Linear Logic - Encoding Classical/intuitionistic Logic in Linear Logic

Encoding Classical/intuitionistic Logic in Linear Logic

Both intuitionistic and classical implication can be recovered from linear implication by inserting exponentials: intuitionistic implication is encoded as !A ⊸ B, and classical implication as !A ⊸ ?B. The idea is that exponentials allow us to use a formula as many times as we need, which is always possible in classical and intuitionistic logic.

Formally, there exists a translation of formulae of intuitionistic logic to formulae of linear logic in a way which guarantees that the original formula is provable in intuitionistic logic if and only if the translated formula is provable in linear logic. Using the Gödel–Gentzen negative translation, we can thus embed classical first-order logic into linear first-order logic.