arXiv subjects

Haruka Kogure

Publications and source records attributed to Haruka Kogure.

10 recordsLinked to original sources

arXiv2026

Modal logics of conjunctively closed provability predicates

We investigate non-normal modal logics corresponding to provability predicates $\mathrm{Pr}_T(x)$ satisfying the derivability condition $\mathbf{C}$: $T\vdash\mathrm{Pr}_T(\ulcorner \varphi \urcorner)\land\mathrm{Pr}_T(\ulcorner \psi \urcorner)\to \mathrm{Pr}_T(\ulcorner \varphi\land\psi \urcorner)$. The modal counterpart of this condition is the axiom scheme $\mathsf{C}$: $\Box A\land\Box B\to\Box(A\land B)$. First, we introduce a new semantics based on closure operators for non-normal modal logics including logics adopting $\mathsf{C}$ as an axiom scheme. We prove modal completeness for several non-normal modal logics studied in this paper with respect to this semantics. Second, we prove the arithmetical completeness theorems for the logics $\mathsf{CN}$, $\mathsf{CNP}$, $\mathsf{CNF}$, $\mathsf{CNPF}$, and $\mathsf{CND}$ by using our new semantics.

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arXiv2026

Finite Kripke models and provability interpretations in quantified modal logic

In this paper, we investigate arithmetical completeness with respect to finite Kripke models of quantified modal logic. We adapt the finite-model embedding techniques of Artemov and Japaridze to two settings involving finite Kripke models. First, for conversely well-founded finite Kripke models of quantified modal logic, we construct a $\Sigma_2$ Fefermanian provability predicate together with an arithmetical interpretation that embeds the model into arithmetic. Second, for finite constant domain Kripke models of quantified modal logic, we construct a $\Sigma_1$ provability predicate satisfying $\mathbf{D2^G}$ and an arithmetical interpretation yielding such an embedding.

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arXiv2025

Arithmetical completeness for some extensions of the pure logic of necessitation

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For $m,n \in ω$, let $\mathbf{NA}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{NA}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate $\mathrm{Pr}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic $\mathbf{NA}_{m,n}$.

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arXiv2025

Provability interpretation of non-normal modal logics having neighborhood semantics

We study provability predicates $\mathrm{Pr}_T(x)$ satisfying the following condition $\mathbf{E}$ from a modal logical perspective: $\mathbf{E}:$ if $ T \vdash φ\leftrightarrow ψ$, then $T \vdash \mathrm{Pr}_T(\ulcorner φ\urcorner) \leftrightarrow \mathrm{Pr}_T(\ulcorner ψ\urcorner)$. For this purpose, we develop a new method of embedding models based on neighborhood semantics into arithmetic. Our method broadens the scope of arithmetical completeness proofs. In particular, we prove the arithmetical completeness theorems for the non-normal modal logics $\mathsf{EN}$, $\mathsf{ECN}$, $\mathsf{ENP}$, $\mathsf{END}$, and $\mathsf{ECNP}$.

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arXiv2025

Modal logical aspects of provability predicates and consistency statements

This paper studies the modal logical aspects of provability predicates and consistency statements for theories of arithmetic. First, we provide an overview of previous works on the correspondence between various derivability conditions for provability predicates and different modal logics. The main technical contribution of the present paper is to establish the arithmetical completeness of the logics $\mathsf{NP}$, $\mathsf{ND}$, $\mathsf{NP4}$, and $\mathsf{ND4}$ by extending Solovay's method and refining Arai's construction of Rosser provability predicates.

