Journal of Symbolic Logic 67 (1):221-234 (2002)

Abstract
We prove that everyn-modal logic betweenKnandS5nis 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 representation problem of finite relation algebras.
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DOI 10.2178/jsl/1190150040
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Hybrid Formulas and Elementarily Generated Modal Logics.Ian Hodkinson - 2006 - Notre Dame Journal of Formal Logic 47 (4):443-478.
Neat Embeddings, Omitting Types, and Interpolation: An Overview.Tarek Sayed Ahmed - 2003 - Notre Dame Journal of Formal Logic 44 (3):157-173.
Algebraic Logic, Where Does It Stand Today?Tarek Sayed Ahmed - 2005 - Bulletin of Symbolic Logic 11 (3):465-516.

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