Lattices of finitely alternative tense logics

Authors: Minghui Ma and Qian Chen
Published in: Studia Logica 109: 1093–1118 (2021). DOI: https://doi.org/10.1007/s11225-021-09942-5

Abstract

A finitely alternative normal tense logic $T_{n,m}$ is a normal tense logic characterized by frames in which every point has at most $n$ future alternatives and $m$ past alternatives. The structure of the lattice $\Lambda(T_{1,1})$ is described. There are $\aleph_0$ logics in $\Lambda(T_{1,1})$ without the finite model property (FMP), and only one pretabular logic in $\Lambda(T_{1,1})$. There are $2^{\aleph_0}$ logics in $\Lambda(T_{1,1})$ which are not finitely axiomatizable. For $nm\geq 2$, there are $2^{\aleph_0}$ logics in $\Lambda(T_{n,m})$ without the FMP, and infinitely many pretabular extensions of $T_{n,m}$.