Mereology

Definition

Mereology is the formal theory of parts and wholes. It axiomatizes the “part-of” relation ($⊑$).

Classical Mereology (Leśniewski, Goodman)

Basic Axioms

• Reflexivity: $x⊑x$
• Antisymmetry: $x⊑y\wedge y⊑x\Rightarrow x=y$
• Transitivity: $x⊑y\wedge y⊑z\Rightarrow x⊑z$

Additional Principles

• Weak supplementation: Proper parts require additional parts
• Strong supplementation: Difference requires a distinguishing part
• Unrestricted fusion: Any collection has a mereological sum

In This Project

Mereological vs. Algebraic Structure

Aspect	Mereology	Algebra
Focus	Constitution	Combination
Question	”What is $S$ made of?"	"What does $S$ produce when combined?”
Operation	Part-of (${⊑}_m$)	Composition ($\circ$)
Property	Idempotent: $x\bigsqcup x=x$	Not necessarily: $S\circ S\ne S$

Key Insight

A skill can be a part of another without being a compositional factor, and vice versa:

Case	Part-of?	Factor?	Example
A	Yes	Yes	Arithmetic is part of algebra AND composes to it
B	Yes	No	Pattern recognition is part of reasoning but doesn’t compose to it
C	No	Yes	Summarization + Translation → Cross-lingual summarization
D	No	No	Cooking and differential geometry

Reconciliation Framework

A skill structure $(\mathcal{S},{⊑}_m,\circ ,1)$ has both:

• Mereological poset $(\mathcal{S},{⊑}_m)$
• Algebraic monoid $(\mathcal{S},\circ ,1)$

Connected by coherence conditions constraining their interaction.

Related Concepts

• ontology — Classification framework
• skills-algebra — Algebraic structure
• composition — The algebraic operation

References

• A mereological reconciliation