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 occurrence 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.