You knew I was going to have to get into this one… 
No. It is literally impossible. If most people use a word to mean X, then X is one of its meanings. Words are, by their nature, arbitrary symbols. They are sounds, or marks on a page or a screen. Nothing more, nothing less. We use them as media of exchange, much the same way we use money.
What things used to cost, and what words used to mean, are irrelevant to what they cost or mean now. How one group uses a word, phrase, or construction has no bearing on the value of that word, phrase, or construction for another group.
This doesn’t mean that today’s usages aren’t related to past usages, or that you can’t relate usages across demographics, but it makes no sense to point to some other time, or some other location, or some other group, and say that their dialect is “wrong” about what a word means or how it should relate syntactically to other words.
Now, if think the word “pig” refers to a horse, then I am truly mistaken about what that word means, but only because people will not understand me when I use the word.
I’m tempted to say you’re begging the question, but I’ll stay out of that snakepit. You say they make a mistake and are ignorant of a word’s correct meaning – but all that boils down to is saying that their usage differs from yours and from that of people you associate with or consider authorities. There is no other basis of comparison. You can’t point to logic, because there is no logical relationship b/t a sound-symbol and its mental associations. It’s a mere trigger. You can’t point to etymology, because words and syntax are constantly evolving and changing.
If person A says phrase X to person B, and person B correctly understands person A’s meaning and does not perceive any error, then who are you or I or anybody else to tell them that there has been some “mistake”?
If everyone understands it (and everyone does – I have never met an adult native speaker of English who would misunderstand the intention of “I ain’t no idiot”), then there is no gap between expression and interpretation. We could declare such usage “incorrect”, but what value would such a declaration have?
Absolutely not.
I encountered a range of reactions from my students as they struggled with academic English. Some were very concerned with what was “correct” and thought they should apply this standard universally. Others thought that being required to use standards other than the ones they used for conversation and casual communication was insulting, and a denigration of their heritage. Others merely resented the workload. Others adapted easily. Etc.
I told them it was simply a written dialect with its own history. Much of what they would learn would help them express themselves more clearly, fully, and precisely. Some of it was just tradition that no longer served a useful purpose. But it was a tool they would need to learn the use of, and when used properly it would serve them well.
Nope. Why would we? It would be like saying “I don’t care how many people pay $2 for a cup of coffee, a cup of coffee does not cost $2.”
Some have raised the issue of clarity on this thread, and that’s a valid issue. Getting students to understand that they were not communicating effectively when they used overly general stock phrases and cliches, ambiguous constructions, arcane and ephemeral (to their readers) lingo, weak or faulty arguments, and confusing punctuation – that was a challenge. I did not allow them to use “random” as a synonym for “unknown” or “irrelevant” in college-level essays, for example, and advised them not to use it in those senses in job applications and interviews; not because that usage was somehow “wrong”, but because it would likely be misunderstood by their audience or cause them as writers to be judged negatively.
Language changes. Different groups have different standards. The standards are real, and ignoring them has real consequences. But there is no inherently “right” dialect, written or spoken, for all times and places. You can declare a usage “wrong” all you like – but it is a meaningless declaration.
