Standard semantic theories predict that non-deictic readings for complex demonstratives should be much more widely available than they in fact are. If such readings are the result of a lexical ambiguity, as Kaplan (in: Almog, Perry, Wettstein (eds) Themes from Kaplan, Oxford University Press, Oxford, 1977) and others suggest, we should expect them to be available wherever a definite description can be used. The same prediction follows from ‘hidden argument’ theories like the ones described by King (Complex Demonstratives: a Quantificational Account, MIT Press, Cambridge, 2001) and Elbourne (Situations and Individuals, MIT Press, Cambridge, 2005). Wolter (That’s That; the Semantics and Pragmatics of Demonstrative Noun Phrases. Ph.D. thesis, University of California at Santa Cruz, 2006), however, has shown that complex demonstratives admit non-deictic interpretations only when a precise set of structural constrains are met. In this paper, I argue that Wolter’s results, properly understood, upend the philosophical status quo. They fatally undermine the ambiguity theory and demand a fundamental rethinking of the hidden argument approach.