BI-SIMULATING IN BI-INTUITIONISTIC LOGIC GUILLERMO BADIA ABSTRACT. Bi-intuitionistic logic is the result of adding the dual of intuitionistic implica- tion to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the first order formulas equivalent to translations of bi-intuitionistic proposi- tional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstr ¨ om and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary chains. Keywords: bi-intuitionistic logic; van Benthem’s characterization theorem; directed bisimulations; model theory. 1. I NTRODUCTION Preservation theorems are results in model theory characterizing (up to logical equiva- lence) formulas of a given syntactic form (cf. [3], p. 295). In the context of the model theory of non-classical logics, we have the example of van Benthem’s celebrated theorem character- izing propositional modal formulas as those first order formulas invariant under bisimulations (cf. [1] for the textbook exposition). In recent work [13], Olkhovikov introduces the notion of asimulations to establish a preservation theorem for intuitionistic propositional formulas following van Benthem’s theo- rem. However, Olkhovikov’s theorem could also have been cast in terms of the more familiar concept of a directed bisimulation. 1 Indeed, it was implicit in the proof of Theorem 5.2 in [5], where directed bisimulations were introduced. The result of adding the dual of intuitionistic implication to intuitionistic logic is known as Heyting-Brouwer or Bi-intuitionistic logic (cf. [10, 4]). 2 The philosophical admissibility of this extension of intuitionistic logic was discussed in [9] and it turns out that, depending on the perspective, it might or might not be a sensitive extension of intuitionistic logic. At any rate, it is a natural model-theoretic extension of the language of intuitionistic logic in the same sense as adding a backward looking ♦ 1 to negation free modal logic with only the modality is. In fact, there is a strong analogy in a precise sense between temporal logic where we have both backward and foward looking operators and bi-intuitionistic logic (cf. [15]). In this paper we use bi-intuitionistic directed bisimulations to give a characterization theorem for bi-intuitionistic propositional formulas analogous to Olkhovikov’s characteri- zation of intuitionistic formulas. As far as we know, this question was open. In contrast to Olkhovikov’s proof, our argument makes no use of the machinery of saturated models 1 See [8] (Ch. 13) for a nice introduction to the concept and how it relates to logics with non-classical implications. 2 For how bi-intuitionistic logic fits in the more general setting of constructive logics see [14]. 1