Lots of hobby-linguists, especially among fans of auxiliary and logical languages, and also among fans of natural languages seem to think that double negatives are an illogical and ugly thing. Fans of logic seem to think it violates some need for clarity of thought, and that this is because it's fundamentally illogical somehow. On the other hand, people interested in naturalism seem to think it is a proof that language is something completely divorced from logic.

Both of these notions are wrong. Of course, they both stem from the thought that not(not(x)) must equal x, and that if it equals x the system no longer is logical.

Granted, language is not fully logical - often, phrases get meanings that are somewhat divorced from the meaning of the subcomponents; but this is mostly because of how we think - we don't think fully logically all the time, but rather in a fuzzy cloud of ideas and associations and connections, and we want to convey a likeness of this cloud, rather than an exact copy of a chain of very clearly delineated ideas. Most of the quirks of language probably come from how our brains work, and how it matches patterns, and will, for instance, associate a phrase more with a situation than with the meaning of the phrase, or somesuch, after some time. Given some thought, it's obvious that the communication between systems that works on such premises, where the protocols for communication are somewhat fluid, where the information often isn't even completely formalizeable, etc, will be somewhat illogical; yet, it often is surprisingly logical, and without some level of logic in it, it would get very difficult to use at all.

But anyways, back to double negation. It's assumed to be illogical, because in usual e.g. boolean algebra, !!x = x; analogously, in arithmetics, -(-(x)) = x; however, why should we assume that the negating function in a given language is -() or !()? Why not assume it is f(x) = 0x? What will be the value of f(f(f(f(x))))? yes, it's gonna be 0000x, which == 0x == 0. (Let's, for readability and parseability and clarity retain the x there.) Is multiplication by zero illogical because it doesn't alternate between x and 0, the way multiplication by -1 alternates between -x and x? Is this proof that maths is illogical?
No, it's not, but by extension, it's also disproof of one common proof that language is illogical, and I hope everyone would stop claiming that double negatives are illogical and therefore bad, or that they are illogical and therefore proof that language is illogical.

There are benefits to such a negating function as well - no need to keep track of how many negations there are stacked, since as long as there's at least one, the result will be negative; whereas for a system where negations cancel out, you need to keep track of whether you're negative or not at any given point. That is rather taxing.

However, I generally think the search for a totally logical language is misguided and fails to appreciate how our neural network handles information and thoughts.