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

TL;DR

This paper demonstrates the equivalence of various classical logic formulations within intuitionistic logic and proves the correctness of the G"odel-Kolmogorov translation using natural deduction, providing detailed formal proofs for educational purposes.

Contribution

It establishes the formal equivalence of classical logic formulations in intuitionistic logic and verifies the G"odel-Kolmogorov translation's correctness with detailed natural deduction proofs.

Findings

Equivalence of classical logic formulations in intuitionistic logic

Correctness of G"odel-Kolmogorov translation for propositional logic

Formal proofs provided in natural deduction format

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.