Some results on the Kripke sheaf semantics for super-intuitionistic predicate logics
``Kripke sheaf semantics'' (which is equivalent to the semantics of predicate Kripke frames with equality) was introduced by the reviewer and \textit{D. Skvortsov} [``Semantics of nonclassical first-order predicate logics'', Mathematical logic, Proc. Summer Sch. Conf. Ded. 90th Anniv. Arend Heyting, Chaika/Bulg. 1988, 105-116 (1990)]. As was stated there, Kripke sheaf semantics is richer than standard Kripke semantics already for logics without equality. The present paper proves a stronger result; namely, it gives simple examples of finitely axiomatizable superintuitionistic predicate logics without equality, which ae complete in Kripke sheaf semantics but incomplete in standard Kripke semantics. In particular, every logic \(H+J+D+\neg\neg\bigl(\exists x p(x)\supset\forall x p(x)\bigr)\), with \(J\) tabular, has these properties (\(H\) denotes Heyting predicate logic, \(D\) is the axiom of constant domains). The paper also contains a correction of an error in the author's earlier paper [ibid. 49, 289-306 (1990; Zbl 0723.03008)].
superintuitionistic predicate logics without equality
scientific article; zbMATH DE number 165914
scientific article