Opposite Day refers to a game usually played by children. One can declare that today is Opposite Day (sometimes retroactively) to indicate something which will be said, or has just been said should be understood opposite to its original meaning (similar to the practice of crossed fingers to automatically nullify promises).
Opposite day causes a self referential paradox. On opposite day, it could be said that "Today is opposite day." Therefore, it is not opposite day, but if one says it is a normal day it would be considered a normal day, which contradicts the fact that it has previously been stated that it is an opposite day.
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