Structural Completeness
While a general classification of structurally complete logics is not an easy task, we have a good understanding of some special cases.
Intuitionistic logic itself is not structurally complete, but its fragments may behave differently. Namely, any disjunction-free rule or implication-free rule admissible in a superintuitionistic logic is derivable. On the other hand, the Mints rule
is admissible in intuitionistic logic but not derivable, and contains only implications and disjunctions.
We know the maximal structurally incomplete transitive logics. A logic is called hereditarily structurally complete, if every its extension is structurally complete. For example, classical logic, as well as the logics LC and Grz.3 mentioned above, are hereditarily structurally complete. A complete description of hereditarily structurally complete superintuitionistic and transitive modal logics was given by Citkin and Rybakov. Namely, a superintuitionistic logic is hereditarily structurally complete if and only if it is not valid in any of the five Kripke frames
Similarly, an extension of K4 is hereditarily structurally complete if and only if it is not valid in any of certain twenty Kripke frames (including the five intuitionistic frames above).
There exist structurally complete logics that are not hereditarily structurally complete: for example, Medvedev's logic is structurally complete, but it is included in the structurally incomplete logic KC.
Read more about this topic: Admissible Rule
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