We develop general strategies for transferring soundness and completeness from more expressive modal languages to less expressive ones, unifying several existing notions of operator definability along the way. For soundness, we exploit semantic insensitivity: if a less expressive language is insensitive to a frame operation, soundness extends to the operation's closure of the original frame class. For completeness, restricting to relational semantics and languages with a single operator, we present strategies for relating the target logic's canonical model to that of a normal modal logic via a truth-preserving translation. Three of those dispense entirely with specifying an accessibility condition for the target logic, inheriting it from a normal modal logic instead.

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