In this work we extend results from ``Graded modalities. I-III'' [see the review above; Part III is to appear] about propositional calculi with graded modalities to the predicative level. Our semantics is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We have not considered semantics with world-relative domains because they do not present any new difficulties with respect to classical case. Our language will have constant and function symbols, but they will have a rigid interpretation. In this instance the terms will be considered ``rigid designators'', that is, their interpretations will be the same in all possible worlds. Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context essentially the same problems would arise as those which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non-denumerable ones is not conceptually problematic.