A negative polarity item (NPI) is a word (or expression) used only in non-positive environments. It's a misnomer because they're not restricted to negative environments, but occur also in questions and hypotheticals. One standard example in English is the indefinite 'any':
I've got some bananas.
I haven't got any bananas.
Have you got any bananas?
If I had any bananas I would eat one.
The indefinite quantifier 'some' only appears in the positive sentence, and the same meaning is expressed by 'any' in the non-positive ones. Note that there is also a stronger meaning of 'any', roughly 'whichever', which can appear positively: I'll eat any bananas I can find. In the weaker sense, 'some' and 'any' mean the same, and swapping them is ungrammatical in positive and negative uses, but both are acceptable in questions and hypotheticals.
* I've got any bananas.
* I haven't got some bananas.
Have you got some bananas?
If I had some bananas I would eat one.
In the latter cases the difference is subtle. It reminds me of the possible use of the subjunctive to indicate an unreal condition. With an unreal, probably only the NPI 'any' is possible. If the situation is conceived as possible, 'some' is somehow more positive than 'any'. Compare:
If there were any tea left in the pot I'd have another cup.
If there's any tea left in the pot I'll have another cup.
If there's some tea left in the pot I'll have another cup.
Another NPI is 'ever'. This is ungrammatical in positive sentences, and there is no positive equivalent:
* I have ever been to Iceland.
I haven't ever been to Iceland.
Have you ever been to Iceland?
If I have ever been to Iceland, I was too young to remember.
We can also count as NPIs some idioms such as budge an inch, be bothered, give a toss. We would predict that these are allowed in questions and hypotheticals, though being idioms, it wouldn't be too surprising if they resisted behaving normally:
* I can be bothered to take the rubbish out.
I can't be bothered to take the rubbish out.
?* Can you be bothered to take the rubbish out?
If I can be bothered to take the rubbish out, I'll check the letter-box.
The hypotheticals I've been using so far are conditional clauses beginning with 'if'. Uncertainty as such is not, however, enough to license an NPI. A positive sentence containing 'might' does not license them. Personally I find it doubtful whether a negative 'might' can take the positive 'some':
I might have some bananas.
* I might have any bananas.
?* I might not have some bananas.
I might not have any bananas.
As well as the clause structures considered so far, many individual verbs and other words license NPIs. Some clearly contain a negative sense, such as 'deny' or 'refuse', while others are less obviously negative, such as 'surprised'. The test is whether they take 'any':
* I assume there are any bananas left.
I deny there are any bananas left.
I refuse to give you any bananas.
I am surprised there are any bananas left.
This implicit negativity has a surprising effect: it can make a grammatical sentence incomprehensible if it contains a triple negative, which appears to be on or beyond the limit of our mental parser to process. Adding an NPI-licensing verb to a double negative makes it much harder to understand:
I am surprised that no-one did the test who didn't want to.
The double negative sentence 'No-one did the test who didn't want to' is quite clear, and you understand it instantly: you don't have to strip off negations to work out who wanted what.

(Note also that this is a true double negation: it negates a negation. So this does not include the single logical negation in 'I ain't seen nobody'. Linguists prefer to call the latter spread negation. There is no processing problem in repeating your one negation over multiple words, and it's standard in many languages and dialects, including Italian, Czech, Cockney, and AAVE.)

Negation can also be expressed by a quantifier such as 'no' or 'none' (or compounds like 'nobody'), and this licenses NPIs both inside its scope and in the main clause outside it:

Nobody who has ever met Chomsky wants to argue with him.
Nobody who has met Chomsky ever wants to argue with him.
The quantifier 'few' has a negative polarity, so it can take 'ever' in both the above positions. Surprisingly, 'every' licenses NPI only inside its scope:
Few who have ever met Chomsky want to argue with him.
Few who have met Chomsky ever want to argue with him.
Everybody who has ever met Chomsky wants to argue with him.
* Everybody who has met Chomsky ever wants to argue with him.
The reason for this was discovered by W. Laduslaw in 1980: NPI is licensed under quantifiers that have downward entailment, that is if it's true for a set then it's true for a subset. So suppose everybody who has met a linguist wants to argue with them. Then, in particular, it's true that everybody who has met Chomsky wants to argue with him. However, if it's true that everybody who's met Chomsky wants to talk to him it doesn't follow that they want to argue with him. So NPI 'ever' is not licensed in the second situation.

Kearns, K., 2000, Semantics, Basingstoke: Palgrave.
Laduslaw, W., 1980, Polarity Sensitivity as Inherent Scope Relations, doctoral diss., Univ. of Texas at Austin.
Liberman, M., Language Log discussion at http://itre.cis.upenn.edu/~myl/languagelog/archives/000500.html

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