A sound interpretation of Leśniewski's epsilon in modal logic KTB

In this paper, we shall show that the following translation $I^M$ from the propositional fragment $\bf L_1$ of Leśniewski's ontology to modal logic $\bf KTB$ is sound: for any formula $ϕ$ and $ψ$ of $\bf L_1$, it is defined as \smallskip (M1) $I^M(ϕ\vee ψ)$ = $I^M(ϕ) \vee I^M(ψ),$ (M2) $I^M(\neg ϕ)$ = $\neg I^M(ϕ),$ (M3) $I^M(εab)$ = $\Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b . \wedge . \Diamond p_b \supset p_a,$ \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.