Parthood (Enclose Relation)

Definition

Let $\mathcal{S},{\mathcal{S}}^{\prime }$ be systems such that ${\mathcal{S}}^{\prime }⊏\mathcal{S}$ (i.e., ${\mathcal{S}}^{\prime }$ is a subsystem of $\mathcal{S}$).

We say that $\mathcal{S}$ encloses ${\mathcal{S}}^{\prime }$, denoting by ${\mathcal{S}}^{\prime }\text{encl}\mathcal{S}$, if additionally:

$\mathfrak{u}({\mathcal{S}}^{\prime })\text{sqin}\mathfrak{e}(\mathcal{S})$

If $\mathcal{S}$ encloses ${\mathcal{S}}^{\prime }$ we say that the latter is a part of the former.

Key Characteristics

• Three-fold relation: Combines mereology (part-whole), functional dependence, and set membership
• Functional component: The enclosed system participates directly in the enclosing system’s dynamics
• Level transition: The enclosed subsystem becomes an element at the next level up
• Mesodynamic role: The part has a role in the mesodynamics of the whole

The Three Aspects of Enclosure

The enclose relation $\text{encl}$ synthesizes three distinct aspects:

1. Mereological relation (part-to-whole): ${\mathcal{S}}^{\prime }$ is spatially or structurally contained in $\mathcal{S}$

2. Functional dependencies: Captured by the subsystem relation $⊏$, where properties of $\mathcal{S}$ depend on properties of ${\mathcal{S}}^{\prime }$

3. Set-theoretic membership: Captured by $\text{sqin}$, where the underlying set of ${\mathcal{S}}^{\prime }$ is an element of $\mathcal{S}$

Distinction from Mere Subsystem Relation

A system ${\mathcal{S}}^{\prime }$ that is merely a subsystem of $\mathcal{S}$ (i.e., ${\mathcal{S}}^{\prime }⊏\mathcal{S}$ but not ${\mathcal{S}}^{\prime }\text{encl}\mathcal{S}$) may have no direct role in the mesodynamics of $\mathcal{S}$:

• It is a part of $\mathcal{S}$ in the sense of occupying a spatial subregion
• But it is not necessarily a functional component

To be enclosed, ${\mathcal{S}}^{\prime }$ must additionally be an element of $\mathcal{S}$, which means that ${\mathcal{S}}^{\prime }$ itself (not merely its elements) participates in the relations that constitute $\mathcal{S}$.

This captures the intuition that in a genuinely nested system, the subsystems at one level become the objects at the next level up.

Role in Nested Systems

The enclose relation is essential for defining nested systems. A nested system $\mathcal{S}$ must satisfy:

$\mathcal{R}\text{encl}\mathcal{S}\text{encl}\mathcal{T}$

for some systems $\mathcal{R}$ and $\mathcal{T}$. This means $\mathcal{S}$ is:

• Simultaneously a whole: It encloses $\mathcal{R}$
• Simultaneously a part: It is enclosed by $\mathcal{T}$

Examples

Cellular Biology

Consider a cell within a tissue:

• Subsystem without enclosure: A spatial region within the cell
  • Occupies space but may not participate in cellular dynamics
• Subsystem with enclosure (parthood): An organelle (e.g., mitochondrion)
  • The organelle $\mathcal{M}$ is a subsystem: $\mathcal{M}⊏\mathcal{C}$
  • The organelle is also an element: $\mathfrak{u}(\mathcal{M})\text{sqin}\mathfrak{e}(\mathcal{C})$
  • Therefore: $\mathcal{M}\text{encl}\mathcal{C}$
  • The mitochondrion actively participates in cellular metabolism

Social Organization

For a company:

• Subsystem without enclosure: An informal working group
  • Exists within the company but has no formal organizational role
• Subsystem with enclosure (parthood): A department
  • The department is functionally dependent on company structure
  • The department is recognized as an organizational unit (element)
  • The department participates in company-level decision-making and workflows

Physical Systems

For a gas in a container:

• Subsystem without enclosure: An arbitrary spatial region
  • We can define it, but it plays no special role in gas dynamics
• Subsystem with enclosure (parthood): A convection cell
  • The convection cell has its own internal circulation
  • The convection cell is treated as a coherent unit in larger-scale flow patterns
  • The convection cell participates as an element in the overall convective dynamics

Relationship to Holonic Systems

The parthood concept bears significant affinity to the notion of holonic systems. A holon, in the sense of Koestler, is simultaneously a whole and a part:

• Exhibits autonomy and agency at its own level of organization
• Serves as a component of larger organizational structures

The enclose relation formalizes this dual nature: a system that is enclosed is simultaneously:

• Composed of parts (has subsystems that it encloses)
• A part of larger wholes (is enclosed by supersystems)

Philosophical Significance

The enclose relation captures a crucial insight about nested hierarchies: level transitions involve both functional dependence and ontological reclassification.

When we move from one level to another:

1. Functional aspect: Higher-level properties depend on lower-level dynamics
2. Ontological aspect: Lower-level systems become higher-level objects

This dual character distinguishes genuine nested systems from mere compositional hierarchies.

Key References

Emergent Nested Systems

C. Walloth (2016) View in Semantic Scholar DOI: 10.1007/978-3-319-27550-5

Discusses the enclose relation in the context of emergent phenomena in nested system hierarchies, emphasizing how “enclosed by, and enclosing” relationships structure complex systems.

Related Concepts

• subsystem - The functional dependence aspect of parthood
• supersystem - What encloses a part
• nested-system - Systems defined by the enclose relation
• hierarchy - Multi-level structures built from enclose relations
• superlevel-system - Level transitions in hierarchical systems

Bibliography Keys

• Walloth2016
• Mesarovic1970
• Mesarovic1976