Maybe, I'm thinking it is more than just the words both and neither:
Some, somebody, someone, something, any, anybody, anyone, anything, either, several, enough, many, most, much, few, least and little are the indefinite pronouns. They all have indefinite antecedents even when the context is completely understood.
"Somebody was at the party."
(Who is somebody? What information might one find in the discourse to explain who somebody is? The way I see it is unless this is a reference to someone named Somebody, this pronoun is truely indefinite)
Everybody, everyone, everything, no, none, nobody, no-one, nothing, all, both, each and neither are something else. They all have definite antecedents either as the whole of something(pun intended), or definable aspects in the case of the words both and each. The only exceptions are when the words are misused.
"Everybody was at the party."
(if the WHOLE which constitutes "everybody" was not previously mentioned in the discourse and every single human being was not present at the party, then the word "everybody" was misused.)
A poacher is walking down the road with a bag over his shoulder. A policeman asks what he's carrying and he tells him he has some piglets.
Policeman "If I guess how many you have in there, will you give me one of them?"
The poacher replies "If you guess how many I have, I'll give you both of them!"
(here, the poacher has two PIGLETS, and that is definite, not indefinite, i.e. the antecedent(what the pronoun refers to) is clear. Both is definitely indicative of two of something in particular, if not, then that is indicative of an incomplete thought)