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