A result on the completeness of superintuitionistic logics (Q1059064)

Superintuitionistic logics which are determined by tight Kripke models (i.e. by posets containing no infinite antichains with intuitionistic valuation on them) are considered. It is proved with essential use of the author's result [Algebra Logika 20, 165-182 (1981; Zbl 0486.03015); Lemma 3.3] that each such logic is countably modellable. (For some contrast let us note that \textit{V. B. Shekhtman} [Studies in nonclassical logics and formal systems, 287-299 (Russian) (''Nauka'', Moscow, 1983)] has constructed a poset with an antichain of cardinality \(2^{\aleph_ 0}\), whose logic is not determined by any countable poset.) As corollaries we receive \textit{A. V. Kuznetsov}'s result [Proc. Int. Congr. Math., Vancouver 1974, Vol. 1, 243-249 (1975; Zbl 0342.02015)] that logics which are determined by (pseudoboolean) algebras with descending chain condition are countably modellable and \textit{S. K. Sobolev}'s result [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 963-986 (1977; Zbl 0368.02062)] that logics which are determined by algebras containing no sublattice \(2^ n\) (n is natural) are the same.