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The Rolling Stones famously sang "I can't get no satisfaction", which is a double-negative. "I can't get any satisfaction" is seen as more grammatical in modern English.

In his YouTube video, Steven Pinker said (6:13):

But "can't" and "any" is just as much of a double-negative as "can't" and "no".

I understand his broader point about one dialect being chosen over another as "the correct one", due to where the political power was, but I don't understand why "can't get any" is a double-negative.

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Yeah, it's not. This isn't the first time Pinker beat this particular drum; an earlier instance is this article, where he elaborates a little bit:

What do "any," "even" and "at all" mean in the following sentences?

I didn't buy any lottery tickets. I didn't eat even a single french fry. I didn't eat junk food at all today.

Clearly, not much: you can't use them alone, as the following strange sentences show:

I bought any lottery tickets. I ate even a single french fry. I ate junk food at all today.

What these words are doing is exactly what "no" is doing in nonstandard English, such as in the equivalent "I didn't buy no lottery tickets"—agreeing with the negated verb. The slim difference is that nonstandard English co-opted the word "no" as the agreement element, whereas standard English co-opted the word "any."

It looks like he half-remembered his own line of argumentation and, in trying to repeat it, turned a very questionable claim into a clearly false one.

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    They’re all negative polarity items; that’s not really a questionable claim. Is a negation and a negative polarity item a double negative? You could at least argue that they are. Since NPIs are limited to negative contexts (including questions), it’s not “clearly false” to consider them negatory in and of themselves. Where they differ from true negations is that they are auxiliary in nature and cannot be used to negate on their own – but that doesn’t necessarily mean they’re not negatory elements. Jan 9 at 23:39
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    Note that, at least in the quote, Pinker isn't saying that "can't ... no" is a double negative; what he's saying is that if you regard it as such, then you should say the same about "can't ... any".
    – TKR
    Jan 10 at 0:47

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