We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński’s pure logic of necessitation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textbf{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">N</mi> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m,n \in \omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mi>ω</mi> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{NA}_{m,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">NA</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, which was introduced by Kurahashi and Sato, be the logic obtained from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">N</mi> </math></EquationSource> </InlineEquation> by adding the axiom scheme <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Box ^n A \rightarrow \Box ^m A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mo>□</mo> <mi>n</mi> </msup> <mi>A</mi> <mo stretchy="false">→</mo> <msup> <mo>□</mo> <mi>m</mi> </msup> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, among other things, we prove that for each <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m,n \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the logic <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textbf{NA}_{m,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">NA</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> becomes a provability logic, that is, there exists a provability predicate <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{Pr}_T(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Pr</mtext> <mi>T</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of T whose T-verifiable modal principles are exactly the logic <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textbf{NA}_{m,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">NA</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>.

Arithmetical Completeness for Some Extensions of the Pure Logic of Necessitation

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发表日期：

2026年5月5日

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Haruka Kogure

摘要

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