A sound interpretation of Le\'sniewski's epsilon in modal logic KTB

TL;DR

This paper presents a sound translation of Leśniewski's epsilon from propositional logic to modal logic KTB, clarifying its interpretation within modal frameworks.

Contribution

It introduces a novel translation scheme for Leśniewski's epsilon into modal logic KTB and proves its soundness, expanding understanding of Leśniewski's ontology in modal contexts.

Findings

Translation scheme is sound for propositional fragment of Leśniewski's ontology.

Provides formal proof of translation's soundness.

Discusses open problems and conjectures related to the translation.

Abstract

In this paper, we shall show that the following translation $I^M$ from the propositional fragment $\bf L_1$ of Le\'{s}niewski's ontology to modal logic $\bf KTB$ is sound: for any formula $\phi$ and $\psi$ of $\bf L_1$, it is defined as \smallskip (M1) $I^M(\phi \vee \psi)$ = $I^M(\phi) \vee I^M(\psi),$ (M2) $I^M(\neg \phi)$ = $\neg I^M(\phi),$ (M3) $I^M(\epsilon ab)$ = $\Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b . \wedge . \Diamond p_b \supset p_a,$ \smallskip \noindent where $p_a$ and $p_b$ are propositional variables corresponding to the name variables $a$ and $b$, respectively. In the last section, we shall give some open problems and my conjectures.