All translations of classical logic into intuitionistic logic What are all possible ways of translating classical logic into intuitionistic logic? That is, if $S$ is the collection of sentences of first order logic, what are all the functions $f : S \to S$ such that $\Gamma \vdash_c \phi$ if and only if $f(\Gamma) \vdash_i f(\phi)$? I know that there is a couple of variants of double negation translation. So, two questions: is there a classification of all possible translations? Are they all logically equivalent to double negation translation?
Remember that the set of theorems of intuitionistic logic is a proper subset of the set of theorems of classical logic. So, if you want to translate every classical theorem to an intuitionistic version you will fail in general.
Remember that the set of theorems of intuitionistic logic is a proper subset of the set of theorems of classical logic. So, if you want to translate every classical theorem to an intuitionistic version you will fail in general.