Embeddings into Free Heyting Algebras and Translations into Intuitionistic Propositional Logic

Abstract

We find a translation with particularly nice properties from intuitionistic propositional logic in countably many variables to intuitionistic propositional logic in two variables. In addition, the existence of a possibly-not-as-nice translation from any countable logic into intuitionistic propositional logic in two variables is shown. The nonexistence of a translation from classical logic into intuitionistic propositional logic which preserves ``and'' and ``or'' but not necessarily ``true'' is proven. These results about translations follow from additional results about embeddings into free Heyting algebras.