We distinguish three different readings of the intuitionistic notions of validity, soundness, and completeness with respect to the quantification occurring in the notion of validity, and we establish certain relations between the different readings. For each of the meta-logicalnotions considered we suggest that the "most natural" reading (which is not the same for all cases) is precisely the one which is required by the recent intuitionistic completeness theorems for IPC.

Intuitionistically. a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights?in particular, the so-called continuity principles?which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer?s reconstruction of mathematics. The predicate ?α (...) is lawless? is not acceptable, the lawless sequences do not form a well-circumscribed intuitionistic set, and quantification over lawless sequences does not make sense. (shrink)