Sahlqvist via Translation

We investigate to what extent Sahlqvist-type results for nonclassical logics can be obtained by embedding into classical logic via some G\"odel-type translations. We prove the correspondence theorem via translation for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also show that canonicity-via-translation can be obtained in a similarly straightforward manner, but only in the special setting of normal bi-Heyting algebra expansions. We expand on the difficulties involved in obtaining canonicity-via-translation outside of the bi-Heyting setting.

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