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=∃z(Pzx∧Pzy)).
- (P1: reflexivity) ∀xPxx
- (P2: anti-symmetry) ∀x∀y((Pxy∧Pyx)→x=y)
- (SSP: strong supplementation) ∀x∀y(¬Pyx→∃z(Pzy∧¬Ozx))
- (UF: unrestricted fusion principle) ∃xα(x)→∃z∀y(Oyz↔∃x(α(x)∧Oyx)), for any formula α(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: ∀x∀y∀z((Pxy∧Pyz)→Pxz) to the foregoing list.
In this talk, the following four definitions of atomicity will be considered where Atom(x)=∀y(Pyx→x=y).
- A0(x)=∀y(Pyx→∃u(Atom(u)∧Puy))
- A1(x)=∀y(Pyx→∃u(Atom(u)∧Puy∧Pux))
- A2(x)=∀y(Pyx→(A1(y)∧∃u(Puy∧Atom(u)∧Pux))
- A3(x)=∀y(Pyx→∃u(Atom(u)∧Puy∧∀u((Atom(u)∧Puy)→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+∀xA3(x) is equivalent to GEM+∀xA0(x) (hence to GEM+∀xAi(x), where i=0 to 3). Such a result means that we won’t get any stronger atomic theory even if we define an atomicity predicate finer than A3(x).