Shouldn't "believe it or not" be "believe it or don't?" I do not see the word "not" being used like that elsewhere.
closed as off-topic by ab2, jimm101, GoldenGremlin, curiousdannii, user140086 Mar 8 '16 at 1:19
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I do not see the word "not" being used like that elsewhere.
For that particular construction, there are others that are similar and easily understood:
Ready or not..., Like it or not..., Whether he knows it or not..., etc.
There is a dropped verb there, not necessary for understanding. You did understand "Believe it or not," did you not?
Believe it or [do] not [believe it]...
Words that are easily understood to be part of a phrase are often omitted. For example, the same can be said this way:
Words easily understood to be part of a phrase are often omitted.
You can explain it as a shortened form of " Believe it or do not believe it". As this is a frequently used sentence introduction "or do not believe it" was reduced to "or not".
Despite the shortened form the sense of this idiomatic formula remains clear.
It seems to follow a slightly more archaic pattern than suggested in the previous answer.
Believe it or [believe it] not.