Synonymous Logics: A Correction Francis Jeffry Pelletier and Alasdair Urquhart August 25, 2006 1 Introduction 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 t 1 and t 2 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 t 2 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 t 2 (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 cor- rect!) 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]. 2 The new example We denote the set of non-negative integers by N, and the set of positive integers by N + . The logics that we define in this section have a single necessity operator , and two constants I and E (they can be read as “initial point” and “end point”). Each logic L X is defined in terms of a set X of positive integers. Given X ⊆ N + , the logic L X is defined to be the smallest normal modal logic containing the following axioms: 1. ♦A → A; 1