  Contextuality means non-existence of a joint distribution for random
variables recorded under mutually incompatible conditions, subject to certain
constraints imposed on how the identity of these variables may change across
these conditions. In simple quantum systems contextuality is indicated by
violations of Bell-type or Leggett-Garg-type inequalities. These inequalities,
however, are predicated on the assumption of no-signaling, defined as
invariance of the distributions of measurement results with respect to other
(e.g., earlier in time) measurements' settings. Signaling makes the
inequalities inapplicable: a non-signaling system with any degree of
contextuality, however high, loses any relation to this concept as soon as it
exhibits any degree of signaling, however small. This is unsatisfactory. We
describe a principled way of defining and measuring contextuality in arbitrary
systems with random outputs, whether signaling is absent or present.

Quantum Physics;Probability