Mereology

Mereology is the formal theory of the part–whole relation. It plays a role in ontology analogous to the one set theory plays in mathematics: a general framework for getting one thing from many, with a settled axiomatisation, a standard model, and a set of well-understood disputes about which axioms to keep.

Its ontological importance is that it offers an alternative to sets as a device of composition, and — if Lewis is right that it is ontologically innocent — a way of getting wholes for free. Whether that is true is one of the most consequential questions in the section.

Parthood as primitive

Write $x ⊑ y$ for " $x$ is part of $y$ ", taken as primitive and read so that everything is part of itself. Proper parthood and overlap are then defined:

$x ⊏ y =_{df} x ⊑ y \land x \neq = y$

$O x y =_{df} \exists z (z ⊑ x \land z ⊑ y)$

The core axioms come in three groups, and it is useful to keep them separate because the disputes attach to different groups.

Ground mereology — parthood is a partial order:

$x ⊑ x (x ⊑ y \land y ⊑ x) \to x = y (x ⊑ y \land y ⊑ z) \to x ⊑ z$

Supplementation — a whole is not exhausted by one proper part. The strong form: if $y$ is not part of $x$ , then $y$ has a part disjoint from $x$ .

$\neg (y ⊑ x) \to \exists z (z ⊑ y \land \neg O z x)$

Unrestricted composition — any things whatever have a fusion. For any satisfiable condition $φ$ , there is a $z$ overlapping exactly those things that overlap something satisfying $φ$ .

Together these give classical extensional mereology (CEM), whose models are complete Boolean algebras with the bottom element removed. Sum, product, and complement are all definable, so the theory is algebraically well behaved.

Extensionality and its critics

Strong supplementation yields extensionality: objects with the same proper parts are identical.

$(\exists z z ⊏ x) \land \forall z (z ⊏ x \leftrightarrow z ⊏ y) \to x = y$

This is the theory's most contested consequence, and the counterexamples are all versions of one complaint: arrangement seems to matter, and mereology cannot see it.

The statue and the clay. The statue and the lump of clay have exactly the same parts at a moment, yet appear to differ — the lump survives being squashed and the statue does not. Extensionality says they are one. This is the problem of material constitution.
Structure. A house and the disassembled pile of its bricks share their parts. So do a living cat and the same matter a moment after death. If arrangement makes a difference to identity, parthood alone does not determine identity.
Repeated parts. A word type contains the same letter twice; a methane molecule contains four hydrogen atoms. Classical mereology has no notion of containing something twice over — this is what drives Lewis's objection to structural universals.

The classical mereologist's reply is that these are not objections to the formal theory but to a combination of the theory with certain identity claims, and that the right response is to deny that the statue and the lump have the same parts at all times, or to deny that they are two. The dispute is joined on the constitution page.

Non-classical alternatives exist and are worth knowing about. Non-extensional mereologies weaken supplementation to allow distinct coincident objects. Non-transitive parthood is occasionally proposed on the ground that a handle is part of a door and the door part of a house, yet calling the handle a part of the house sounds wrong — though the usual diagnosis is that "part" here is being used in a functional sense that is not the mereological one, and that the mereological relation is genuinely transitive.

The theory's origins are worth a note: Leśniewski developed mereology in the 1910s as a nominalistically acceptable alternative to set theory, and Leonard and Goodman rediscovered it as the "calculus of individuals" in 1940 for the same motive. It was designed from the start to be the ontologically cheap way of getting wholes.

Ontological innocence

Lewis's claim in Parts of Classes is that mereology is ontologically innocent: in accepting some things you have thereby accepted their fusion, because the fusion is nothing over and above them. Describing the parts and then adding the whole is not describing two things; it is describing the same reality twice.

"Given a prior commitment to cats, say, a commitment to cat-fusions is not a further commitment. The fusion is nothing over and above the cats that compose it. It just is them. They just are it."

If this is right, unrestricted composition costs nothing and the Special Composition Question has an easy answer. If it is wrong, universalism is committed to an enormous number of additional entities.

The claim is made precise by composition as identity, which comes in strengths:

Strong CAI (Baxter): the many are the one, literally. Identity is not restricted to one-one cases. This makes innocence trivially true, at the cost of a notion of identity that violates Leibniz's Law — the cats are many and the fusion is one, so they differ in a property.
Weak CAI: composition is not identity but is analogous to it — it shares identity's key features (the whole is nothing extra, the composition relation is unique). Innocent-sounding but the analogy does the disputed work.
Moderate views: composition is a sui generis relation of ontological dependence, and the fusion is a genuine addition to being that happens to be fully grounded in its parts. Not innocent, but not costly either.

A useful test: does the fusion of the cats have any feature not fixed by the cats and their arrangement? If not, calling it an addition to being looks like double-counting. If so — and the statue/clay case suggests it might, since the statue has modal properties the clay lacks — innocence fails.

Sets and sums

The contrast with set theory is developed on the sets and pluralities page, and its moral belongs here too. Fusion is flat and idempotent: $a + a = a$ , and re-fusing gives nothing new, so there is no ascending hierarchy. Membership is neither: ${a} \neq = a$ , and ${{a}} \neq = {a}$ , which is exactly what generates the cumulative hierarchy. Parthood is transitive; membership is not.

This is why mereology can plausibly claim innocence while set theory cannot. A fusion adds no structure to its parts; a set does — which is precisely what makes sets useful and mereology cheap.

Where mereology sits

Mereology is a piece of formal apparatus rather than an ontological thesis, but it is not neutral. Adopting classical extensional mereology settles, by itself, that there are arbitrary fusions and that coincident objects are impossible — two substantive conclusions reached without further argument. That is a general hazard the criterion of commitment page warns about: contentious decisions get made in the choice of formalism and then look like results.

The next page takes up the question the axiom of unrestricted composition answers by fiat: under what conditions do some things compose a further thing at all? That is the Special Composition Question.