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The Third Opinion Paradox arises from the fact that the Third Opinion Project states that Third Opinions are available under the project only if there are exactly two editors involved in the dispute. Consider the following scenario:
- User A and User B are having a dispute at article X. They are the only two editors involved in the dispute.
- A or B lists their dispute at the Third Opinion Project to get a neutral third opinion.
- User C happens to look at the Third Opinion project page, sees the dispute about X, and thinks that the dispute is interesting.
- With no intention to issue a Third Opinion, and without removing the dispute from the WP:3O page, User C goes to article X and expresses his or her opinion about the subject of the dispute.
- User D, who frequently issues Third Opinions, sees the dispute on the Third Opinion project page, thinks about issuing an opinion under the Third Opinion Project, and looks at article X and its talk page to get the background on the dispute.
- User D sees User C's comments and, without issuing a Third Opinion, removes the dispute from the Third Opinion dispute list because more than two editors are involved in the dispute.
Thus the paradox: User C's opinion was a third opinion but was not a Third Opinion but was a Third Opinion.
The same result would have occurred if User C had become involved in the dispute without ever knowing that the dispute was listed at the 3O project page or before the dispute was listed there. The fact that User C was or was not biased or was or was not neutral would have no effect: the dispute would still be removed for having more than two editors.
The reason is, basically, simple: the Third Opinion Project offers third opinions, not fourth opinions, and is intended for the situation where two editors who are debating a point in full good faith just get "stuck" and can't come to a decision. Once a third editor has entered the dispute, the dispute is no longer stuck. (Indeed, even if a "genuine" Third Opinion is given, it cannot "unstick" them by itself; the dispute will still not be settled until they come to consensus.)