In the paper it is argued that for some linguistic phenomena, current discourse representation structures are insufficiently finegrained, both from the perspective of serving as representation in NLP and from a truth conditional perspective. One such semantic phenomenon is uniqueness. It is demonstrated that certain elements are forced to have a unique interpretation, from a certain point in discourse onwards. This could be viewed as the semantic counterpart of surface order. Although it has always been acknowledged that the left-to-right order of constituents influences the meaning of an utterance, it is, for example, not reflected in standard Discourse Representation Theory ([Kamp, 1981]). In the paper, an alternative representation for unique constituents will be proposed, resulting in asymmetry of certain conjoined conditions in a DRS-representation.