SINGLE AXIOMS FOR ATOMISTIC AND ATOMLESS ... - Project Euclid
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SINGLE AXIOMS FOR ATOMISTIC AND ATOMLESS ... - Project Euclid
ROBERT E. CLAY. It is part of the folk-lore of the subject, that Lesniewski's mereology is neutral with respect to the existence of atoms. It has long been ... Page 2 ...
345 Notre Dame Journal of Formal Logic Volume XVI, Number 3, July 1975 NDJFAM
SINGLE AXIOMS FOR ATOMISTIC AND ATOMLESS MEREOLOGY ROBERT E. CLAY
It is part of the folk-lore of the subject, that Lesniewski's mereology is neutral with respect to the existence of atoms. It has long been known that one could have a system of atomless mereology by adding the following axiom or its equivalent to any mereological axiom system. [A] :A tA .=>. [3B]. B ε pr(A). Similarly one could have a system of atomistic (completely atomic) mereology by adding .=>. C= B.
[A]::AεA.^:.[3B]:.Bεe\(A):[C]:Cεe\(B) If we define the name " α t m " by [J3].\ JSεαtm .=: BεB:[c]:
Cεe\{B) .=>. C = B
this simplifies to [A]:AεΛ .3. [3B].Bεe\(A)
.Bεcftm.
Rickey, cf. [4], p. 90, introduced the functor " a t " defined by [AB]:BεcΛ(A) .=. Bεe\(A) . £ ε a t m which further reduces the characteristic axiom of atomistic mereology to [A]:AεA .=>. [3B].
Bεσt(A).
Using Rickey's functor " a t " Sobociήski axiomatized atomistic mereology in [4]. Lejewski gave the first single axioms for atomistic and atomless mereology in [2]. In this paper we shall give shorter single axioms for both systems. Lejewski's single axiom for atomistic mereology is LI
DX2 [AB].\Aεe\(B) .=:AεA.BεB:[D]:Dεex(B).Ώ.Dεex(A) X15 [A]:AεA .3. [3F] .Fεe\(A). ~(AεF) This rather curious numbering is designed to facilitate the second half of the proof of the equivalence of the two systems. First we shall derive XI, DX2, and X15 from El. E2 E3 E4 DEI £5 PR 2. 3.
