A Joint Logic of Problems and Propositions

In a 1985 commentary to his collected works, Kolmogorov informed the reader that his 1932 paper On the interpretation of intuitionistic logic “was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types—propositions and problems.” We construct such a formal system as well as its predicate version, QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The axioms of QHC are obtained as a result of a simultaneous formalization of two well-known alternative explanations of intiuitionistic logic: (1) Kolmogorov’s problem interpretation (with familiar refinements by Heyting and Kreisel) and (2) the proof interpretation by Orlov and Heyting, as clarified and extended by Gödel.