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# On Modal Logics between K × K × K and $S5 \times S5 \times S5$

R. Hirsch, I. Hodkinson and A. Kurucz
The Journal of Symbolic Logic
Vol. 67, No. 1 (Mar., 2002), pp. 221-234
Stable URL: http://www.jstor.org/stable/2695006
Page Count: 14
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## Abstract

We prove that every n-modal logic between Kn and S5n is undecidable, whenever n ≥ 3. We also show that each of these logics is non- finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov-Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite relation algebras.

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