 Healey (1989), who was among the first to make use of the biorthogonal decomposition theorem for interpretational purposes, has proposed a somewhat different modal scheme. After using the biorthogonal decomposition theorem, Healey modifies the set of possible properties in order to fulfill a variety of additional desiderata, e.g. pertaining to the relationship between composite systems and their subsystems. The structure of definite-valued properties that emerges from these conditions is rather complicated. Some progress has been made since Healey’s book was published (see Reeder and Clifton 1995) but, in general, it remains difficult to see what the set of definite-valued properties is according to this approach.