Editing Admissible rule (section)

==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.<ref>Rybakov (1997), Thms. 5.5.6, 5.5.9</ref> On the other hand, the [[Grigori Mints|Mints]] rule
:<math>\frac{(p\to q)\to p\lor r}{((p\to q)\to p)\lor((p\to q)\to r)}</math>
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 any 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 respectively 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<ref name="hsc"/>
::[[File:Tsitkin frames.svg]]
Similarly, an extension of ''K''4 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).<ref name="hsc"/>

There exist structurally complete logics that are not hereditarily structurally complete: for example, [[intermediate logic|Medvedev's logic]] is structurally complete,<ref>Prucnal (1976)</ref> but it is included in the structurally incomplete logic ''KC''.