Boole's algebra isn't Boolean algebra

Boolean algebra governs the logical calculus of observations. But in many applications it doesn't make sense to consider all pairs of observations to be compatible. Rather, the fundamental structures originally studied by Boole are piecewise Boolean algebras. I will present a more practical description of these somewhat unwieldy structures in terms of domains, and discuss examples in computability theory and especially quantum theory, in a way that is accessible to a broad audience of computer scientists, mathematicians and logicians.