Modal interpretations constitute a particular approach to associating dynamical variables with physical systems in quantum mechanics. Given the “quantum logical” constraints that are typically adopted by such interpretations, only certain sets of variables can be taken to be simultaneously definite-valued, and only certain sets of values can be ascribed to these variables at a given time. Moreover, each allowable set of variables and values can be uniquely specified by a single “core” projector in the Hilbert space associated with the system. (...) In general, the core projector can be one of several possibilities at a given time. In most previous modal interpretations, the different possible core projectors have formed an orthogonal set. This paper investigates the possibility of adopting a non-orthogonal set. It is demonstrated that such non-orthogonality is required if measurements for which the outcome can be predicted with probability 1 are to reveal the pre-existing value of the variable measured, an assumption which has traditionally constituted a strong motivation for the modal approach. The existing framework for modal interpretations is generalized to explicitly accommodate non-orthogonal core projectors. (shrink)