Bartlett: Did you or did you not throw the watering can?
Fitch: I did not!
Bartlett: Yes or no?! Did you throw the watering can?
Fitch: No!
Bartlett: Answer the question!!!!
Fitch: I didn't throw it!
Bartlett: So ... he denies it! ... Very well ... would you be surprised to hear that you'd thrown the watering can?
A pause.
Fitch: ... Yes.
Bartlett: And do you deny not throwing the watering can?
Fitch: Yes.
Bartlett (triumphantly): Ha!!!
Fitch: No!!!
Bartlett: Very well, Mr Fitch ... would it be true to say that you were lying ... if you denied that it was false to affirm that it belied you to deny that it was untrue that you were lying?!
Fitch: Er ...
Bartlett: You hesitate, Mr Fitch! An answer, please, the court is waiting! Ah ha ha hah! Ah ha ha hah!
Fitch: Yes.
Bartlett: Shit. No further questions, m'lud.

-- from the 'Judge Not' skit by John Cleese, first performed in the Cambridge Footlights review of 1963

Negation in logic is simple. Two negations cancel out leaving a positive, with three negations two cancel out leaving one negation, and so on indefinitely. Natural language doesn't work like that. We can usually handle double negations, but beyond that the brain hits a cognitive limit.

The point of the exchange in the courtroom skit is that of course there's a processing limit: we can't be expected to take in a whole stream of negatives and unpeel them instantaneously to give a confident yes or no answer. But the surprising thing is that the limit is two. We can't even understand three:

You can't deny that it isn't strange.
Now if you thought you immediately understood the above sentence, you're probably wrong, or bluffing, or guessing, or lucky. In fact if you google for patterns such as {can't deny * isn't} or {can't deny * didn't} to pick up examples, then read the results for context, you'll find the final negative is almost always redundant. People use "You can't deny that it isn't strange" to mean "You can't deny that it's strange", i.e. it is strange.

One classic example is "No head injury is too trivial to ignore". It is obvious what it means, or is supposed to mean, but try and puzzle out the logic of it.

There are a number of important points to make about this.

One. The limitation is one of language. Of course you can easily work it out logically. You can get pencil and paper, and cross out pairs of negatives, and rewrite what remains in positives, until you get a clear answer. Or you can try to do this in your head (though there's a good chance you'll get confused if you do.) Formal logic is another talent of the human brain, but it takes time: there's a delay of several seconds, and more if you get tangled. But language comprehension is instantaneous. In general, you know what a sentence means the instant you hear or read it. With triple negatives, you don't.

Two. Words like "deny" are negative too. The count of negatives includes not only "not", "none", "never", "no-one", "nothing", but numerous verbs containing negative semantic aspects, such as "deny", "miss", "avoid", "fail", "refuse"; also adjectives like "surprised", adverbs like "hardly", "scarcely", quantifiers like "few", and prefixes like "un-" and "mis-". These are all called negative polarity items (q.v. for more detail), and there are a couple of simple tests for them in English. Simple positive sentences don't contain the words "ever" or "any", while negatives and questions can: *I ever went to Paris, I didn't ever go to Paris, Did I ever go to Paris? I ate some peanuts, I didn't eat any peanuts, Did I eat any peanuts? An NPI word behaves like a negative: I hardly ever ate any peanuts, I deny I ever ate any peanuts.

Three. Many languages (e.g. Italian, Spanish, Czech) and dialects of English (Cockney, AAVE) allow an ordinary single negation to be marked across multiple words: I didn't eat none. This is commonly called double negative, but isn't actually a negation of a negation, so linguists often use a different term for it, such as spread negation. Speakers of Italian or AAVE have no difficulty using it, and it doesn't enter into the problem I'm discussing here. A true double negation is something like "I don't deny throwing it", "I could scarcely fail to notice", "I refuse to ask no-one", "You can't say that it isn't strange".

Four. This is not a scalar matter. It is not that one negation takes a bit of processing effort, two take more, three take more still, and so on up. Most of the time doubles are just fine: those at the end of my previous paragraph don't confuse us. But the third one throws a spanner in the works. Probably more often than not a triple negation means the same as a double one, because the third one can't be processed and is just dropped. The question is then why the triple construction arises at all.

