Refutations, proofs, and models in the modal logic K

Studia Logica 70 (2):193 - 204 (2002)
Abstract
In this paper we study the method of refutation rules in the modal logic K4. We introduce refutation rules with certain normal forms that provide a new syntactic decision procedure for this logic. As corollaries we obtain such results for the following important extensions: S4, the provability logic G, and Grzegorczyk''s logic. We also show that tree-type models can be constructed from syntactic refutations of this kind.
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Tomasz Skura (2011). On Refutation Rules. Logica Universalis 5 (2):249-254.
Tomasz F. Skura (2005). Intuitionistic Socratic Procedures. Journal of Applied Non-Classical Logics 15 (4):453-464.
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