Undecidable problems for propositional calculi with implication

In this article, we deal with propositional calculi over a signature containing the classical implication. with the rules of modus ponens and substitution. For these calculi, we consider several decision problems such as recognition of derivations, extensions, completeness and axiomatizations. The main result of this article is that the problem of recognizing extensions is undecidable for every propositional calculus, and the problems of recognizing axiomatizations and completeness are undecidable for propositional calculi containing the axiom x ->(y -> x). As a corollary, for all formulas A, the problem of derivability of A is undecidable.