What is the syntactic category of 'there are'?
I have a background in philosophy and logic, and I don't know much about syntax. I guess 'there are' is not a determiner. I would simply call 'there are' a quantifier, and something like 'there are Fs' a quantifier phrase, but I want to know the name they receive in the literature on Syntax.
Solution 1:
You're right -- "there are" is not a determiner. In, e.g., "There are two snails here." "there" is a noun phrase subject and "are" is a finite auxiliary verb. The term 'quantifier' is sometimes used for "two", "some", and other words that concern quantity.
I don't see a lot of difference between the terminologies of logic and linguistics, but in the example you've chosen, the meaning is expressed in a different way. In translating back and forth from predicate logic to English, we have to change between a noun phrase "there" in English and a quantifier "there (is)" in logic, but that doesn't make the two the same, of course. They correspond.
Solution 2:
My first guess is that this can be analyzed as a locative inversion construction, where the order of subject and verb's complement (usually a adverb with locative meaning) can be inverted.
For example, you can say the rain falls down (non-inverted form) or down falls the rain (inverted form).
You would then analyze a sentence like There are three dogs on the grass by considering the non-inverted homologue Three dogs are there in the grass. As with other syntactic alternations in English, the constituents keep the same lexical categories (noun phrase and adverbial phrase), and the same semantic roles (copula subject is three dogs and copula complement is there in the grass), but the arguments have "demoted" syntactic roles: the subject in the non-inverted version becomes the complement in the inverted version, while the locative complement becomes the adjunct in the inverted version. The only difference in meaning is that there is bleached of its deictic meaning in the inverted version.
So the adverb "there" has no semantic role, it is an adverb, and the subject of the sentence. Subjects with no semantic role are called expletive subjects, so that is the most informative label you could give "there" when discussing its grammatical role. Adverbs normally cannot be the subject of a sentence (one exception being the locative inversion construction), so that's why it's important to note that this is a locative inversion construction.