Rautenberg THE LATTICE OF NORMAL MODAL LOGICS

Most material below is ranked around the splittings of lattices of normal modal logics. These splittings are generated by finite subdirect irreducible modal algebras. The actual computation of the splittings is often a rather delicate task. Refined model structures are very useful to this purpose, as well as they are in many other respects. E.g. the analysis of various lattices of extensions, like ES5, ES4.3 etc becomes rather simple, if refined structures are used. But this point will not be touched here. The variety TBA (cf. below), which corresponds to S4, is congruencedistributive. Hence any every tabular extension L ⊇ S4 has finite many extensions only. Around 1975 it has been proved by several authors, that also the converse is true: If L ⊇ S4 has finitely many extensions only, then L is necessarely tabular. These and other results will be extended to much richer lattices. For simplicity we state the results explicitely only for the lattice N of modal logics with one modal operator. But most of them carry over literally to the lattice N of k-ramified normal modal logics (k modal operators). It is easy to construct examples of 2-ramified modal logics (tense logics) which are POST -complete but not tabular. Whether this will be possible for normal (unramified) modal logic seems to be an open problem. In view of results below it seems very unlikely that such examples exist.