We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev . We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok—Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (finitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.
"Canonical rules." J. Symbolic Logic 74 (4) 1171 - 1205, December 2009. https://doi.org/10.2178/jsl/1254748686