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. Tarski (1935) proved the structural fact that fixes the subject: read with fusion as a genuinely unrestricted principle, the models of classical extensional mereology are exactly the complete Boolean algebras with the zero element deleted, so parthood is inclusion, fusion is join, and the missing zero is the only difference. Taking fusion instead as a first-order schema gives the Boolean algebras minus zero, and everything then follows from Boolean algebra. That theory is decidable, since Tarski (1949) showed the elementary theory of Boolean algebras is decidable and Ershov's invariants give a quantifier-elimination proof; so mereology, unlike set theory or arithmetic, has no incompleteness phenomenon. The atomless case is sharper still: the theory of atomless Boolean algebras is complete and ℵ₀-categorical, its unique countable model being the free Boolean algebra on countably many generators, so atomless mereology settles every sentence in its language. Extensionality is not a separate assumption but a consequence: strong supplementation together with transitivity yields that individuals with the same proper parts are identical. The axioms do not decide atomicity — atomistic and atomless models both exist — and Lewis (1991) showed mereology reaches set theory only when supplemented with plural quantification and a primitive singleton function.

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

Limitations. Fusion is idempotent, commutative and associative, so a whole carries no record of how it was assembled: the fusion of the bricks and the fusion of the wall are the same individual, and mereology cannot distinguish a structure from its material or represent order, multiplicity or repetition. There is no ordered pair and no iteration, so no hierarchy is generated — forming fusions never leaves the Boolean algebra one started in, and Cantor's theorem has no mereological analogue, which is why mereology cannot serve as a foundation without a set-theoretic or plural supplement. Decidability comes from this poverty. The operations are partial for want of a null individual: x × y is undefined for disjoint x and y, and x − y is undefined when x is part of y, so complement exists only relative to the universe. Extensionality forbids distinct coincident objects, so statue and clay cannot be told apart. Parthood is atemporal and bivalent in the classical axioms, so change of parts and indeterminate parthood are both inexpressible without adding indices or a non-classical logic. And the axioms decide neither atomism nor atomlessness.

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