A semantical investigation into Leśniewski's axiom of his ontology

Abstract

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol ɛ, is called a quasi ɛ-struoture iff (a) the universe A of A consists of sets and (b) a ɛ b is true in A ↔ (∃[p) a = {p } & p ε b] for every a and b in A, where a(b) is the name of a (b). A quasi ɛ-structure A is called an ɛ-structure iff (c) {p } ε A whenever p ε a ε A. Then a closed formula σ in L is derivable from Leśniewski's axiom ∀ x, y[x ɛ y ↔∃ u (u ɛ x)∧∀ u; v(u, v ɛ x→ u ɛv)∧∀ u(u ɛ x→ u ɛ y)] (from the axiom ∀ x, y(x ɛ y → x ɛ x)∧∀ x, y, z(x ɛ y ɛ z → y ɛ x ɛ z)) iff σ is true in every ɛ-structure (in every quasi ɛ-structure).