Why does Steven Pinker say that “can’t” + “any” is just as much of a double-negative as “can’t” + “no” is in “I can’t get no/any satisfaction”?

"Can't get any" is not a double negative because "any" is just the version of "some" in a negative context or question context (the so-called non assertive contexts). So, as you say "I don't have any chalk." when someone asks you "Do you have chalk?", and not "I don't have some chalk.", you also say "I can't get any satisfaction out of that hobby." when someone asks "Can you get any satisfaction out of that hobby?" (and you don't get any).

The word any, when used in a negation, is a Negative Polarity Item (NPI), which is defined by the paper Definite Descriptions and Negative Polarity as words that “seem happy under negation and are sometimes unhappy without negation”.

When you have a double negative, “standard” English dictates that you need to use a single negative but not both (or else you get litotes):

• I can’t get satisfaction.
• I can get no satisfaction.

In contrast, you don’t get a grammatical sentence with any:

• I can get any satisfaction.

“I can get any satisfaction” would be grammatical as part of a longer sentence but note then how it doesn’t mean “no satisfaction”. Note also that there’s another sense of any (“free choice any”) with a completely different meaning that would be grammatical in similar sentences because it’s not a NPI (“I can get any TV” meaning that no TVs are out of your budget).

I decided to see how this compares with a dialect where double negatives emphasize the negative instead of canceling it.

I did a lemma-grouped search for VERB no in COCA to get a baseline for American English. I looked at a random sample of 100 for the phrase “[MAKE] no”. All the hits I saw were single negatives. The same was true for the same search in iWeb.

I searched CORAAL for AAVE examples using the regex bma[kd][^.,|]* nob which is a slightly broader search but ultimately trying to find similar examples. 13 or so hits were “double negatives”, such as “don’t make no sense”. There were about 5-6 examples with a single negative (such as “that makes no sense”). This shows that the single negative “no” is at least sometimes used among speakers of the dialect, though it’s impossible to know why from this alone. In searches for other verbs (such as one looking for “can ... no”), I haven’t been able to find any examples with the single negative “no” but I can’t tell if that’s because there’s not enough data or if it’s ungrammatical in the dialect.

There’s a paper that suggests that words like “no” are NPIs in AAVE but they don’t follow all the same rules as NPIs like “any” in “standard” English, though I didn’t have time to read it fully.

The problem is that "Double Negative" doesn't have any specific meaning. Like "Split Infinitive", it's a product of popular grammatical mythology, not logic.

First, there is a lot more negation around than one might have suspected.
As it says here,

... examples include syntactic constructions (This is it, isn’t it? Not any big ones, he didn’t), variation (so didn’t I; ain’t got none), morphology (-n’t, -free, un-), (morpho)phonology (do/don’t), intonations (‘Riiight’), and lexemes sporting negation overt (not, never, nix), incorporated (doubt, lack), calculated (few), entailed (prohibit), or presupposed (only).

And these interact together in all kinds of ways. It's not a case of two negatives yielding a positive; English is not algebra. Sometimes negatives do cancel, as in

• That is not an unexpected result.

where the two negatives form a rhetorical phrase to say something like
You're right, but we already knew that.

Plenty of other negatives don't cancel out. For instance,

• Not in here, you can't do that

does not give the addressee permission to do that in here.

And then there's the problem of multiple negations. One can get trapped by using a negative phrase without realizing it; one negative too many results in what Language Log calls a Misnegation, like

The government rushed to investigate the case thoroughly, eager to dispel any notion that it did not take lightly the killing of one of its citizens.

As far as NPIs being negative, think of them as all being locked in the negative field together. You wouldn't know that iron was magnetic but copper wasn't if you didn't know about magnetic fields. Plenty of languages just license any negatives inside a negative field, like Spanish No saben nada 'they don't know anything', literally "not know-3pl nothing". AAVE is like that, but whitebread English licenses NPIs instead.

So, I wouldn't worry about what Steven Pinker said. He said it in a different context, was talking about something else, and was probly just trying to break the news gently.

Summary

Steven Pinker does not say that “can’t get any” is a double negative, merely that it is no less of one than “can’t get no”. I think he is exaggerating, to make the point that grammatical rules are arbitrary rather than logical.

What Pinker actually says

The questioner writes

I don't understand why "can't get any" is a double-negative.

Steven Pinker does not claim that it is a double negative, only that

‘“can’t” and “any” is just as much of a double negative as “can’t” and “no”.’

What I think he means

I take him to mean that English works equally well whether the rule for totally negating “can get” is to say “can’t get no” or to say “can’t get any”, and he goes on to point out that it is a historical accident that “can’t … any” came to be preferred by those laying down the rules.

He is explaining the difference between descriptive and prescriptive rules, and says that

many of the prescriptive rules make no sense whatsoever

and uses the rules forbidding “split infinitives” and “double negatives” to illustrate this.

He is making the point that prescriptive grammar is more arbitrary and less logically compelling than people often suppose it to be, and that dialects which do not conform to it can be just as complex and expressive.

I do not, however, think it fair to say (as he does not explicitly) that the rule forbidding double negatives makes no sense whatsoever; in fact it codifies a convention which may be useful in some contexts.

We can define “double negative” so that the rule makes sense

What I think is confusing about Pinker’s claim is that “no” is in fact considerably more strongly associated with negation than “any” is. If we choose to therefore classify “can’t” and “no” as negative, but not “any”, and define a “double negative” as a construct using two words we classify as negative, then “can’t get no” is a double negative while “can’t get any” is not.

The rule does not forbid all double negatives and may be useful

Moreover, what the rule actually means is not to avoid a double negative, but to avoid using it to express simple negation: there are two negatives in “don’t say he’s not welcome”, “say not the struggle nought availeth”, “let nobody say you are not a sensualist” and perhaps even “thou shalt not bear false witness”, but all are acceptable. The point is that these are interpreted so that one negative negates part of the utterance containing another negative, while the negatives in “can’t get no” support one another.

If we want to construct complicated sentences, perhaps in a mathematical or philosophical context, in which we can negate arbitrary constructs, we need to agree when one negative negates a construct, and when two negatives belong together to form a simple negation. A simple rule to achieve this is to forbid the latter case – but this comes at the expense of classifying many common utterances as incorrect.