Abstract
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 ${p}_{\mathrm{Hardy}}\ensuremath{\approx}0.167$. 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-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky inequality. We apply the same method to build a paradox based on the Clauser-Horne-Shimony-Holt inequality.
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