Kolmogorov and Kuroda translations into basic predicate logic

Abstract

Abstract Kolmogorov established the principle of the double negation translation by which to embed Classical Predicate Logic ${\operatorname {CQC}}$ into Intuitionistic Predicate Logic ${\operatorname {IQC}}$. We show that the obvious generalizations to the Basic Predicate Logic of [3] and to ${\operatorname {BQC}}$ of [12], a proper subsystem of ${\operatorname {IQC}}$, go through as well. The obvious generalizations of Kuroda’s embedding are shown to be equivalent to the Kolmogorov variant. In our proofs novel nontrivial techniques are needed to overcome the absence of full modus ponens in Basic Predicate Logic. In [3] we argued that ${\operatorname {IQC}}$ is not the logic of constructive mathematics. Our doubts were far from new. New was that we put forward an alternative, ${\operatorname {BQC}}$. One concern is that ${\operatorname {BQC}}$ is too weak for serious mathematics, or even trivial. This paper is one step to alleviate such concerns.