In hindsight, the debate about presupposition following Frege’s discovery that the referential function of names and definite descriptions depended on the fulfillment of an existence and a uniqueness condition was curiously limited for a very long time. On the one hand, it was only in the 1960s that linguists began to take an interest and showed that presupposition was an allpervasive phenomenon far beyond this philosophers’ pet definite descriptions. And on the other hand, and this is our real concern, it is now only too obvious that the uniqueness condition is too restrictive to be applicable to the general case. An utterance of “The cat is on the mat” should not imply that there is only one cat and one mat in the whole world. The obvious move is to limit the uniqueness condition to some notion of utterance context.