A sound interpretation of Le\'sniewski'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 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 (M1) $I^M(\phi \vee \psi)$ = $I^M(\phi) \vee I^M(\psi),$ (M2) $I^M(\neg \phi)$ = $\neg I^M(\phi),$ (M3) $I^M(\epsilon ab)$ = $\Diamond p_a \supset p_a . \wedge . \Box p_a \supset \Box p_b . \wedge . \Diamond p_b \supset p_a,$ where $p_a$ and $p_b$ are propositional variables corresponding to the name variables $a$ and $b$, respectively. We shall give some comments including some open problems and my conjectures.