Synonymous Logics: A Correction

In a paper devoted to problems of synonymy and translation between modal logics [2], the authors attempted to show that there are two modal logics so that each is exactly translatable into the other, but they are not translationally equivalent. Unfortunately, there is an error in the proof of Theorem 4.7 of this paper, that purports to prove this result, as Lloyd Humberstone discovered [1]. In §4 of the published article, the authors define two modal logics, T∗ and KU∗, each of them formulated with a necessity operator ¤A, together with a propositional constant C. Theorem 4.7 claims that there are translations t1 and t2 between the two logics, so that each logic is exactly embedded in the other, but that the two logics are not translationally equivalent. However, as Humberstone has pointed out, there is a fatal mistake in the proof that the translation t2 embeds the logic T∗ exactly in the logic KU∗. The formula C → [¤(¤C ∧ C) ∧¤C ∧ C] is provable in KU∗ by the rule of necessitation and propositional logic; however, this formula is t2(C→¤¤C), and this is not provable in T∗, since in that logic no special axioms are assumed for the propositional constant C. In the remainder of this article, we give an alternative (and hopefully correct!) proof of the same result, namely that there are two normal modal logics that are exactly translatable into each other, but not translationally equivalent. The two logics are defined by adapting a construction from [3].