Using "the" vs "a"

Question

I'm debating with a friend about using "the" and "a" in a sentence. The sentence in question is:

This person is famous for wearing a hat.

I personally prefer using "a" like I used above, but my friend is certain that the "a" should be replaced with "the". The person is Nefertiti, by the way and we are talking about Nefertiti's bust. I think that "a" should be used since "the" implies that I have mentioned it before, but of course I may be wrong.

Edit: What if I was playing a "guessing" game where I'm giving hints, does this change anything?

Answer 1

In contrast to the answer by Recurse,

This case would be an irregular usage of the

Not really irregular at all; the just signifies that the speaker has a specific instance of the set in their mind (or a specific subset, in plural). "The current US president" is a specific instance of the set of US presidents; "the soup of the day" implies there is exactly one soup being advertised as "soup of the day" in the salient context. "I'll have the coke" means the one single coke you see in your friend's fridge when everything else is beer. None of these need a prior mention, just a reasonable expectation that the particular single instance would be recognised by the listener.

Thus... I'd judge "Nefertiti is famous for the hat" as strange: if you need to tell me what she's famous for, then I likely don't already know it, and couldn't recognise which hat you were referring to.

"Nefertiti is famous for the hat she always wore" is okay, because I now have the specific instance of a hat in my mind: the (single) one that she always wore, as identified by those words.

"Nefertiti is famous for her hat" is also okay, with a similar meaning.

I would think all of these considerations apply to the Jeopardy-like guessing game as well, and I would not think "This person is famous for a hat" a good hint.

(Completely irrelevantly, if I heard "This person is famous for a hat" on Jeopardy, my answer would likely be "What is the Sorting Hat?", because some hats have personalities, too, and get famous despite being just a piece of clothing.)

But rephrase a bit, and things change. Without the pragmatic requirement of "famous": "Here's Nefertiti's bust, of course she's wearing the hat." is perfectly fine, as this sentence presupposes that the listener would know which hat is relevant to Nefertiti.

Also, the OP's "This person is famous for wearing a hat" is perfectly fine, as we do not need the listener to be able to identify which hat we are talking about.

EDIT: CGEL, Chapter 5, Section 6.1 (emphasis mine):

In general, use of the definite article presupposes the existence of the entity, set or quantity that the addressee is expected to be able to identify.

Answer 2

Of course, her hat would've been the preferred word, but in answer to the question,

This case would be an irregular usage of the, as despite the hat not being mentioned in the past, the hat isn't just an unknown one. It is well documented and famously known.

Nefertiti is famous for the hat

sounds more accurate than

Nefertiti is famous for a hat.

Similar to how one says

This church is well-known for the clock tower

rather than

This church is well-known for a clock tower