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arXiv2025

Doubly partially conservative sentences

The purpose of the present paper is to analyze several variants of Solovay's theorem on the existence of doubly partially conservative sentences. First, we investigate $Θ$ sentences that are doubly $(Γ, Λ)$-conservative over $T$ for several triples $(Θ, Γ, Λ)$. Among other things, we prove that the existence of a $Δ_{n+1}(\mathsf{PA})$ sentence that is doubly $(Σ_n, Σ_n)$-conservative over $T$ is equivalent to the $Σ_{n+1}$-inconsistency of $T$ over $\mathsf{PA}$. Secondly, we study $Θ$ sentences that are hereditarily doubly $(Γ, Λ)$-conservative over $T$ for several triples $(Θ, Γ, Λ)$.

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arXiv2025

A variety of partially conservative sentences

We study the existence of a $Θ$ sentence which is simultaneously $Γ$-conservative over consistent RE extensions $T$ and $U$ of Peano Arithmetic for various reasonable pairs $(Γ, Θ)$. As a result of this study, we prove the existence of a sentence which is essentially $Θ$ and exactly hereditarily $Γ$-conservative over any single theory $T$ for various reasonable pairs $(Γ, Θ)$. This is an affirmative answer to Guaspari's question.

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arXiv2023

On the conservation results for local reflection principles

For a class $Γ$ of formulas, $Γ$ local reflection principle $\mathrm{Rfn}_Γ(T)$ for a theory $T$ of arithmetic is a scheme formalizing the $Γ$-soundness of $T$. Beklemishev proved that for every $Γ\in \{Σ_n, Π_{n+1} \mid n \geq 1\}$, the full local reflection principle $\mathrm{Rfn}(T)$ is $Γ$-conservative over $T + \mathrm{Rfn}_Γ(T)$. We firstly generalize the conservation theorem to nonstandard provability predicates: we prove that the second condition $\mathbf{D2}$ of the derivability conditions is a sufficient condition for the conservation theorem to hold. We secondly investigate the conservation theorem in terms of Rosser provability predicates. We construct Rosser predicates for which the conservation theorem holds and Rosser predicates for which the theorem does not hold.

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arXiv2023

Interpolation properties for the bimodal provability logic $\mathbf{GR}$

We study interpolation properties for Shavrukov's bimodal logic $\mathbf{GR}$ of usual and Rosser provability predicates. For this purpose, we introduce a new sublogic $\mathbf{GR}^\circ$ of $\mathbf{GR}$ and its relational semantics. Based on our new semantics, we prove that $\mathbf{GR}^\circ$ and $\mathbf{GR}$ enjoy Lyndon interpolation property and uniform interpolation property.

math.LO↗

arXiv2023

Arithmetical completeness theorems for monotonic modal logics

We investigate modal logical aspects of provability predicates $\mathrm{Pr}_T(x)$ satisfying the following condition: $\mathbf{M}$: If $T \vdash φ\to ψ$, then $T \vdash \mathrm{Pr}_T(\ulcorner φ\urcorner) \to \mathrm{Pr}_T(\ulcorner ψ\urcorner)$. We prove the arithmetical completeness theorems for monotonic modal logics $\mathsf{MN}$, $\mathsf{MN4}$, $\mathsf{MNP}$, $\mathsf{MNP4}$, and $\mathsf{MND}$ with respect to provability predicates satisfying the condition $\mathbf{M}$. That is, we prove that for each logic $L$ of them, there exists a $Σ_1$ provability predicate $\mathrm{Pr}_T(x)$ satisfying $\mathbf{M}$ such that the provability logic of $\mathrm{Pr}_T(x)$ is exactly $L$. In particular, the modal formulas $\mathrm{P}$: $\neg \Box \bot$ and $\mathrm{D}$: $\neg (\Box A \land \Box \neg A)$ are not equivalent over non-normal modal logic and correspond to two different formalizations $\neg \mathrm{Pr}_T(\ulcorner 0=1 \urcorner)$ and $\neg \big(\mathrm{Pr}_T(\ulcorner φ\urcorner) \land \mathrm{Pr}_T(\ulcorner \neg φ\urcorner) \bigr)$ of consistency statements, respectively. Our results separate these formalizations in terms of modal logic.

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