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.