A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.
We would like to thank Alex Citkin and James G. Raf-tery for many useful comments and remarks that improved the presentation of the paper. We are also very grateful to the referees for valuable suggestions and pointers to the literature. The second author was supported by the research Grant 2017 SGR 95 of the AGAUR from the Generalitat de Catalunya, by the I+D+i research Project PID2019-110843GA-I00 La geometria de las logicas no-clasicas funded by the Ministry of Science and Innovation of Spain, by the Beatriz Galindo Grant BEAGAL18/00040 funded by the Ministry of Science and Innovation of Spain, and by the MSCA-RISE-Marie Skłodowska-Curie Research and Innovation Staff Exchange (RISE) project MOSAIC 101007627 funded by Horizon 2020 of the European Union.
Publisher Copyright:
© 2022, The Author(s).