26,27
] and stipulate that functions are a subtype of dispositions,
as illustrated by Spear et al.’s [
24
] current BFO account of functions, which we will
assume and present briefly below. A disposition is a causal property that inheres in
(binary:INH) a bearer (e.g. objects) and that is linked (binary:REL) to a realization,
namely to a specific possible behavior of the bearer of the disposition. To be realized in
a process, a disposition needs to be triggered (binary:TRI) by some other process.
Classical examples include fragility (the disposition to break when pressed with a force)
and solubility (the disposition to dissolve when put in a solvent). Characteristically,
dispositions may exist even if they are not realized or even triggered: for instance, a glass
is fragile even if it never breaks or even if it never undergoes any shock.
A dispositional view of functions would distinguish sharply between a function (as
a subtype of disposition) and its functioning (as the realization of the disposition). For
instance, the biological function of this heart to pump blood is its disposition that can be
realized in a process of blood pumping (in which the heart participates) when the heart
receives blood. Such a dispositional view of functions is explicitly advocated by Spear
et al. [
24
]. They submit that functions are dispositions that their bearers come to have
either through natural selection (in the case of biological entities) or through intentional
design (in the case of artifacts).
Although it has been subject to criticism [28,29], a dispositional theory of functions
would fit well with our purpose to consider functional parthood with an emphasis on
biological objects. In the first place, many existing accounts of biological functions
would seem to “understand that functions are a kind of dispositional and causal effect”
[30, p. 236]. Moreover, a dispositional approach to functions may help to scrutinize
closely the functionality of biological objects. For instance, Vogt [
18
] develops a domain
3 Vieu [8, p. 154] states: “the notion of function used here is perhaps too general and arguably weak,
simply relying on a ‘functioning as’ primitive predicate, very lightly axiomatized. The real functional role of
the part within the whole (…) remains unanalyzed in this proposal.”
granularity framework for the life sciences and proposes an additional subtype of “causal
unity” that is used in the elucidation of the BFO category of “object”. That is: “Causal
unity via bearing a specific function unifies an entity through the function that the entity
bears, with its functional component parts bearing sub-functions” [18, p. 10]. Granted
that a disposition is a causal property, this function-related notion of causal unity may be
better clarified in terms of a dispositional perspective on functions. Finally, a
dispositional theory of functions can have the added advantage of being integrated well
into a broader dispositional framework for biomedical ontologies, as illustrated by
analysis of pathological dispositions such as diseases and risks [31].
2.2. A theory of disposition-parthood
Barton et al. [
25
] develop a theory of part-whole relations (binary:DP) between
dispositions in terms of mereological parthood (binary:P). They define a “disposition
complex” as a disposition that has some proper “disposition part” (binary:DPP) and
propose three formal constraints on disposition-parthood. First, the bearer of a
disposition part is a (proper or improper) part of the bearer of the disposition complex:
(DP-BER) [DP(d1,d2)∧ INH(d1,x)∧ INH(d2,y)]→ P(x,y)
Second, if a disposition complex is realized in a process, then at least one of its proper
disposition parts is realized in a part of this process:
(DP-REL) [∃d1 DPP(d1,d2)∧ REL(d2,x)]
→ ∃d3,y[DPP(d3,d2)∧ REL(d3,y)∧ P(y,x)]
Third and finally, if a disposition complex is triggered, then at least one of its proper
disposition parts is triggered:
(DP-TRI) [∃d1 DPP(d1,d2)∧ TRI(d2,x)]
→ ∃d3,y[DPP(d3,d2)∧ TRI(d3,y)]
3. A dispositional perspective on functional parthood
We will consider functional parthood based on a dispositional account of functions and
a theory of parthood between dispositions. Here we will adopt a monist approach to
functional parthood, partly because it would seem to be a traditional characterization of
specific part-whole relations in biomedical ontologies (e.g. RO [
20
]). In particular, we
will espouse the simple view [
3,9,32
] that functional parthood (binary:FP) between
particular objects is defined as the conjunction of mereological parthood (P) between
them and some “playing-a-functional-role-in” relation (binary:FR) between them.4 To
put it formally:
4 For another monist approach, Johansson [33,34] opines that an apparently binary functional parthood
relation implicitly assumes a third relatum (refer to Varzi [32] for criticism). In this paper we respect the
common understanding of functional parthood as a binary relation. Notably, it would have the added advantage
of fitting better with the existing theory of binary parthood relations between dispositions [
25
].
(D1) FP(x,y) =def. P(x,y)∧ FR(x,y)5
To consider functional parthood dispositionally, let us consider Vieu’s [
8
] cell/heart
example of functional parthood. Suppose that this cardiomyocyte (cardiac muscle cell)
c1 is a functional part of this heart h1. According to our monist approach, c is a
mereological part of h and c1 also plays a functional role in h1. Let Fc1 be the function of
c1 to contract. Let Fh1 be the function of h1 to pump blood through the body. Given a
dispositional account of functions, Fc1 and Fh1 are dispositions. More specifically, the
fact that Fc1 is a dispositional part of Fh1 is supported by the observation that three formal
requirements of disposition-parthood are satisfied with respect to Fc1 and Fh1:
(DP-BER) The bearer (i.e. c1) of Fc1 is a (proper) part of the bearer (i.e. h1) of
Fh1.
(DP-REL) If Fh1 is realized in a process (of h1 pumping blood through the
body), then Fc1 is realized in a part of this process (namely the contracting
process of c1). (See Section 4 for thoughts on parthood between processes.)
