Finite Model Property in Weakly Transitive Tense Logics

Abstract

The finite model property (FMP) in weakly transitive tense logics is explored. Let $\mathbb {S}=[\textsf{wK}_t\textsf{4}, \textsf{K}_t\textsf{4}]$ be the interval of tense logics between $\textsf{wK}_t\textsf{4}$ and $\textsf{K}_t\textsf{4}$. We introduce the modal formula $\textrm{t}_0^n$ for each $n\ge 1$. Within the class of all weakly transitive frames, $\textrm{t}_0^n$ defines the class of all frames in which every cluster has at most n irreflexive points. For each $n\ge 1$, we define the interval $\mathbb {S}_n=[\textsf{wK}_t\textsf{4T}_0^{n+1}, \textsf{wK}_t\textsf{4T}_0^{n}]$ which is a subset of $\mathbb {S}$. There are $2^{\aleph _0}$ logics in $\mathbb {S}_n$ lacking the FMP, and there are $2^{\aleph _0}$ logics in $\mathbb {S}_n$ having the FMP. Then we explore the FMP in finitely alternative tense logics $L_{n,m}=L\oplus \{\textrm{Alt}_n^F, \textrm{Alt}_m^P\}$ with $n,m\ge 0$ and $L\in \mathbb {S}$. For all $k\ge 0$ and $n,m\ge 1$, we define intervals $\mathbb {F}^k_{n,m}$, $\mathbb {P}^k_{n,m}$ and $\mathbb {S}^k_{n,m}$ of tense logics. The number of logics lacking the FMP in them is either 0 or $2^{\aleph _0}$, and the number of logics having the FMP in them is either finite or $2^{\aleph _0}$.