Degree of Kripke-incompleteness of Tense Logics

TL;DR

This paper generalizes Blok's dichotomy theorem on Kripke-incompleteness from normal modal logics to tense logics, showing that the degree of incompleteness is either 1 or continuum, and characterizes strictly Kripke-complete logics.

Contribution

It extends the dichotomy theorem to lattices of tense logics and characterizes strictly Kripke-complete logics via iterated splittings.

Findings

Dichotomy theorem applies to lattices of tense logics.

Iterated splittings characterize strictly Kripke-complete logics.

The degree of Kripke-incompleteness is either 1 or continuum.

Abstract

The degree of Kripke-incompleteness of a logic $L$ in some lattice $\mathcal{L}$ of logics is the cardinality of logics in $\mathcal{L}$ which share the same class of Kripke-frames with $L$. A celebrated result on Kripke-incompleteness is Blok's dichotomy theorem for the degree of Kripke-incompleteness in $\mathsf{NExt}(\mathsf{K})$: every modal logic $L\in\mathsf{NExt}(\mathsf{K})$ is of the degree of Kripke-incompleteness $1$ or $2^{\aleph_0}$. In this work, we show that the dichotomy theorem for $\mathsf{NExt}(\mathsf{K})$ can be generalized to the lattices $\K$, $\LT$ and $\NExt(\ST)$ of tense logics. We also prove that in $\K$, $\LT$ and $\NExt(\ST)$, iterated splittings are exactly the strictly Kripke-complete logics.