Labelled Tableau Calculi for Weak Modal Logics

Many normal and regular modal logics have simple formalizations in terms of labelled tableaux (cf. [3], [4]). But these modal logics have direct characterisation in terms of Kripke frames, and labels are naturally modelled on this kind of semantics. It is an interesting question whether this well known method can be extended to some congruent and monotonic modal logics, which are not characterisable by Kripke frames. Fortunately, they are determined by neighbourhood frames, a kind of more general relational semantics. So the main problem is how to apply the method of labels to cover logics with different interpretation of modalities. After short recollection of basic facts concerning respective modal logics and neighbourhood frames, we will offer analytic tableau calculi for some logics axiomatizable by combinations of axioms D, T, 4, 5 and the rule RN (necessitation) over the weakest congruent logic E and monotonic logic M. The source of inspiration for our tableau rules are cut-free sequent calculi from [5].