  Contextuality was originally defined only for consistently connected systems
of random variables (those without disturbance/signaling).
Contextuality-by-Default theory (CbD) offers an extension of the notion of
contextuality to inconsistently connected systems (those with disturbance), by
defining it in terms of the systems' couplings subject to certain constraints.
Such extensions are sometimes met with skepticism. We pose the question of
whether it is possible to develop a set of substantive requirements (i.e.,
those addressing a notion itself rather than its presentation form) such that
(1) for any consistently connected system these requirements are satisfied, but
(2) they are violated for some inconsistently connected systems. We show that
no such set of requirements is possible, not only for CbD but for all possible
CbD-like extensions of contextuality. This follows from the fact that any
extended contextuality theory \T is contextually equivalent to a theory \T' in
which all systems are consistently connected. The contextual equivalence means
the following: there is a bijective correspondence between the systems in \T
and \T' such that the corresponding systems in \T and \T' are, in a
well-defined sense, mere reformulations of each other, and they are contextual
or noncontextual together.
