Syllogistic as a Fragment of Elementary Ontology

Abstract

Five interpretations of syllogistic in Leśniewski's ontology are presented in the paper. Two of them are weak (φ^w) and strong (φ^s) interpretations of syllogistic (without negative terms) in ontology. The remaining are called here intermediate ones.

The first of them (φ^m), formulated by Iwanuś, is also an interpretation of syllogistic without negative terms. The last two intermediate interpretations (φⁿ, φ^o) are interpretations of syllogistic with negative proposed by the author.

Assuming a help functor of coexistence:

coex(x,y) ↔ (ex(y) → ex(x)). (ex(nx) → ex(ny))

the φⁿ interpretation has the following form:
xay ↔ coex(x,y). x y
xey ↔ coex(x,ny). x ny
xiy ↔ (coex(x,ny) → Σz(zεx. zεy))
xoy ↔ (coex(x,y) → Σz(zεx. zε‾x))
The interpretation φ^o is a variant of φⁿ.

It is proofed that syllogistic by the interpretation φ^m is a fragment of elementary ontology and that the syllogistic with negative terms by interpretations φⁿ and φ^o is inferentially included in this system with no empty universe.