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Given the two separate facts:
John Doe is an uninteresting name.
Usain Bolt is an interesting name.
which of the following statements is correct:
A - "John Doe is an uninteresting name. Usain Bolt is"
or
B - "John Doe is an uninteresting name. Usain Bolt isn't"
In other words, why does the following equation seem true?
"John Doe is an uninteresting name. Usain Bolt is" = "John Doe is not an interesting name. Usain Bolt is."
(even though they are logically different).
Your equation is not true.
For Usain Bolt to be an interesting name, the ellipsis is
John Doe is an uninteresting name. Usain Bolt isn't [an uninteresting name].
In your equation, the ellipsis again omits the noun, but this time it's a different noun.
John Doe is not an interesting name. Usain Bolt is [an interesting name].
When you omit the noun in ellipsis, it must remain the same noun as was previously referenced. Consequently "John Doe is an uninteresting name. Usain Bolt is" makes no sense. You would need "Usain Bolt is too," which isn't what you want to say.
In each case, to reverse the previous sentence, since the noun must remain the same it is the verb which must change.
I basically agree with Andrew Leach. (And I upvoted him. :-) Let me add a few words too long to fit in a comment.
When you omit words from a sentence in this manner, as Andrew says, the reader can only assume that you meant the same words as in the previous clause or sentence. "John Doe is an uninteresting name. Usain Bolt is." The reader can only assume you mean "Usain Bolt is an uninteresting name." Which of course isn't what you are trying to say.
Prefixing a word with "un" is not the same as saying "not". It's a new word. It is perfectly reasonable to say that something is "not uninteresting". Indeed, saying something is "is not uninteresting" is not necessarily the same as saying that it "is interesting". It would be taken as much milder. I am reminded of the time that a friend of mine said to a woman that she was "not unattractive". I laughed and told him that he must be a real hit with the ladies when he used that sort of grand flattery. Telling someone she "is not ugly" is not the same as saying she "is pretty", and his comment was basically saying the former.
Update
RE David Schwartz comment
If the "rule" was not that when words are omitted like this, the reader assumes that the omitted words are the same as those found in the previous clause or sentence, how could it be clear from context? How would the reader know what to fill in?
In this case, you are thinking that the reader should/could assume that you mean "Usain Bolt is an interesting name." But how would he know that your point is that Usain Bolt is the opposite of John Smith, and not the same? Just because you think Usain Bolt is an obviously interesting name doesn't mean your reader does. Perhaps where he comes from there are more people named Usain Bolt than there are people named John Smith. Perhaps he finds the noble trade of blacksmith fascinating, and his religious faith is deeply inspired by the Gospel of John, but he sees nothing interesting in either Usain or Bolt.
More realistically, take a case that is not about words themselves. Suppose you wrote, "John Smith is a tall man. Usain Bolt is. Fred Stover is. Roger Miller is." Could you safely assume that the reader would instantly grasp that you mean "Usain Bolt is short. Fred Stover is of average height. Roger Miller is a dwarf." How would he know? The only way he could tell "from context" is if you had already told him the heights of these people earlier, in which case this sentence is superfluous.
Maybe there are cases where someone could figure out things like this from context. But the point of writing is supposed to be to convey information to the reader, not to make him guess what you meant.
No, the equation is not true. Double negatives are not allowed in formal standard English, only in dialect and song lyrics and folk tales: "The tar baby, he don't say nothin'." And "I ain't got nobody that I can depend on", a blues song recorded by Bessie Smith and Carlos Santana (not together) and many others.
"John Doe is not an interesting name. Usain Bolt is." is the correct English to express the ideas in sentences 1 and 2. I would add the conjunction "but" between those two sentences, though: John Doe isn't an interesting name, but Usain Bolt is.
"John Doe is an uninteresting name. Usain Bolt is." is not the correct English for sentence 1 and 2.
"John Doe is an uninteresting name. Usain Bolt isn't." is grammatical English and the meaning is clear, but it's not good style because it expresses a positive -- "Usain Bolt is an interesting name" -- in a negative way; however, context might make that acceptable.
"Still, using a double negative in formal prose is a definite no-no (!) and will lead only to confusion for most readers, who will try to reconfigure the double negative into something algebraically positive." Grammar of double negatives
