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This is the principle of Liberty: the physicist can use whatever framework he chooses when describing a system

However, this raises the question whether it would not be more straightforward to require only ##\operatorname{Re}D(\alpha,\beta)=0## for all ##\alpha\neq \beta##, which would be enough to ensure that applicability of classical logic and probability theory within a single framework. This is not done, because of the observation by Lajos Diós that the (canonical) composition of two statistically independent quantum systems ##A## and ##B## would not necessarily satisfy that condition (for given frameworks for ##A## and ##B## satisfying that condition). But the composition of such statistically independent quantum systems is not otherwise discussed in any substantial way in typical presentations of consistent histories.

Doesn't this make consistent histories even more incomplete? If composition of statistically independent quantum systems is important enough to justify using the stronger consistency condition, then it should also be important enough to warrant some substantial discussions of their role in that quantum logic.