Union-splittings, the Axiomatization Problem, and the Rule Dichotomy Property in Modal Logic
Tenyo Takahashi

Abstract:
This thesis studies logical properties in lattices of modal logics, focusing on union-splittings, the axiomatization problem, and the rule dichotomy property. We use semantic approaches to investigate these topics by working with the theory of stable canonical rules and formulas. Under the modal duality, we apply universal algebra to modal algebras and combinatorial methods to modal spaces.
We reformulate and extend the theory of stable canonical rules and formulas by introducing the notion of definable filtration, which will be the semantic foundation for much of the thesis. Building on this, a new combinatorial method, the Refinement Construction, is developed to prove the finite model property for a large class of logics and rule systems, generalizing the finite model property of union-splittings in NExtK, K4-stable logics, and stable rule systems.
We then give a semantic characterization of union-splittings in the lattice NExtK and show that both being a union-splitting and a splitting are decidable in NExtK. This yields two more decidable properties in NExtK, namely, being a decidable formula and having a decidable axiomatization problem. These results answer the open questions [WZ07, Problem 2] and [CZ97, Problem 17.3] in the affirmative.
Finally, we study admissibility and the rule dichotomy property in the weak transitive logic wK4 and the basic modal logic K. We refine the notion of rule dichotomy property, and show that stable canonical rules have the rule dichotomy property over wK4 but fail over K. The latter supports Jeřábek’s remark that the rule dichotomy property is a very strong property and thus is likely to fail for many logics [Jeř09].
The last chapter applies descriptive set theory to study the cardinality of sets of logics without assuming the Continuum Hypothesis and resolves the open questions [JL18, Question 6.4 (ii)] and [BBM25, Section 8 (1)].