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Recently I received a rejection to my submission of an article to a journal. Despite three positive reviews, an additional reviewer wrote a short response with an example to claim that a proof in my paper was wrong. The editor claimed to "carefully" check the example and for this reason rejected my paper (no resubmission).

Now, this example is not a counterexample, which I can show with a single sentence (it doesn't satisfy the property I assumed in my theorem). So what should I do?

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    Did you email the editor with your one sentence explanation why the counterexample isn't a counterexample? – user60356 Aug 16 '16 at 14:13
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    This would be a ridiculous sort of corruption. If the editor for some reason didn't want your paper to be published, they had the ability to desk reject it instantly upon receiving it, without having to give any phony explanation or waste any reviewer's time. – Nate Eldredge Aug 16 '16 at 14:32
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    And as a political note, a PhD student who makes a habit of calling senior researchers "incompetent" (let alone "corrupt") isn't likely to get too far in the business, whether or not he is correct. I realize you are anonymous here, but all the same, you might want to get used to toning it down. "X seems to be mistaken" is fine, if you have done your due diligence and are very sure of what you are saying. "X is incompetent" is over the line in almost every situation. – Nate Eldredge Aug 16 '16 at 16:38
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    @Kuhndog: I agree that reviewers and editors should have sufficient experience and expertise to judge the paper. What I am saying is that even such experts make mistakes. If you are saying that anyone who ever makes such mistakes is ipso facto unqualified to review or edit, then you are setting an impossible standard, and certainly not the standard that the academic community sets. – Nate Eldredge Aug 16 '16 at 17:29
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    @Kuhndog: Not sure about CS, but in math, an incorrect proof is definitely grounds for straight rejection, regardless of the paper's other good qualities. Correctness comes before everything else. If the error is quite minor and the reviewer can see that's it's easy to correct, you might get away with being asked to revise. But generally, the process of getting the proof right is supposed to happen before the peer review process, not as part of it. The editor doesn't want to be stuck in the middle of arbitrarily many rounds of "is-this-better-how-about-now"... – Nate Eldredge Aug 16 '16 at 20:15
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Ok—Your paper was rejected because a reviewer made a mistake. I'll use my powers of Bayesian Induction to guess the editor just defaults to this reviewer, and there's not academic fraud issue. Everyone in academia has a drawer in their desk of decent papers that didn't get published just because.

You already emailed your advisor. You took the first step. That's good. I hope you didn't imply the editor's against you, but if you did, so be it.

Here's what you should you do now:

  1. Go into the bathroom, take a deep breath, look in the mirror, and say: "this is going to be OK." Good papers get initially rejected for stupid reasons all the time.

  2. Show the statement of the theorem, the purported counterexample, and your one-sentence refutation to your advisor or another colleague and get their reaction, to check whether you might have misunderstood the issue.

  3. Did the editor communicate directly with you? How did they communicate with you? If it was by email, write an email back. Write a three-sentence email about why the reviewer's counterexample isn't a counterexample. Show your proposed email to your advisor for feedback before sending it to the editor.

  4. You will wait to get a yes-no answer.

  5. If you still get rejected, submit the paper to another journal.

  6. If all else fails, put the paper up on arxiv and try to present it at a conference.

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    Good advice, but in between (1) and (2), I would suggest showing the proof and the purported counterexample to the OP's advisor or another colleague, just in case the OP has misunderstood the issue. – Nate Eldredge Aug 16 '16 at 16:37
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    In some fields, a conference is not inferior to a paper (sometimes it's the other way around), so I suggest you change item 5 to "try to submit it to some other kind of venue where acceptance is easier". – einpoklum Aug 17 '16 at 13:19
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When you are writing a paper, your task is not to prove something. It is to communicate to others that you have proven it. If you make some theoretical mistake in your paper, that's your problem and you should correct it. If you write a paper which does not communicate your results well enough and misleads your readers to false conclusions, this is also your mistake, and it is your job to correct it.

And yes, writing is hard. We love to jot down our train of thought in the way which makes most sense to us, and then we expect that others will understand. In reality, a good paper tailors its writing to the readers. The best authors design their text as a track for the reader's train of thoughts, knowing the possible junctions of misunderstanding and gently leading the reader away from them.

