I interpreted “the middle” as a point and its near surroundings, which explains some of the disagreement. (All of your specific examples in your original post were near the midpoint, which didn’t clarify which interpretation you intended.)
I think that the more fundamental rule isn’t about middles, and (as demonstrated here) that’s easily misinterpreted without including many qualifiers and specifics. “Larger intervals are more likely, to the extent that the distribution is flat” is more fundamental, but there are so many ways to define large intervals that it doesn’t seem very useful here. It all depends on what you call a very unusual point—if it’s the middle that’s most unusual to me, then my version of the doomsday argument says “we will probably either die out soon or last for a very long time, but not last exactly twice as long as we already have.” (In this case, my large interval would be time-that-civilization-exists minus (midpoint plus neighborhood of midpoint), and my small interval would be midpoint plus neighborhood of midpoint.)
I interpreted “the middle” as a point and its near surroundings, which explains some of the disagreement. (All of your specific examples in your original post were near the midpoint, which didn’t clarify which interpretation you intended.)
I think that the more fundamental rule isn’t about middles, and (as demonstrated here) that’s easily misinterpreted without including many qualifiers and specifics. “Larger intervals are more likely, to the extent that the distribution is flat” is more fundamental, but there are so many ways to define large intervals that it doesn’t seem very useful here. It all depends on what you call a very unusual point—if it’s the middle that’s most unusual to me, then my version of the doomsday argument says “we will probably either die out soon or last for a very long time, but not last exactly twice as long as we already have.” (In this case, my large interval would be time-that-civilization-exists minus (midpoint plus neighborhood of midpoint), and my small interval would be midpoint plus neighborhood of midpoint.)