S5 x S5 x S5 Lacks the Finite Model Property

摘要

It follows from algebraic results of Maddux that every multi-modal logic L such that [S5, S5,..., S5] contains L contains S5~n is undecidable, whenever n ≥ 3. This implies that the product logic S5 x S5 x S5 does not have the product finite model property. Here we answer a question of Gabbay and Shehtman by showing that S5 x S5 x S5 also lacks the 'real' finite model property (fmp). We prove that every logic L from the above interval lacks the fmp. (In algebraic setting: If V is a variety of n-dimensional diagonal-free cylindric algebras which contains all the representables then V does not have the finite algebra property.)