The average author, and sometimes even the best author, cannot write a perfect text from scratch. Readers will misunderstand things every now and then. And a reviewer is a very thorough reader. If your reviewer misunderstands something in your paper, then it is likely that a large swath of later readers (who don't try to follow your text anywhere as closely) will misunderstand it in the same way. This is a major flaw in your paper, and has to be fixed. Luckily, such fixes are very easy with a bit of thought.

In your case, the problem is obviously that the assumption you made was not salient at the time the reader reached your proof. You have to change that. For example, in the place where you list your assumptions, add a sentence or two discussing why the assumption was made and what are its consequences. It will force the reader to think about the assumption and notice it, and keep it in mind for the rest of the paper. Should he think of his counterexample later, he will notice himself that it is not a counterexample, as long as it is obvious how it hurts the assumption. If it is not obvious, it is a good idea to discuss (somewhere after the proof) how examples of this type are not counterexamples, because there is a twist which makes them hurt your assumption.

After the paper has been corrected in this way, you can decide whether to resubmit to the same journal (point out to the editor that it was a misunderstanding and you have made changes to the text to clarify the point) or to another venue.

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When a paper is rejected, do not argue about it. Submit to another journal.

If the paper is really good, it is the loss of the rejecting journal.

The loss to you of publishing in a less-important journal is minuscule compared to the aggravation you suffer by arguing and worrying about a rejection.

If (over time) you become known to editors as an argumentative submitter, it will only count against you.

I remember many years ago, the professor in the office next to mine was an editor on a major journal. He once remarked that the most unpleasant part of that duty was arguments received from authors of rejected papers.

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    While I think this is generally good advice, if a paper was truly rejected on a factually incorrect basis, as claimed by the OP, then I think this ought to be addressed with the editor. Simply stating in a clear measured way "The claimed mistake in my proof was incorrect, and here's why" should not lead to you being considered argumentative. – user24098 Aug 17 '16 at 14:29
  • Also, if the review process is problematic, and several authors complain, then none of them would be singled out as "argumentative" - which is why I agree with @dan1111 that "honesty is the best policy" in OP's case. – einpoklum Aug 17 '16 at 21:37
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    This advice can be very misleading, and is certainly culture-dependent. Rejections come in many flavors, including "rejection with possibility to resubmit" - which often just means major revisions demanded. Incorrect reviews do happen, and successful appeals happen, too. You just have to calibrate to know when it is worth arguing. If this is a genuine clear error in review, it will be much faster to appeal than to be re-reviewed. – AJK Aug 18 '16 at 1:43
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I'll make two practical suggestion:

  1. It is unlikely the editor would be willing to entertain the possibility of accepting your paper anyway, for a combination of several factors:

    • lack of time: academics are mostly super-busy always.
    • feeling it's beyond what his/her duty entails - considering how s/he is typically doing it without being paid
    • conceit: some people are too high and mighty to be bothered with correcting such mistakes
    • apathy: some people do not feel guilty for letting something like that stand

    now, this doesn't mean you shouldn't try appealing to the editor, like @JackStClaire suggests, but expect failure with high probability.

  2. Get people in your field to glance at your paper and the review, without having told them there's something wrong with the review. Or - tell them something like "I'm worried about what happens in the case of XYZ, what do you think?" - if they are not able to come out and say "oh, that's not a counter-example, it doesn't meet the preconditions" - that means you should probably have explained within the paper itself why that would not be counter-example. Yes, maybe it seems stupid and obvious to you; but apparently it doesn't to other people

but most importantly:

  1. Remember how unfair this has been to you, in case you end up being an editor or a reviewer yourself. When you are in that position, make a vow to be extra careful in your work and to accept appeals of reviews and considering them on the merits.

... and that goes for all you readers of this Academia.SX page as well!

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Don’t get involved in the arguments, as the chances are that you will have him as a reviewer again. Next time he may just reject you without any good reason, as a personal grudge can develop between you and him. If you really want to publish, send it to another journal.

One of my acquaintances had his paper published after it was previously rejected by several journals, and guess what, the paper was accepted in a much better journal than all those which rejected it.

Carry on. Don’t lose hope. It works like this.

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