An “observable” is just some measurable property of a system. In other words it corresponds to some experimental apparatus which interacts both with the system under study and with us so as to provide us with a signal which we interpret as information about the measured value of the property, (or in reverse allows us to adjust its “settings” so that after interacting with it the system has some specified range of property values).

Often, the interaction with us involves our reading some value off a scale. And normally, when reading a measured value off a scale, we proceed by counting intervals of a certain unit size on the scale, then after the last full unit mark has been counted looking at fractional units and so on on finer and finer scales until we reach the sensitivity limit or experimental error bound of the instrument. And so any practical measurement can be reduced to a sequence of yes-or-no questions (or equivalently true-false propositions).

So it might make sense to start by just considering first just those simple observables with just two possible values, say 1 for ‘yes’ or ‘true’, and 0 for ‘no’ or ‘false’.

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