||We present a logical type of proof of contextuality for a two-qubit state. We formulate a paradox that cannot be used by a two-qubit system with local measurements to test contextuality while it is possible by using entanglement measurements. With our scheme we achieve P-Hardy approximate to 0.167. Our approach uses graph theory and the exclusivity principle to give an interpretation of a logical type of proof of quantum correlations. We review the Hardy paradox and find a connection to the Klyachko-Can-Binicioglu-Shumovsky inequality. We apply the same method to build a paradox based on the Clauser-Horne-Shimony-Holt inequality.