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Sohei Iwata

Publications and source records attributed to Sohei Iwata.

5 recordsLinked to original sources
arXiv2025

Cut-free sequent calculi for the provability logic D

We say that a Kripke model is a GL-model if the accessibility relation $\prec$ is transitive and converse well-founded. We say that a Kripke model is a D-model if it is obtained by attaching infinitely many worlds $t_1, t_2, \ldots$, and $t_ω$ to a world $t_0$ of a GL-model so that $t_0 \succ t_1 \succ t_2 \succ \cdots \succ t_ω$. A non-normal modal logic D, which was studied by Beklemishev (1999), is characterized as follows. A formula $φ$ is a theorem of D if and only if $φ$ is true at $t_ω$ in any D-model. D is an intermediate logic between the provability logics GL and S. A Hilbert-style proof system for D is known, but there has been no sequent calculus. In this paper, we establish two sequent calculi for D, and show the cut-elimination theorem. We also introduce new Hilbert-style systems for D by interpreting the sequent calculi. Moreover, we show that D-models can be defined using an arbitrary limit ordinal as well as $ω$. Finally, we show a general result as follows. Let $X$ and $X^+$ be arbitrary modal logics. If the relationship between semantics of $X$ and semantics of $X^+$ is equal to that of GL and D, then $X^+$ can be axiomatized based on $X$ in the same way as the new axiomatization of D based on GL.

math.LO↗
arXiv2023

The persistence principle over weak interpretability logic

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.

math.LO↗
arXiv2021

Topological semantics of conservativity and interpretability logics

We introduce and develop a topological semantics of conservativity logics and interpretability logics. We prove the topological compactness theorem of consistent normal extensions of the conservativity logic $\mathbf{CL}$ by extending Shehtman's ultrabouquet construction method to our framework. As a consequence, we prove that several extensions of $\mathbf{CL}$ such as $\mathbf{IL}$, $\mathbf{ILM}$, $\mathbf{ILP}$ and $\mathbf{ILW}$ are strongly complete with respect to our topological semantics.

math.LO↗
arXiv2019

Fixed-point properties for predicate modal logics

It is well known that the propositional modal logic $\mathbf{GL}$ of provability satisfies the de Jongh-Sambin fixed-point property. On the other hand, Montagna showed that the predicate modal system $\mathbf{QGL}$, which is the natural variant of $\mathbf{GL}$, loses the fixed-point property. In this paper, we discuss some versions of the fixed-point property for predicate modal logics. First, we prove that several extensions of $\mathbf{QGL}$ including $\mathbf{NQGL}$ do not have the fixed-point property. Secondly, we prove the fixed-point theorem for the logic $\mathbf{QK} + \Box^{n+1} \bot$. As a consequence, we obtain that the class $\mathsf{FH}$ of Kripke frames which are transitive and finite height satisfies the fixed-point property locally. We also show the failure of the Craig interpolation property for $\mathbf{NQGL}$. Finally, we give a sufficient condition for formulas to have a fixed-point in $\mathbf{QGL}$.

math.LO↗