  We present a logical type of proof of contextuality for a two-qubit state. We
formulate a paradox that cannot be verified by a two-qubit system with local
measurements while it is possible by using entanglement measurements. With our
scheme we achieve $p_{\rm Hardy} \approx 0.167$, which is the highest
probability obtained for a system of similar dimension. Our approach uses graph
theory and the global exclusivity principle to give an interpretation of
logical type of proofs of quantum correlations. We review the Hardy paradox and
find connection to the KCBS inequality. We apply the same method to build a
paradox based the CHSH inequality.
