On Blass translation for Leśniewski's propositional ontology and modal logics

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 $\mathbf{K}$ \it by means of Hintikka formula\rm . 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, $\mathbf{K}$ $+$ $\square (\square \varphi \supset \varphi )\supset \square \varphi$. The faithfulness also holds for normal modal logics, e.g., $\mathbf{KD}$, $K4$, $KD4$, $\mathbf{KB}$. We shall conclude this paper with the section of some open problems and conjectures.

25 pages