A sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB

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