Source
Achille Varzi, “Mereology”, Stanford Encyclopedia of Philosophy, LIT-124, first published 2003, substantive revision of 27 June 2026, read in full in NOTE-163 (all sections and all 48 notes; the figures through their captions).
What was actually shown
The ladder. Core Mereology M is (P.1) Reflexivity, (P.2) Transitivity and (P.3) Antisymmetry over a primitive P. MM adds (P.4) Supplementation, PPxy → ∃z(Pzy ∧ ¬Ozx). EM adds (P.5) Strong Supplementation, ¬Pyx → ∃z(Pzy ∧ ¬Ozx). GEM is EM with every instance of Unrestricted Sum3, the systems of Leśniewski and Leonard–Goodman, “also known as Classical Mereology”. The same theory is MM with Unrestricted Sum2, since (P.5) is then derivable. EM or MM with Sum1 is strictly weaker (Fig. 8).
Three sums (§4.2). S1 is a minimal upper bound. S2, after Leśniewski, Tarski and Lewis, requires every part of z to overlap x or y. S3, after Leonard–Goodman, Simons and Casati–Varzi, requires z to overlap exactly what overlaps x or y. Given Transitivity, S2 implies S3; S3 implies S2 only with (P.5).
Three routes and one failure (§4.4). From M the only adequate pairings are Supplementation with Sum2, Strong Supplementation with Sum3, and Complementation with Sum1 (Hovda 2009; Cotnoir and Varzi 2019). The Fig. 4 model satisfies Supplementation and Unrestricted Sum3 and is not a model of GEM (Pietruszczak 2000a; Pontow 2004). That contradicts Simons (1987: 37), Casati and Varzi (1999) and the entry’s 2003 edition, corrected in Spring 2009 (note 29).
Related results the entry reports.
- Given (P.1) and (P.2), Supplementation entails Antisymmetry. MM with products for overlappers already contains EM (Simons 1987: 30f).
- GEM is finitely axiomatizable in first-order logic (Tsai 2018) and decidable (Tsai 2013a), whereas M, MM and EM are undecidable.
- GEM’s parthood is “virtually isomorphic” to inclusion on the non-empty subsets of a set. Precisely, every model is isomorphic to a Boolean subalgebra, not necessarily complete, of a complete Boolean algebra with zero removed (note 34; Pontow and Schubert 2006, Thm 34).
- Atomistic GEM with κ atoms has 2^κ − 1 elements, so it has models of sizes 1, 3, 7, 15 and 2^ℵ₀ and none of size 2, 4, 6 or ℵ₀.
- Every GEM model has a top element, so GEM is incompatible with junk.
What could have come out otherwise. Simons’s pairing could have sufficed. A single countermodel shows it does not.
What this does not say
- That GEM is the right mereology. The result is about which axiom sets coincide. The entry treats every principle beyond the core, and the core itself, as contested (THEORY-589).
- That the result holds outside its setting. It is stated in first-order logic with identity and a single primitive P, with infinitary principles as schemas over open formulas, which cover only denumerably many specifiable sets. Note 9 concedes that free logic or plural quantification might suit sums better.
- That “sum” and “fusion” are interchangeable. Some authors reserve “sum” for S2 and “fusion” for S3 (note 23). A claim that something is the “sum” of some parts is not determinate until the definition is fixed.
- That the record has checked it. The entry gives citations and at most one-line derivations, and NOTE-163 did not check the sources.
- Anything about quantum or systems “mereology”. Note 18 excludes quantum part–whole formalisms, and note 1 the systems-theory and ecological senses.
Connections
- Whether mereology’s axioms are meaning postulates is THEORY-589, from the same entry.
- Fiat and bona fide boundaries (LIT-1024, THEORY-622) add topology to a mereology of this kind, with extensional mereology and unrestricted fusion taken for simplicity.
- Hilbert-space factorization (THEORY-017) is the quantum part–whole question the entry leaves out.