A1 is not a conservative extension of S4 but of S5

In [1], D. W. Hart and C. Mcginn considered two logics A1 and A2. These logics embody part of a tradition about a priori knowledge and necessity. They proved that A2 is a conservative extension of a well-known modal logic S5 but left the problem whether A1 is a conservative extension of S4 open. In this note, we shall show that A1 is not a conservative extension of S4 but of S5, and also correct an inadequate proof.