Here we present the most general framework for $n$-particle Hardys paradoxes, which include Hardys original one and Cerecedas extension as special cases. Remarkably, for any $nge 3$ we demonstrate that there always exist generalized paradoxes (with the success probability as high as $1/2^{n-1}$) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardys inequalities, which enable us to detect Bells nonlocality for more quantum states.
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