Hardy's non-locality paradox is a proof without inequalities showing thatcertain non-local correlations violate local realism. It is `possibilistic' inthe sense that one only distinguishes between possible outcomes (positiveprobability) and impossible outcomes (zero probability). Here we show thatHardy's paradox is quite universal: in any (2,2,l) or (2,k,2) Bell scenario,the occurence of Hardy's paradox is a necessary and sufficient condition forpossibilistic non-locality. In particular, it subsumes all ladder paradoxes.This universality of Hardy's paradox is not true more generally: we find a new`proof without inequalities' in the (2,3,3) scenario that can witnessnon-locality even for correlations that do not display the Hardy paradox. Wediscuss the ramifications of our results for the computational complexity ofrecognising possibilistic non-locality.
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