  We analyze the structure of the so called non-signaling theories respecting
relativistic causality but allowing correlations violating bounds imposed by
quantum mechanics such as CHSH inequality. We discuss relations among such
theories, quantum mechanics, and classical physics. In particular we
reconstruct the probability theory adequate for the simplest instance of a
non-signaling theory, the two non-signaling boxes world, and exhibit its
differences in comparison with classical and quantum probabilities. We show
that the question whether such a theory can be treated as a kind of
"generalization" of the quantum theory of the two-qubit system cannot be
answered positively. Some of its features put it closer to the quantum world,
for example measurements must be destructive, on the other hand the Heisenberg
uncertainty relations are not satisfied. Another interesting property
contrasting it from quantum mechanics is that the subset of "classically
correlated states," i.e.\ the states with only classical correlations, does not
reproduce the classical world of two two-state systems. Our results establish a
new link between quantum information theory and the well-developed theory of
quantum logics and can shed new light on the problem why quantum mechanics is
distinguished among non-signaling theories.
