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arXiv · 2503.01191

Completeness theorems for modal logic in second-order arithmetic

Sho Shimomichi·Yuto Takeda·Keita Yokoyama
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Abstract

This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in $\mathrm{RCA}_0$, and that, over $\mathrm{RCA}_0$, $\mathrm{ACA}_0$ is equivalent to the strong completeness theorem for modal propositional logic using canonical models. We also consider a simpler version of the strong completeness theorem without referring to canonical models and show that it is equivalent to $\mathrm{WKL}_0$ over $\mathrm{RCA}_0$.

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BibTeXRIS

Sho Shimomichi, Yuto Takeda, Keita Yokoyama. 2025-03-03. Completeness theorems for modal logic in second-order arithmetic. https://arxiv.org/abs/2503.01191

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