Most Properties are Undecidable for Transitive Tense Logics

A logics' property is decidable in a class of logics if there exists an algorithm that decides whether a finitely axiomatizable logic in the class has the property. Many properties are undecidable for bimodal logics but decidable for linear tense logics, which leads to a general question on how the interactions of modalities affect the decidability of properties. In this paper, we study the decidability of properties for transitive tense logics and show that most properties are undecidable in the lattice NExt(K4t) of transitive tense logics, including Kripke completeness, the finite model property, and decidability. Our proof method adapts Chagrov's approach of constructing a reduction from an undecidable problem of Minsky machines to the decision problem for logics' properties, yielding a general scheme of proving the undecidability of these properties.