How can we distinguish complements from specifiers?

Question

I know this is a problem in the history of Linguistics. The most famous example I can think of is the Determiner Phrase vs. Noun Phrase debates. I'm trying to figure out, if you have evidence that Phrase A always shows up only once and directly before Phrase B, how can you identify whether Phrase B is a complement of Phrase A (1) or whether Phrase A exists in a specifier relation to Phrase B (2)? Of course, the same can be said in reverse, where Phrase A could be the complement of Phrase B, or Phrase B could be the specifier of phrase A, but those symmetrical alternate examples can be satisfied by answers to their symmetrical counterparts.

(1)

(2)

In both cases, the lexical head of A has the strength to agree with B, so any sort of agreement tests can't help. Nor can any c-command tests, since A c-commands B in both scenarios. Perhaps I could use a c-command test to see if B C-commands A in some way. Still, I'm not sure of a good test to distinguish these two structures.

Answer 1

It may be too difficult to answer this question at this level of abstractness and generality. Depending on the specific words and the specific categories in an example, you might be able to find different ways of testing.

For example, a case in point that I once came across is the German phrase nah an der Küste (close to the shore). The question here is whether we have a PP "an der Küste" (by the coast), modified by "nah" (close, near), or whether we have a PP complement to an adjective "nah" (near to it). The solution is that it has to be the first case, because otherwise you would be able to use "nah" as the head of an attribute to a noun; "?? ein an der Küste nahes Haus" (a house near the coast). So here, the possible distribution of an adjective gives the clue, but I don't see a more general phrase structure test.