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.
Read more about this topic: Linear Logic
Famous quotes containing the words classical and/or logic:
“Several classical sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“The logic of worldly success rests on a fallacy: the strange error that our perfection depends on the thoughts and opinions and applause of other men! A weird life it is, indeed, to be living always in somebody else’s imagination, as if that were the only place in which one could at last become real!”
—Thomas Merton (1915–1968)