What is the difference between
Do you mind closing the window?
and
Do you mind my closing the window?
Does the first sentence mean that the speaker asked the listener to close the window? Does the second sentence mean that the speaker asks whether he/she can close the window?
Does the difference lie in that the action, "close the window", is done by different person?
Thank you very much!
Best Answer
In OP's first example there's an implied your (or you) after mind...
...corresponding to...
Note that 4 has becoming increasingly common over recent decades, particularly in BrE. Syntactically, everything after the word mind is a noun phrase. As a general principle, using the possessive form (my, your, his) in such contexts is "dated", but it survives more in "Do you mind [NP]" because it's a relatively formal construction anyway (and people tend to stick to older forms in formal contexts).
Also note that we tend not to use mind in this sense unless it's part of a question ("Does your husband mind our/us meeting like this?") or a negation ("I don't mind your/you phoning so late at night"). That's to say although "I mind your/you smoking while I'm eating" is grammatically "valid", most native speakers simply wouldn't say it.
Let's consider the more general context and introduce another syntactically similar verb...
I've included 5 there as a clearer example of a simple "noun phrase". Although currently most native speakers (particularly, AmE) still use possessive my in OP's example (because semantically as well as grammatically, it's a formal usage), they're much more likely to use the accusative/object forms in 7/9/11 above, rather than 6/8/10.
In light of this ongoing shift away from the possessive in constructions of this general form, I think it's probably more useful to interpret OP's first sentence as a "cut-down" version of my example 2, and the second one as a slightly dated version of my example 4. You'll still encounter the possessive in various contexts, but I don't think you ever need to produce it yourself.
EDIT: In the interests of "balance", it's well worth considering StoneyB's answer to a very closely-related question, where the final "advice to learners" paragraph recommends using the possessive. I'm not going to disagree with that, provided you wish to master "formal" usage, and bearing in mind that it would be slightly "dated, odd" to use the possessive in informal contexts such as example 6 above.