In this note I give an account of the relations between different notions of weakened quantification, which can be defined in intuitionistic logic. The reason for publishing it is the fact that the last gap in our knowledge of these relations has been filled in an article by Kleene [5].

For the intuitionistic calculus of propositions I refer to my article [2]; besides the rules of substitution and of inference I need only a few formulas from this calculus, which I shall repeat below. The simplest way of formalizing the intuitionistic predicate calculus is obtained by adjoining to the calculus of propositions the symbols, rules, and axioms as given by Hilbert and Ackermann [4], Chapter 3, §§4-5, pp. 53-57. (Reference is made to this book as HA.) I repeat only the axioms (e) and (f) and the rule γ.