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In this article, we deal with propositional calculi over a signaturecontaining the classical implication $\to$ with the rules of modus ponens andsubstitution. For these calculi we consider few recognizing problems such asrecognizing derivations, extensions, completeness, and axiomatizations. Themain result of this paper is to prove that the problem of recognizingextensions is undecidable for every propositional calculus, and the problems ofrecognizing axiomatizations and completeness are undecidable for propositionalcalculi containing the formula $x \to ( y \to x )$. As a corollary, the problemof derivability of a fixed formula $A$ is also undecidable for all $A$.Moreover, we give a historical survey of related results.

Undecidable problems for propositional calculi with implication