  In the Contextuality-by-Default theory random variables representing
measurement outcomes are labeled contextually, i.e., not only by what they
measure but also under what conditions (in what contexts) the measurements are
made, including but not reducible to what other measurements are made
"together" with the given one. We propose in this paper that the quantum
observables generating these random variables be labeled contextually as well,
making the sets of the observables in different contexts disjoint. A quantum
observable is defined as a pair consisting of the observable's label and a
self-adjoint operator in a Hilbert space. If a system is consistently connected
(i.e., obeys "non-disturbance," "non-invasivenes," or "non-signaling"
conditions), the observables measuring the same property in different contexts
have the same operator. A set of random variables possessing a joint
distribution is represented by commuting observables. The reverse, however, is
not true: random variables from different contexts do not have a joint
distribution irrespective of whether the corresponding observables commute. We
illustrate this view of observables by deriving the Tsirelson bound for
consistently connected cyclic systems of rank 3.
