Abstract
We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G 0(these logics are the smallest normal modal logics containing K and the schemata □A → □□A, □A → A and □⋄□A → (□ → □A); □A → □□A and A→□⋄A; □A → □□A and □(□(A→ □A) → A) → □A resp.) with the following properties: the calculi for S4.04 and G 0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.
In addition we show that G 0is not compact (and therefore not canonical), and we proof with the tableau-method that G 0is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G 0is decidable and also characterised by the class of all frames for G 0.
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Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.
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Amerbauer, M. Cut-free tableau calculi for some propositional normal modal logics. Stud Logica 57, 359–372 (1996). https://doi.org/10.1007/BF00370840
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DOI: https://doi.org/10.1007/BF00370840