12/03 - Rodolfo C. Ertola Biraben
Modal aspects in bi-intuitionistic logic
In the context of a Heyting algebra extended with the dual ─ of the relative pseudocomplement →, we study ¬D and D¬, where ¬ stands for usual intuitionistic negation and D stands for co-negation. Operation D may be defined, for any x, as (x → x) ─ x. Operations ¬D and D¬ behave very much like necessity and possibility, respectively. The result has all the modal properties of modal system B. We prove facts considering .
In 1942 Moisil considered DD and ¬¬ for necessity and possibility, respectively (see ). Now, for DD modal axiom K is not available.
We consider the extension with the corresponding S4 axiom and prove that it is conservative. In both cases we study the modalities.
We also compare the given framework with the extension with the connective Δ in Fuzzy Logic (see e.g. ).
We mostly present our results from the algebraic point of view. However, we also include some logic considerations. Adding axioms to intuitionistic logic is not enough, i.e. it is necessary to add a rule. Also, two different logics appear, depending, from a semantical point of view, on whether a truth-preserving or a truth-degree-preserving consequence is chosen.
 Hajek, P. (1998), Metamathematics of Fuzzy Logic, Kluwer.
 Moisil, G. C. (1972), Essai sur les logiques non chrysippiennes, Éditions de l'academie de la république socialiste de Roumanie.
 Simpson, A. K. (1994), The proof theory and semantics of intuitionistic modal logic, PhD thesis, University of Edinburgh.