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Which is correct:

I have a doctor appointment this afternoon.
I have a doctor's appointment this afternoon.

My preference is "doctor appointment," and my reasoning is because no one ever says, "I have a dentist's appointment." And yet, I hear most people use the latter.
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It's "doctor's appointment".

I like your reasoning, but in fact "dentist's appointment" is also correct. Moreover, it isn't entirely true that no-one ever says it. "Dentist's appointment" is said in England, for example. Your reasoning could be correct in your part of the world, of course.

Rommie
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Another way to say this is
"an appointment at the doctor's (surgery)"
perhaps this is why we say a "doctor's appointment" - doctor's [surgery] appointment.

Note; we say "an appointment AT the doctor's" and "an appointment WITH the doctor"

P.S. Dentists are a little more easily dealt with - we can say a 'dental appointment'Emotion: big smile
Cheers

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Thanks, Rommie, for the reply; but I'm still confused. The apostrophe makes it possessive. Whose appointment is it: mine or the doctor's? I must be missing something.
im kind of confuse as well
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 Mike in Japan's reply was promoted to an answer.