I understand the following phrase as follows:

how to not be seen – how to avoid being seen altogether

how not to be seen – how to avoid outfits or circumstances in which you do not want to be seen

But it’s rather ambiguous and perhaps it’s the other way around or just depends on context.

I’ve seen Order of "not" with infinitive, but it still feels like there is a difference in this case and that the meanings are different.

If you’re curious, this comes from a Monty Python sketch.

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    I think that both mean the same thing, but 'how not to X' is the standard usage. – Kate Bunting Dec 16 '17 at 9:07
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    Broadly yes, the first means how to avoid being seen altogether, as a saboteur or assassin might wish to. However yes, your second is ambiguous. It could mean either how to avoid outfits or circumstances in which you do not want to be seen but it could also mean exactly the same as the first. Mental brackets might help, at least in the second case - which was where the problem lay, wasn't it? Either way might need re-phrasing but do you see a clear difference in thought between how not to (be seen) and how not (to be seen)? – Robbie Goodwin Dec 16 '17 at 20:43

Both mean the same, but whether the first is poor grammar merely having entered popular usage depends on whether "not" is seen as an adverb. Modern dictionaries like the Oxford view it as such, but since they are not linguistic governing authorities, an etymological approach may be more reasonable.

When "not" first came about in the 13th century it was seen as a negative particle rather than an adverb, corroborating the historical inefficacy of the first phrase. Note that the word "not" itself was an innovation on the Middle English ancestor of the noun "naught" which comes from the Old English ancestor of "no wight" (i.e. no thing).

Given that the word "not" is only about seven centuries old, and the lack of an authority on the English language comparable to the académie française, it is reasonable to suggest that the exact rules on its usage are still not fixed.

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Basically (ugh, sorry) to avoid ambiguity, you need to use the first form; the second has unfortunately acquired a separate meaning as @robbie-goodwin pointed out.

For serious “unambiguity”, you could even rephrase the first as something like how to remain unseen or how to avoid being seen. That's almost negative ambiguity :o)

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I would agree with the Oxford dictionary that "how to not be seen" is ungrammatical, in part because the infinitive is split. As "not" negates "to be" rather than "seen", "How to be not seen" is better, but clumsy, and I agree that "How to be unseen" is not only more elegant, but more precise.

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