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On Blass Translation for Leśniewski’s Propositional Ontology and Modal Logics

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Abstract

In this paper, we shall give another proof of the faithfulness of Blass translation (for short, B-translation) of the propositional fragment \(\mathbf{L}_1\) of Leśniewski’s ontology in the modal logic \(\mathbf{K}\) by means of Hintikka formula. And we extend the result to von Wright-type deontic logics, i.e., ten Smiley-Hanson systems of monadic deontic logic. As a result of observing the proofs we shall give general theorems on the faithfulness of B-translation with respect to normal modal logics complete to certain sets of well-known accessibility relations with a restriction that transitivity and symmetry are not set at the same time. As an application of the theorems, for example, B-translation is faithful for the provability logic \(\mathbf{PrL}\) (= \(\mathbf{GL}\)), that is, \(\mathbf{K}\) \(+\) \(\Box (\Box \phi \supset \phi ) \supset \Box \phi \). The faithfulness also holds for normal modal logics, e.g., \(\mathbf{KD}\), \(\mathbf{K4}\), \(\mathbf{KD4}\), \(\mathbf{KB}\). We shall conclude this paper with the section of some open problems and conjectures.

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Acknowledgements

I would like to thank Prof. Andreas Blass, Prof. Emeritus Mitio Takano, the late Prof. Stanislaw Świerczkowski and the late Prof. Emeritus Arata Ishimoto for their valuable comments, stimulation and encouragement. I would also like to thank, especially, the late Prof. Anne Troelstra and Prof. Dirk van Dalen for their encouragement. Further I appreciate anonymous referees for their comments to make this paper better.

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  1. Depertment of Medical Molecular Informatics, Meiji Pharmaceutical University, Tokyo, Japan

    Takao Inoué

  2. Graduate School of Science and Engineering, Hosei University, Tokyo, Japan

    Takao Inoué

  3. Department of Applied Informatics, Faculty of Science and Engineering, Hosei University, Tokyo, Japan

    Takao Inoué

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Correspondence to Takao Inoué.

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Inoué, T. On Blass Translation for Leśniewski’s Propositional Ontology and Modal Logics. Stud Logica 110, 265–289 (2022). https://doi.org/10.1007/s11225-021-09962-1

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