both / either

[licinio]

I've always thought both and either had separate meanings, until I saw these two sentences:

Monday or Tuesday. Both are convenient to me. Either of them is fine.

I thought that using "both" meant taking two possibilities together, but this example suggests that "both" has a wider meaning and can be used regardless of picking only one. Agree?

[bhaisahab]

I've always thought both and either had separate meanings, until I saw these two sentences:

Monday or Tuesday. Both are convenient to me. Either of them is fine.

I thought that using "both" meant taking two possibilities together, but this example suggests that "both" has a wider meaning and can be used regardless of picking only one. Agree?

No, both sentences mean the same. Either of them is fine.

[5jj]

Both (of the two possibilities) are fine. I probably wish to come on only one day (Monday or Tuesday), but, if required, I could come on two days.

Either (of the two possibilities) is fine. I don't mind which, but I am going to come on only one day.

In practice, they convey pretty much the same message,

[5jj]

Either of them is fine.

Indeed, both are fine.

[licinio]

By a stroke of luck, he's always been able to write with either hand.

I take the cue from your replies, and ask: would it make sense if I said "with both hands"?

[5jj]

Yes.

In other contexts, 'with both hands' could mean using the two hands at the same time, but the intended meaning is clear in your sentence.