We study the finite model property of subframe logics with expressible transitive reflexive closure modality. For \(m>0\) , let \({\textsc {L}}_m\) be the logic defined by axiom \(\lozenge ^{m+1} p\rightarrow \lozenge p\vee p\) . We construct quotient filtrations for the logics \({\textsc {L}}_m\) , which implies that these logics and their tense counterparts have the finite model property. Then, we construct selective filtrations of the canonical models of \({\textsc {L}}_m\) , which implies that all canonical subframe logics containing \({\textsc {L}}_m\) have the finite model property.
