"I'm a friend of Bill's." You're a friend of Bill's what?

I don’t understand this construction but I see it a lot.

“I’m a friend of Bill.” Possessive formed by “of”.

Makes sense.

“I’m one of Bill’s friends.”

Possessive formed by noun plus apostrophe “a”, another possessive form.

Again, makes perfect sense.

“I’m a friend of Bill’s.”

Makes no sense. "Bill’s is a possessive. You’re a friend of Bill’s what?

Bill’s acquaintance?

how is it any different than, “I’m a friend of his”?

mc

ps: my dad is a fiend of Bill’s

“I’m a friend of Bill’s” can be a way of saying you’re a member of Alcoholics Anonymous, as in Bill W., one of the authors of the Big Book.

It doesn’t matter if it “makes sense” logically, that’s not how language works - it’s perfectly idiomatic, and probably has been for centuries.

The choice is between “I’m a friend of Bill” and “I’m a friend of Bill’s”, or even “I’m Bill’s friend”.

We should continue to discuss this. But either way “Bill’s” modifies “friend” - there is no Bill’s what. Perhaps it would be simpler to call this genitive rather than possessive.

ETA there is apparently an explanation of this sort of double genitive in the OED, so it has a history:

In this particular construction, “Bill’s” is not simply a possessive (although the form is identical). The " 's" is part of a double genitive (or double possessive) construction, attested in English since the 1300s.

While it might look a little unusual and seem illogical, “a friend of Bill’s” is perfectly cromulent English. Zombywoof’s point is well taken.

While true, it’s still fun to try and figure out if there’s any unwritten rule behind it, or if it’s just a one-off anomaly.

It seems to me that we use the “of [possessive noun]” form a lot. We say “That is a favorite quote of John’s,” not “…quote of John.” In fact, it seems we use the possessive case in nearly all instances of possession.

The times when we don’t use it are when it’s not really possession, like with “king of Spain.” We can technically say “Spain’s King,” but it’s not like Spain owns the king.

At least, that’s what I can think of. Can anyone else think of examples of when we use “of [possessive noun]” vs. “of [objective noun]”?

Here is it:

The first quote is from 1300, so this isn’t a recent innovation.

From a play of Bill’s:

If they say they’re a “friend of Bill W.”, they’re a member of Alcoholics Anonymous.

If they say they are a “friend of Bill W.” then they love Calvin and Hobbes.

“This is a favorite quote of John” and “This is a favorite quote of John’s” mean two different things to me. The first means that it’s a favorite quote of something John said. The second means it’s a quote that John favors, from some other source.

Sometimes it helps to sketch out the parts of the sentence which are implied or unsaid.
“I’m a friend of Bill’s”
could be expanded into
“I’m a friend [, a member of the set] of Bill’s [friends]”.

YMMV.

Asa linkfrom the Wiki article explains, there are some instances where the construction is the also only one that is unambiguous:

So are you actually confused? Like you literally don’t know what it means when somebody says that? I daresay you do know what it means, hence no confusion, hence a pointless OP.

Just who the hell is Bill and why is he so popular?

I thought OP was pretty clear that he didn’t understand the implicit grammatical construction, and he was asking for an explanation; and he was not making the occasional prescriptivist claim that an apparently illogical but idiomatic usage was literally confusing.

Hence a pointlessly rude comment.

Maybe I misread it then, but it seems like a question that has its own answer then. It’s like asking why bridges span water. Because that’s what bridges do? What kind of answer is he hoping for? You see, sometimes people say this because they say it that way. Question answered.

It makes me think of a line from The Young Ones. A couple of girls had been invited to the guys’ squalid, smelly flat.

Girl one: “It smells like a gent’s”.
Girl two: “A gent’s what?”