A semantical investigation into Leśniewski's axiom of his ontology

✍ Scribed by Mitio Takano

• Publisher: Springer Netherlands
• Year: 1985
• Tongue: English
• Weight: 342 KB
• Volume: 44
• Category: Article
• ISSN: 0039-3215
• DOI: 10.1007/bf00370810

✦ Synopsis

A structure zr for the language ~, which is the first-or4er language (without equality) whose only nonlogical symbol is the binary predicate symbol ~, is called a quasi s-structure iff (a) the universe lzr of .4 consists of sets and (b) a r b is true in zCe~(3p) [a = {p} &p e b] for every a and b in Izr where a(b) is the name of a (b). A quasi s-structure zr is called an c-structure iff (e) {p} e lzr l whenever p ~ a IzCl. Then a closed formula a in L is derivable from Legniewski's axiom Vx, y Ix r y ~-*3u(uzx)AVu, v(u, vex-->uzv)AVu(usx->uey)]! (from the axiom Vx, y(xzy -+x ~ x)AVx, y, z(x e y e z~y s 9 e z)) fff a is true in every s-structure (in every quasi e-structure). In this paper we shall be concerned with Legniewski~s axiom 2: VxVy Ix ~ y ~ 3u(u ~ x) ^ VuVv(u 9 x^ v 9 x~u s v) ^ Vu(u ~ x-~u ~ y)] of his ontology, and a somewhat weaker axiom if: VxVy (x ~ y ~x ~ x) ^ VxVyVz(x ~ y^y ~ z.->x s z) ^VxVyVz(x s y^y s z-->~ ~ x)