(DP-TRI) If Fh1 is triggered (by a process of h1 receiving blood from the lungs),
then Fc1 is triggered (by this process).
Intuitively, the claim that Fc1 is a dispositional parthood of Fh1 would be
explanatorily relevant to the “functional integrity” [
2
] or the “causal unity via bearing a
specific function” [
18
] of h1, since it is a part-whole relation between functions. c1 plays
a functional role in h1 because Fc1 is a disposition part of Fh1. This can be then
generalized into the following axiom stating that if a material entity x plays a functional
role in another material entity y, then there is a function (unary:F) of x and a function of
y such that the former is a disposition part of the latter:
(A1) FR(x,y) → ∃z,w(F(z)∧ F(w)∧ INH(z,x)∧ INH(w,y)∧ DP(z,w))
We make three remarks on A1. First, one may consider making A1 stronger by
replacing the conditional therein by the biconditional (“↔︎ ”). However, the legitimacy of
such a change would depend on the details of a dispositional account of functions, which
is beyond the scope of this paper. Such a change might also require taking classes into
consideration, as type-level functions are vital in prior works on functional parthood
[
8,22
] and the identity of particular dispositions is determined by type-level triggers and
realizations [
27
]. Second, from the right hand side of A1 and DP-BER, we can deduce
that P(x,y). That is, “playing-a-functional-role-in” (FR) implies mereological
parthood (P). In other words, from D1 and A1, we simply get:
FP(x,y)↔︎ FR(x,y).
Said differently, as soon as we adopt A1, there is no difference between
“being-afunctional-part” and “playing-a-functional-role-in”. Third and finally, as the existential
quantifier in A1 shows, our approach is compatible with the claim that a material entity
might have several functions: for example, a lamp has both a lighting function and a
decorative function. In other words, not all functions of a material object would be
relevant for functional parthood presently concerned that is involved in this object.
5 Vieu [
8
] rightly maintains that a fuller analysis of functional parthood requires adding time as a third
argument to a binary functional parthood relation. We will use a binary FP relation for simplicity and leave its
temporalization for future work.
4. Discussion and concluding remarks
We have proposed a dispositional perspective on functional parthood while being
motivated by its utilization in the biomedical domain. We will now provide a few
important pointers to its future development (besides the temporalization of the binary
FP relation mentioned in Section 3). First and foremost, further investigation is required
into the theory of disposition-parthood that underlies our approach. Barton et al. [
25
]
identify several subtypes of disposition-parthood, but their classification is not meant to
be exhaustive. A more comprehensive account of disposition-parthood will enable us to
give a more meticulous analysis of the “functional role” involved in functional parthood,
such as a classification of FR relations based on subtypes of the DP relation.6
In addition, we will need to address a long-standing problem of the transitivity of
functional parthood [
8,22,32-34,36
] (as well as other candidate mereological principles
such as “supplementation axioms” [37]). Functional part is transitive in many cases, such
as the cuff/sleeve/jacket example [38] and the carburetor/engine/car example [
8
].
Functional parthood may not be transitive, however.7 There are at least two directions of
research in which we will be able to proceed so as to tackle such a fundamental issue
regarding functional parthood. The first line of inquiry is about DP-REL in the theory of
disposition-parthood (see Section 2.2). This axiom is deeply rooted in a part-whole
relation between processes (recall “P(y,x)” therein). This means that close scrutiny of
disposition-parthood will necessitate a well-developed account of parthood between
processes, which will in turn necessitate a well-established ontology of processes. This
task will comprise, for example, dealing with a vexed granularity problem of how to
individuate processes, which has been extensively explored in philosophy [39,40] and in
formal ontology [41,42].
The second line of inquiry concerns non-functional causal unity of material entities.
As we alluded to in Section 2.1, Vogt [
18
] argues for causal unity via bearing a specific
function. This is nonetheless proposed as an addition to the three kinds of causal unity
specified in BFO: causal unity via internal physical forces, causal unity via physical
covering, and causal unity via engineered assembly of components. It would be plausible
to think that these non-functional types of causal unity are closely entwined with
nonfunctional part-whole relations embedded in material entities (see, for details, Masolo et
al.’s [43] exploration using mereological parthood). Therefore, a dispositional approach
to functional parthood may require closer examination of such non-functional and more
fundamental part-whole relations by means of disposition-parthood.
6 A positive by-product of a more advanced theory of disposition-parthood would be to solidify the
dispositional account of functions (such as Spear et al.’s [
24
]), given Burek’s [35] desideratum that an ontology
of functions should support function decomposition by analysis of sub-functions. That is to say, such
subfunctions would be some of the disposition parts of the overall function.
7 For instance, Lyons’s [38] handle/door/house example is frequently taken to be a case of the
nontransitivity of functional parthood in the relevant literature, but it is nonetheless highly controversial. To take
one example, Vieu [
8
] contends that it would become a case of the transitivity of functional parthood by
replacing the word “door” by another term “door handle”: “changing ‘handle’ in ‘door handle’ contributes to
specifying what is the role of the handle with respect to the house, i.e., a component of a door, which is
(assumed to be) a component of the house” (p. 149). This observation motivates her to adopt a lexical approach
to functional parthood. From an ontological (monist, in particular) perspective, by contrast, it might possibly
imply that functional parthood involved in the handle/door/house example may not preserve transitivity given
one construal of a functional role, but it may preserve transitivity given another construal. We will leave closer
examination of this example for the future.
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