  Hardy's non-locality paradox is a proof without inequalities showing that
certain non-local correlations violate local realism. It is `possibilistic' in
the sense that one only distinguishes between possible outcomes (positive
probability) and impossible outcomes (zero probability). Here we show that
Hardy'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 for
possibilistic 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 witness
non-locality even for correlations that do not display the Hardy paradox. We
discuss the ramifications of our results for the computational complexity of
recognising possibilistic non-locality.