There are idioms that have a redundant negative, and these are sometimes of triple form. For example, "I shouldn't be surprised if it didn't rain tomorrow". To me this has a rather old-fashioned sound to it, shown by "should" for "would", and by the use of the subjunctive in the subordinate clause. "It rained tomorrow" and "It didn't rain tomorrow" are nonsense on their own, in the factual indicative mood; but under "if" they are subjunctive (which looks like past tense), expressing the hypothetical. The second "not" seems to go with this. If you update it by taking out the subjunctives, it reads fine in the positive ("I would be surprised if it rains tomorrow"), but now the negative ("I would be surprised if it doesn't rain tomorrow") can have either the same meaning or the opposite meaning as the positive. Possibly only some dialects of English allow this redundant subjunctive "not". But for those that do, it seems to be an idiom, that is an illogical but established way of saying it, and the people saying it aren't actually confused.

I wonder if this is also true of the "can't deny it isn't" construction. I wouldn't say this myself, but it is so consistently the same as "can't deny it is" by the people who do say it that I suspect it might be now fixed as an idiom. The examples are often so clear that's a bit unkind to say they're all confused. Here are some where this triple boils down to its logical single:

Even you can't deny ornithology isn't exactly a field that will change the fate of the human race.
( = same as "Ornithology isn't exactly a field... and you can't deny that.")
I'm not bashing it [smoking], but you can't deny it isn't healthy.
( = It isn't healthy, and you can't deny that.)
But here are the more common kind, with idiomatic or redundant "not":
You can't deny he isn't the most credible news name.
( = He is the most credible news name, and you can't deny that.)
It is sad but we can't deny it isn't true.
( = It is true, and it is sad but we can't deny it.)
This isn't about any depression, though I can't deny it isn't related.
( = It is related, and I can't deny that.)
Two more points. I'm not completely sure I've classified the above sentences right: it's quite hard to feel confident. Anyway, secondly, the problem isn't just with an accumulation of negatives across a sentence, but only when they structurally dominate one another, that is when one negative has logical scope over another. In "This isn't about any depression, though I can't deny it isn't related", the first "isn't" isn't a problem. We understand the clause "This isn't about any depression", process it whole, then go on to the next one, beginning "though...". The first "isn't" doesn't negate anything in the "though"-clause, so it doesn't add to the burden. Another example is
Even people that don't believe he was the Savior can't deny he didn't exist.
The noun phrase "people that don't believe he was the Savior" has a negative with internal scope, but it doesn't negate anything outside it. We process this negative, understand what kind of people are being talked about, then go on to use it as the subject of the main clause "... can't deny ...". But this is followed by a redundant negative. What is true is that he existed, and they can't deny it. We could say the same thing this way:
Even people that don't believe he was the Savior can't deny he existed.
This, though it still has three negatives in it, isn't hard to process, because it's not trying to express negation of a negation of a negation. The negations are in different parts of the sentence and only get stacked two deep at most. By contrast, here's what appears to be a genuine attempt at a quadruple negation, and is as expected incomprehensible:
Nobody can't deny we didn't try to attak.
Finally, I have to admit it's not always triples. Some doubles are widely misused in the opposite of their logical sense, like "impossible to underestimate" in the examples below. However, in almost all cases there are other words that are implicitly negative, and in the following I've made them bold. The Language Log blog has been collecting many examples of these under the name of overnegation, and there a number of learned linguists have been trying to work out why these constructions are so common.
NYC is the only US city where less than 50% of the households do not own a car.
Although his attendance at school was still very poor, Stanley never failed to miss a movie at the local theaters.
This is sure to be a killer tournament, don't fail to miss it!
It's impossible to underestimate the value of early diagnosis of breast cancer.
I challenge anyone to refute that the company is not the most efficient producer in North America.
The Skilling indictment demonstrates in no uncertain terms that no executive is too prominent or too powerful and that no scheme to defraud is too complex or too fancy to avoid the long arm of the law. [Ack! - Gr.]
The double-negative expression "fail to miss" almost always means just "miss". Just about the only logically correct usage of it on the Web is the famous Douglas Adams passage about how to fly.

By the way, why is "challenge" marked as implicitly negative? Remember the NPI test: it can take "any", as in "I challenged her to produce any evidence", but a simple positive can't: *"I asked her to produce any evidence". To challenge is not simply to ask, but to claim that someone can't.

Another cognitive limit in language is the central embedding.

Some representative Language Log threads with more examples and analysis:

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