KNOCKING DOWN BOXES: THE FMP FOR K ⊕ ◻ m + k p → ◻ m p $\mathbf {K}\oplus \Box ^{m+k} p\to \Box ^m p$ bold upper K circled plus white medium square Superscript m plus k Baseline p right arrow white medium square Superscript m Baseline p

Abstract It is a long-standing open problem whether modal logics of the form K ⊕ ◻ n p → ◻ m p $\\mathbf {K}\\oplus \\Box ^np\\to \\Box ^mp$ bold upper K circled plus white medium square Superscript n Baseline p right arrow white medium square Superscript m Baseline p for n > m > 1 $n>m>1$ n greater than m greater than 1 have the finite model property (FMP). We solve this by showing that any modal logic axiomatized by formulas of the form ◻ n p → ◻ m p $\\Box ^np\\to \\Box ^mp$ white medium square Superscript n Baseline p right arrow white medium square Superscript m Baseline p where n > m > 1 $n>m>1$ n greater than m greater than 1 has the FMP.