This is really what the UDA argument forces us to do for getting the physical. But still the machine cannot know it, so that the machine perspective is quite different. That is, not only we get probabilities, but we get quantum probabilities. Unfortunately, in the process, we loose the necessitation rule. So we cannot use directly the result in the litterature for isolating an arithmetical interpretation of quantum logic.
Open problem remains.
That is close to the difference between experiment and experience. Now I have read Ziegler page on quantum logic. It is very good and Ziegler explains the relation between quantum logic and quantum probability in a rather clear way. Only modal quantum logic is missing.