Post complete and 0-axiomatizable modal logics

The study of dual spaces of finitely generated free algebras over equational classes of modal algebras or Heyting algebras provides a tool which permits the resolution in a uniform way of a considerable number of otherwise disconnected problems, which concern normal modal logics and intermediate logics (see, as regards the former, [8] and [2], and, as regards the latter, [3] and [4]). In the present paper we study the O-canonical models for the propositional normal modal logics, which are, up to isomorphism, the dual spaces of the free algebras without generators over the equational classes of modal algebras. This analysis allows us, via a result due to Makinson-Segerberg, to determine in a rather visualizable way the Post numbers of all the propositional normal modal logics which are standard in the literature; moreover it allows us to solve the problem of finding, for each cardinal number (Y such that 1s (YG 2%, the cardinality of the set of logics whose Post number is CY. Finally, we investigate the lattice A’(K) of the modal logics which are axiomatizable by means of formulas without propositional variables; among other things, we show that A’(K) is not dually strongly atomic and contains a sublattice isomorphic to (R , S) . All these results strengthen the impression of the great complexity of the lattice A(K) of all normal modal logics.