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This paper provides an analysis of Czech bare vs. demonstrative NPs and in particular of their referential uses involving situational uniqueness. Contrary to the traditional view that bare NPs correlate with uniqueness and demonstrative NPs with anaphoricity, I argue that the relevant classification involves two types of uniqueness: inherent uniqueness, correlated with bare NPs, and accidental uniqueness, correlated with demonstrative NPs. The notions of inherent and accidental uniqueness are formalized using situation and modal semantics. An extension to generic, anaphoric, and non-specific NPs is proposed.