# Atomicity under non-transitive mereology

## Hsing-Chien Tsai

The non-transitive mereology (NTM) to be addressed in this talk is axiomatized by the following axioms and axiom schema (\(Oxy=\exists z(Pzx\land Pzy)\)).

- (P1: reflexivity) \(\forall xPxx\)
- (P2: anti-symmetry) \(\forall x\forall y((Pxy\land Pyx)\to x=y)\)
- (SSP: strong supplementation) \(\forall x\forall y(\neg Pyx\to\exists z(Pzy\land\neg Ozx))\)
- (UF: unrestricted fusion principle) \(\exists x\alpha(x)\to\exists z\forall y(Oyz\leftrightarrow\exists x(\alpha(x)\land Oyx))\), for any formula \(\alpha(x)\) where z and y do not occur free.

Note that we will get the so-called General Extensional Mereology (GEM) if we add the transitivity axiom: \(\forall x\forall y\forall z((Pxy\land Pyz)\to Pxz)\) to the foregoing list.

In this talk, the following four definitions of atomicity will be considered where \(Atom(x)=\forall y(Pyx\to x=y)\).

- A0(x)=\(\forall y(Pyx\to\exists u(Atom(u)\land Puy))\)
- A1(x)=\(\forall y(Pyx\to\exists u(Atom(u)\land Puy\land Pux))\)
- A2(x)=\(\forall y(Pyx\to(A1(y)\land\exists u(Puy\land Atom(u)\land Pux))\)
- A3(x)=\(\forall y(Pyx\to\exists u(Atom(u)\land Puy\land\forall u((Atom(u)\land Puy)\to Pux)))\)

It is easy to see that all of these definitions are equivalent under GEM. But we will show that some pairs of them are not equivalent under NTM. Furthermore, we will also show that NTM+\(\forall xA3(x)\) is equivalent to GEM+\(\forall xA0(x)\) (hence to GEM+\(\forall xAi(x)\), where i=0 to 3). Such a result means that we wonâ€™t get any stronger atomic theory even if we define an atomicity predicate finer than A3(x).