This article presents a paradox regarding the presuppositional properties of the cleft sentences and proposes a solution for it. On the one hand, Figueiredo and Silva & Menuzzi (2014, 2015) demonstrate that clefts are favored by a presupposition of unicity when used as answers to WH questions, which suggests that they are characterized by such a presupposition (cf. WEDGWOOD, 2005, WEDGWOOD et al. 2006, MENUZZI, 2012, BURING and KRIZ, 2013). On the other hand, Moretto (2016) and Moretto & Menuzzi (2016) show that certain clefts are compatible with only and therefore with the assertion of exhaustivity; in this case, they are compatible only with a weaker presupposition, of existence (cf. HORN, 1981, 1996). The explanation I propose for this paradox is this: an assertion of identity provides a complete answer only when there is a contextual presupposition of unicity - which is not inherent property of WH questions (contra POLLARD & YASAVUL, to appear).