Remarks on unsharp quantum observables, objectification, and modal interpretations

In this contribution I wish to pose a question that emerged from discussions with Dennis Dieks regarding my talk at the Utrecht Workshop. The question is whether a variant of a modal interpretation is conceivable that could accommodate property ascriptions associated with nonorthogonal resolutions of the unity and nonorthogonal families of relative states as they occur in imperfect or genuinely unsharp measurements. To explain this question I will review a recent formulation of the quantum measurement problem in the form of an insolubility theorem that incorporates the case of unsharp object observables as well as certain types of unsharp pointers. In addition to demonstrating the necessity for some modification of quantum mechanics, this allows me to specify the logical position of the modal interpretations as a resolution to the measurement problem and to indicate why I think their current versions are not yet capable of dealing adequately with unsharp quantum observables. The technical tools that will have been explained along this line of reasoning will finally serve to make precise the notion of (unsharp) value ascription that I would find desirable for a modal interpretation to