Hereditary structural completeness of weakly transitive modal logics
Simon Lemal
Abstract:
In a deductive system, a rule is said to be admissible if the tautologies of the system are closed under its applications, and derivable if the rule itself holds in the system. Although every derivable rule is admissible, the converse is not true in general. A classical problem in the area is to determine which deductive systems have the property of all admissible rules being derivable, i.e. are structurally complete. Early results on this problem suggest that even though a full characterisation of the structurally complete modal and superintuitionistic logics is out of reach, it might be possible to characterise the hereditarily structurally complete systems, those which are not only structurally complete themselves but whose finitary extensions are too. Hereditarily structurally complete intermediate logics were characterised by Citkin (1978). This result was generalised to logics extending the transitive modal logic K4 by Rybakov (1995). Carr (2022) revisited Rybakov’s result and corrected some of its errors. This thesis gives a full characterisation of the hereditarily structurally complete extensions of the modal logic wK4 of weakly transitive frames. The logic wK4 is a “close neighbour” of K4. It inherits many of its properties, yet there are essential differences. We also give a description of the n-universal models of wK4 and compare it to the n-universal models of K4.