Mereology

Origin. Leśniewski developed mereology (1916). Formal theory of parts and wholes. Alternative to set theory for some purposes. Developed by Leonard, Goodman, Simons, and others. Foundation for ontology and spatial reasoning.

Models. Part-whole relations. Set membership: a ∈ {a,b} (a is in the set). Parthood: a ≤ a+b (a is part of the whole). No "set of all sets" problem. Extensionality: same parts → same whole. Atomistic vs. atomless mereology.

Formalism.

Basic relation: x ≤ y (x is part of y)

Core axioms:

1. Reflexivity: x ≤ x
2. Antisymmetry: x ≤ y ∧ y ≤ x → x = y
3. Transitivity: x ≤ y ∧ y ≤ z → x ≤ z

Defined notions:

• Proper part: x < y ≡ x ≤ y ∧ x ≠ y
• Overlap: x ○ y ≡ ∃z. z ≤ x ∧ z ≤ y
• Disjoint: x ⊥ y ≡ ¬(x ○ y)
• Underlap: x ⊔ y ≡ ∃z. x ≤ z ∧ y ≤ z

Classical mereology (General Extensional Mereology):

• Supplementation: x < y → ∃z. z ≤ y ∧ z ⊥ x
• Extensionality: ∀z.(z ○ x ↔ z ○ y) → x = y
• Unrestricted fusion: ∃x.φ(x) → ∃y.∀z.(z ○ y ↔ ∃x.(φ(x) ∧ z ○ x))

Atomism: Atom: x with no proper parts. Atomistic: everything is fusion of atoms. Atomless: everything has proper parts.

Symbols.

Metatheory. Classical mereology = Boolean algebra minus zero element. Decidable (Boolean algebra decidable). Model: powerset minus empty set. Weaker mereologies: drop axioms. Atomless: models exist (intervals). Extensionality controversial philosophically.

Applies to. Ontology engineering. Spatial representation. Philosophy (material objects). Knowledge representation (parthood). Medical ontologies (body parts). Engineering (component structure). Natural language (mass nouns).

Limitations. Unrestricted fusion controversial (any objects have a sum?). No structure beyond part-whole. Temporal parts debated. Vague objects problematic. Extensionality fails for some (structured wholes). Complement relative to universe only.

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