Cut-free Tableau Calculi for some Propositional Normal Modal Logics*

TLDR: It is proved with the tableau-method that G0 is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G0is decidable and also characterisedBy the classof all frames for G0.

Abstract: We give sound and complete tableau and sequent calculi for the propositional normal modal logics $4.04, K4B and G O (these logics axe the smallest normal modal logics containing K and the schemata (:)A -+ ODA, DA --+ A and D A ; DA --+ D(qA and D(D(A --+ DA) -+ A) -+ DA resp.) with the following properties: the calculi for S4.04 and G O are 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 0 is not compact (and therefore not canonical), and we proof with the tableau-method that G O is chaxacterised by the class of all finite, (transi- tive) trees of degenerate or simple clusters of worlds; therefore G O is decidable and also chaxacterised by the class of all frames for G 0.