Classical logic and intuitionistic logic: equivalent formulations in natural deduction, G\"odel-Kolmogorov-Glivenko translation

Abstract

This report first shows the equivalence bewteen several formulations of classical logic in intuitionistic logic (tertium non datur, reductio ad absurdum, Pierce's law). Then it establishes the correctness of the G\"odel-Kolmogorov translation, whose restriction to the propositional case is due to Glivenko. This translation maps a formula $F$ of first order logic to a formula $F^{\lnot\lnot}$ in such a way that $F$ is provable in classical logic if and only if $F^{\lnot\lnot}$ is provable in intuitionistic logic. All formal proofs are presented in natural deduction. These questions are well-known proof theoretical facts, but in textbooks, they are often ignored or left to the reader. Because of the combinatorial difficulty of some of the needed formal proofs, we hope that this report may be useful, in particular to students and colleagues from other areas.