大川 裕矢

Abstract We focus on the persistence principle over weak interpretability logic. Our object of study is the logic obtained by adding the persistence principle to weak interpretability logic from several perspectives. Firstly, we prove that this logic enjoys a weak version of the fixed point property. Secondly, we introduce a system of sequent calculus and prove the cut‐elimination theorem for it. As a consequence, we prove that the logic enjoys the Craig interpolation property. Thirdly, we show that the logic is the natural basis of a generalization of simplified Veltman semantics, and prove that it has the finite frame property with respect to that semantics. Finally, we prove that it is sound and complete with respect to some appropriate arithmetical semantics.

Abstract We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic $$\textbf{IL}$$. We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.

<title>Abstract</title> We investigate the fixed point properties for sublogics of the interpretability logic $\textbf {IL}$. In our previous work, it was proved that a sublogic $\textbf {IL}^{-}(\textbf {J 2}_+, \textbf {J 5})$ has the fixed point property (FPP) and that a sublogic $\textbf {IL}^{-}(\textbf {J 4}, \textbf {J 5})$ has a newly introduced weaker property $\ell $FPP. In this paper, we provide countably many sublogics of $\textbf {IL}^{-}(\textbf {J 2}_{+}, \textbf {J 5})$ (resp. $\textbf {IL}^{-}(\textbf {J 4}, \textbf {J 5})$) having FPP (resp. $\ell $FPP).