By "expresses the category D" I mean, preferably, that there is solid evidence/argumentation for a given morpheme to be analyzed as overtly heading a Determiner projection. I would limit such expressions in this case to those communicating meaning like the definite article the, but not demonstratives like that.
By not displaying inverse scope, I mean that an equivalent translation of (1) would support the interpretation in (2) but not in (3). Perhaps the same for (4) with the interpretations paraphrased in (5) and (6):
- Some student saw every professor
- There is a (single) student (say, John) such that he saw every professor
- For every professor x, there is a (different) student that saw x
- There is a solution to every problem
- There is a (single) solution such that it applies to every problem
- For every problem x, there is a (different) solution for x