Fuzzy partial propositional logic, recently proposed by Behounek and Novak, provides a simple framework for dealing with graded propositions with possibly undefined truth degrees. A natural next step is an extension to predicate logic and investigation of basic notions of the theory of fuzzy partial sets. I will present first steps in this direction (joint work with Martina Dankova, work in progress). In particular, I will discuss basic partial fuzzy quantifiers, elementary relations and operations on fuzzy partial sets, and their representation in terms of totally defined fuzzy sets.