  From behavioral sciences to biology to quantum mechanics, one encounters
situations where (i) a system outputs several random variables in response to
several inputs, (ii) for each of these responses only some of the inputs may
"directly" influence them, but (iii) other inputs provide a "context" for this
response by influencing its probabilistic relations to other responses. These
contextual influences are very different, say, in classical kinetic theory and
in the entanglement paradigm of quantum mechanics, which are traditionally
interpreted as representing different forms of physical determinism. One can
mathematically construct systems with other types of contextuality, whether or
not empirically realizable: those that form special cases of the classical
type, those that fall between the classical and quantum ones, and those that
violate the quantum type. We show how one can quantify and classify all
logically possible contextual influences by studying various sets of
probabilistic couplings, i.e., sets of joint distributions imposed on random
outputs recorded at different (mutually incompatible) values of inputs.
