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This paper supports the view that the distinction between strong NPs and weak NPs rests on the presence or the absence of the uniqueness component. When a uniquely identifiable set is presupposed, the NP is perceived as strong, producing the proportional reading which indicates a certain proportion out of a unique set, but, when such a presupposition is absent, the NP is received as weak, yielding the cardinal reading that refers to some quantity that is not based on a unique set. Drawing the distinction between strong/weak NPs on the basis of a uniquely identifiable set, we can naturally address why the existential there-be construction refuses strong NPs. The cognitive status related to the existential there-be construction is type identifiability which is placed much lower than unique identifiability, the status of which strong NPs must satisfy. In terms of the maxim of quantity, type identifiability that is weaker in scale of the givenness hierarchy than unique identifiability implies that it is not uniquely identifiable and this implicature of the existential there-be construction causes incompatibility with strong NPs.