# A Sound Interpretation of Leśniewski's Epsilon in Modal Logic KTB

## Authors

• Takao Inoue Meiji Pharmaceutical University, Department of Medical Molecular Informatics, Tokyo, Japan; Hosei University, Graduate School of Science and Engineering Tokyo, Japan https://orcid.org/0000-0002-2080-7480

## Keywords:

Le´sniewski’s ontology, propositional ontology, translation, interpretation, modal logic, KTB, soundness, Grzegorczyk’s modal logic

## Abstract

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 $$\phi$$ 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. In the last, we shall give some comments including some open problems and my conjectures.

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2021-11-09

## How to Cite

Inoue, T. (2021). A Sound Interpretation of Leśniewski’s Epsilon in Modal Logic KTB. Bulletin of the Section of Logic, 50(4), 455–463. https://doi.org/10.18778/0138-0680.2021.25

Research Article