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

Takao Inoue

doi:10.18778/0138-0680.2021.25

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

DOI:

https://doi.org/10.18778/0138-0680.2021.25

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 $L_{1}$ of Leśniewski's ontology to modal logic $K T B$ is sound: for any formula $ϕ$ and $ψ$ of $L_{1}$ , it is defined as

(M1) $I^{M} (ϕ \lor ψ) = I^{M} (ϕ) \lor I^{M} (ψ)$ ,

(M2) $I^{M} (\neg ϕ) = \neg I^{M} (ϕ)$ ,

(M3) $I^{M} (ϵ a b) = ◊ p_{a} \supset p_{a} . \land . ◻ p_{a} \supset ◻ p_{b} . \land . ◊ p_{b} \supset p_{a}$ ,

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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