Restricted Positive Quantification Is Not Elementary

We show that a restricted variant of constructive predicate logic with positive (covariant) quantification is of super-elementary complexity. The restriction is to limit the number of eigenvariables used in quantifier introductions rules to a reasonably usable level. This construction suggests that the known non-elementary decision algorithms for positive logic may actually be best possible. 1998 ACM Subject Classification F.4.1 Mathematical Logic, I.2.3 Deduction and Theorem Proving