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October 13, 2025Open Access

Completeness theorems for modal logic in second-order arithmetic

Key Points

The weak completeness theorem for modal propositional logic is provable in RCA_0, enhancing understanding of logical foundations.
ACA_0 is equivalent to the strong completeness theorem for modal logic using canonical models, reflecting deep connections in mathematical logic.
A simpler strong completeness theorem, without canonical models, shows equivalence to WKL_0 over RCA_0, highlighting foundational interrelations.
These findings may imply foundational insights into the nature of modal logic and its relationships within second-order arithmetic.

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 RCA₀, and that, over RCA₀, ACA₀ 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 WKL₀ over RCA₀.

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Cite This Study

Shimomichi et al. (Mon,) studied this question.

synapsesocial.com/papers/68ece2abd1bb2827d12971d2 https://doi.org/https://doi.org/10.48550/arxiv.2503.01191

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