The representation and determinable structure of quantum properties

Orthodox quantum theory tells us that properties of quantum systems are represented by self-adjoint operators, and that two properties are incompatible just in case their respective operators do not commute. We present a puzzle for this orthodoxy, pinpointing the exact assumptions at play. Our solution to the puzzle specifically challenges the assumption that non-commuting operators represent incompatible properties. Instead, they represent incompatible levels of specification of determinates for a single determinable. This solution yields insight into the nature of so-called quantum indeterminacy and demonstrates a new and fruitful application of the determinable-determinate relation in quantum theory.