What is the difference between the following sentences (in meaning)?
Are "present continuous" and "be going to" interchangeable?
I'm going to visit my brother on Friday.
I'm visiting my brother on Friday.
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1) The present continuous is often used for arrangements, and when we have prepared for the event somehow. So here it emphasizes that my brother and I are both prepared to meet, we have already talked about it and decided, maybe made particular plans for that day.
In another example, if you say "I'm flying to Paris tomorrow.", it means you have bought a ticket, taken time off work, etc.
2) Future with 'be going to' is usually used for plans and intentions. So if you say that you are going to meet your brother, you intend to do it, but you might not have taken the necessary steps for this to happen.
You can say something like "I am going to retire at the age of 60, buy a house and plant a cherry tree in my backyard." You might be 30 when you say that, it shows that you have thought about this and think of doing this, not that you have already bought a house. But if you say "I am retiring next year", that is pretty fixed.
The two are really close and often interchangeable. In the examples you are giving, I do not see a reason to prefer one over the other.
If something is not yet happening, it is not present, nor should present continuous be used. Thus only your first example is grammatically correct. Correct constructions for the second would include:
I will be visiting my brother on Friday.
I plan to visit my brother on Friday.
That said, native speakers do sometimes say things like, "I'm visiting my brother on Friday." Folks will understand what you mean, and not think much of it. But the only way that would actually be grammatically correct is if you were a time traveler, perhaps talking to a friend several days ago on your time-phone.
I believe we make allowable exceptions for generalities. Here you are describing a specific event that will happen on Friday. But where something is more tenuous to begin with, a college student might state before the semester starts:
Have you already started? Maybe you have registered, bought books and supplies, or read the course syllabuses? Your friend may not know or care about any of those things. He just wanted to know if you might be in any of the same classes, and you provided him information along those lines. So it is sort of irrelevant if you have yet spent a single minute in any of your classes.
Future idea: The present continuous can be used to express a future, not only to express something happening right at the moment. See the two examples below:
A) I'm going to school tomorrow. I'm seeing my sister this afternoon. VERSUS I'm going to school now. I'm seeing my sister now.
b) Also, sometimes, to be more emphatic in the present, we use GOING TO plus a verb, which is also a "real future": I'm going to go to school now. A future with go that is about to begin. VERSUS I'm going to go to school tomorrow. A future that will begin tomorrow.
Simple future is WILL: and usually expresses intention: I'll go to school tomorrow and am studying now.
1) I'm going to school now or tomorrow. 2) I'm going to go to school now or tomorrow. VERSUS I'll go to school tomorrow are not the same.
The future with will either expresses an intention or contrasts with what you are presently doing but does not express something definitively. I'll do that work later I intend to do it later); now, I'm studying for exams.
Also, the use of WILL expresses that you will do something for someone: I'll pick the kids up at schools. I'll open the window now because it's too hot.