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Akama et al. systematically studied an arithmetical hierarchy of the law ofexcluded middle and related principles in the context of first-orderarithmetic. In that paper, they first provide a prenex normal form theorem as ajustification of their semi-classical principles restricted to prenex formulas.However, there are some errors in their proof. In this paper, we provide asimple counterexample of their prenex normal form theorem, then modify it in anappropriate way. In addition, we characterize several prenex normal formtheorems with respect to semi-classical arithmetic.

Prenex normal form theorems in semi-classical arithmetic