On Blass translation for Le\'sniewski'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 $\bf L_1$ of Le\'{s}niewski's ontology in the modal logic $\bf K$ \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 $\bf PrL$ (= $\bf GL$), that is, $\bf K$ $+$ $\Box (\Box \phi \supset \phi) \supset \Box \phi$. The faithfulness also holds for normal modal logics, e.g., $\bf KD$, $\bf K4$, $\bf KD4$, $\bf KB$. We shall conclude this paper with the section of some open problems and conjectures.