In this paper an alternative Skolemization method is introduced that, for a large class of formulas, is sound and complete with respect to intuitionistic logic. This class extends the class of formulas for which standard Skolemization is sound and complete and includes all formulas in which all strong quantifiers are existential. The method makes use of an existence predicate first introduced by Dana Scott.

Those formulas which are valid in every Kripke model having constant domain whose base is the ordered set R of real numbers (or, the ordered set Q of rational numbers) are characterized syntactically.

An intermediate predicate logicS + n (n>0) is introduced and investigated. First, a sequent calculusGS n is introduced, which is shown to be equivalent toS + n and for which the cut elimination theorem holds. In § 2, it will be shown thatS + n is characterized by the class of all linear Kripke frames of the heightn.