On Intuitionistic Logic's Rival to Classical Logic

Intuitionistic logic based on the thoughts of intuitionism in mathematics,arguing that something exists if and only if it can be constructed,and that mathematics is the basis of logic.Intuitionistic logic objects to the principle of the excluded middle and reductio ad absurdum,holding that both logicism and formalism are wrong.What is the relation between intuitionistic logic and classical logic,different interpretations might make different answers.Besides,intuitionistic logic faces many problems,though M.Dummett had defended it from philosophy,there are still a lot of rebukes.