Admissible and Derivable Rules in Intuitionistic Logic

A well-known problem in intuitionistic logic is the existence of valid but not derivable rules. This problem seems to be related with some constructive features of intuitionism (disjunction and existence property) but appear also in modal logics. We study here a particular case of this phenomenon, admissible rules in propositional calculus. G.E.Mints in [Mi 72] give sufficients conditions for admissible rules to be derivable. H.Friedman in [Fr 75] states the problem of the decidability of admissibility and V.V.Rybakov solves it in [Ry 84, Ry 86] using semantical and algebraic methods and the Gödel translation of intuitionistic logic into modal logic. We proposed here another approach to the problem closed to [Mi 72] but pointing out a particular class of substitutions useful when dealing with admissibility (section 2). We make use of these substitutions in this paper to extend the results of G.E.Mints. This approach leads to results we do not expose here. They are essentially a decidable characterization of admissibility using the intuitionistic sequent of calculus (precisely a kind of “retro-derivation” in sequent calculus), see [Ro 91]. We began this work by studying W.Dekkers conference notes. He rediscovered independently result from Mints in a particular case (admissibility and derivability are the same in the “→” fragment). These results are part of my Ph.D.Thesis supervised by M.Parigot.