Is "Can I have a cup of coffee?" polite? What if I ask a store employee, "Can I have something?"
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To Sam's comment, the more pedantically-minded English speakers will state that beginning a question with Can I is to question one's own ability rather than to make a request. To satisfy those people, you're better off using may:
If you want to be polite about it, add a please:
If you want to be especially considerate, like when you need to get some coffee from a person who's had a bad day, make the hassle of getting the coffee the focus of the request instead of the coffee itself:
Finally, if you're familiar with the person, you can just skip the question altogether without offense:
It's common to assume a person who's looking for coffee is suffering from caffeine withdrawal, allowing for a good deal of leniency when considering how polite the request for coffee is.
Can has an abruptness to it that could come off sounding demanding.
would be better, and it never hurts to say please:
Yes, it is polite. We all use the modal verbs for polite requests. 'can' is not as polite as 'could' or 'would' but it's commonly used in situations like this. 'will' is even possible. The nuances of each can depend upon a number of factors; familiarity, the degree of deference one wants to express, ... .
The notion that 'can' means "ability" in these polite request situations is the result of an old, poorly thought out prescription. Corpus studies show that 'may' is rarely used adult to adult, compared to 'can/could' for these types of polite requests.
The meaning of most polite requests using 'can/could' is "is it possible ...". It has little to nothing to do with capability or ability for a rational person wouldn't consider whether a cafe/restaurant has the ability to make someone a cup of coffee.
He doesn't have to be an English teacher, Mohsen. All English native speakers understand these things. A real English teacher describes the language as it is, as it really works, as the vast majority of users understand and use the language.
I can't recall from The Longman Grammar of Written and Spoken English, and I don't have my copy at hand, whether 'could' was more common or 'can' was more common. What I do remember is that both were very common. The reason for that is that 'can' is a polite request. If you are concerned which to use, then opt for the more polite and as has been noted by others, add a 'please'.
These are all more than polite enough for the situation you describe:
If you want to get really polite:
With respect, it would be helpful to all if those pedantically minded English speakers would avail themselves of a dictionary and for good measure, a corpus study of English.
The first would show them that can holds more than the one meaning they seem to be fixated on, ability.
The second would show that one wouldn't be better off in the sense that one would be way out of step with their language peers.
This, of course, in no suggests that may is in any way excluded from use as a way to ask permission. It most assuredly has its place in certain situations. I'm only pointing up how common it actually is in actual native speaker use.
Of course, all modal choice is largely an individually driven choice. But that choice shouldn't be influenced by inaccurate information.
Both the dictionary and the corpus studies show that there is absolutely no factual reason to satisfy pedantically minded English speakers. If you run into such a person, ask them if they can provide a reason that has some basis in reality.
This is a slightly tongue-in-cheek BE vs AE thing.
American English -
British English -
This may be the only example in the language where the AE form is more deferential than the BE!
I am careful to be polite, and use the politer "May I" in general practice but sometimes I need to know if my unusual choice of flavors is even available and use "can" instead.
Thus while Can I..." is sometimes questioned - hopefully humorously - it is perfectly appropriate and understood in a coffee shop, because it may well be that the shop is out of a particular flavor. Politeness may be inferred from things "appropriate and understood."