Singular definite descriptions like "the hat" generally only make sense in contexts that provide enough information for hearers to deduce which object, in this case which hat, the speaker intends to refer to. As a result, without relevant preceding discourse such descriptions are usually unacceptable unless the context of utterance provides a unique potential witness for the description, say, a single salient hat. There are, however, systematic exceptions to this. Consider the letters in the word "starters", for instance, and imagine I ask you to look at the "the 't' after the 'r'". Even though "the 'r'" should be incoherent (since there are two 'r's), it is clear that "the 't' after the 'r'" refers to the second of the two 't's, and thus the first of the two 'r's. Similarly, "the book on the shelf" (as opposed to the ones on the table), "the woman in the painting" (as opposed to the women in real life), and "the guy who threw the beer at the lead singer" (as opposed to the guys who didn't throw any beers) may all be felicitous in contexts with multiple shelves, multiple paintings, and multiple beers. In each example, it is as if the two definite descriptions in fact define each other: e.g., the book, y, on a shelf, x, where x is the shelf y is on.
In view of examples like these, I argue that definite descriptions are interpreted in two steps, and that these two steps don't always happen in immediate succession. In the first step, all possible referents for the description are collected; in the second, they are counted. If the number of potential referents is one, all is well; otherwise, the description is ill-formed. What is interesting about the descriptions above is that these two processes take effect at different points in the respective semantic computations. Technically, I'll say the scopes of these definite articles are split. This allows descriptions to interact with one another in ways that have not previously been predicted to be possible. I'll go on to show that this idea sheds light on an old ambiguity in the interpretation of superlatives, and suggest that a handful of quantificational adjectives might actually be piggybacking on the "delayed" cardinality test of the definite determiner. Finally, I step back and demonstrate how we might maintain traditional notions of semantic compositionality, according to which words are interpreted in one place at one time, in light of the bipartite meaning of 'the'.