Following a positively quantified statement such as A few of the children sang the chorus, a plural pronoun is likely to refer to the set of children who sang (the reference set). Negative natural language quantifiers (NLQs) such as few or not many on the other hand are more likely to be followed by reference to the complement set of children who did not sing. According to the Presupposition-denial account of negative NLQs, the complement set is available for pronominal reference following these expressions because they imply a shortfall between the amount denoted and a presupposed larger amount. Focus on the shortfall set is effectively focus on the complement set. Previous support for this account is largely based on a series of experiments which show that complement set focus is also possible following positive NLQs, if a previously mentioned character expects a larger amount thereby creating a shortfall between the character s expectations and the amount denoted by the NLQ. It is not clear however whether the shortfall implied by a negative NLQ must be based on expectation per se, or whether the NLQ-based implication is more general. We report three experiments which show that a shortfall can also be created between an NLQ and a character s desire for a particular quantity. Our results suggest that the implication of negative NLQs that a larger amount is denied need not be based on expectation but may be less specific.