Example: How many electricians do you need to screw in a lightbulb.
One. They are perfectly competent human beings.
For lack of something more constructive to do, I'm going to prove that anti-jokes are jokes, and thus cannot be anti-jokes. However, taking apart humor is never funny; it just makes the existing humor less funny. But sacrifices must be made.
So, what does a joke require to be "funny"?
Example: Assassins do it from behind.
If you don't know what the joke is referring to, or why it refers to something in a such way, then one cannot find the humor within it. In this, one must know that assassins are stealthy killers. And that "doing it" can be taken as a euphemism, in addition to the literal meaning. Without these two pieces of knowledge carried "into" the joke beforehand, one would not have found this humorous.
Example: What's a miracle? When a busload of lawyers drives off a cliff.
What's a tragedy? There was an empty seat.
This is only true for some jokes, and is quite similar to the above. Some jokes find their humor in making light of or satire of some stereotypes. This joke requires one to have knowledge of the stereotype that "Lawyers = bad". Otherwise, one would definitely not find the joke entertaining.
Chains: Some jokes attach themselves in chains. That is, strings of jokes all based upon the same thing which play off the same general idea, just with differing punchlines. Examples include dumb blonde jokes, lightbulb jokes, and dead baby jokes.
Surprise: All jokes lose their "funniness" after a few iterations. Why? Because the punchline is not new any more. Much of the original hilarity from a joke stems from the unexpectedness of the punchline. Even if you know generally what the punchline will be, as long as you do not know it verbatim, you'll still find it funny.
On to the "anti-jokes". How well do they match up with "real" jokes?
Let's take a new anti-joke, lest I risk being repetitive.
Doctor, doctor! I have a carrot in my ear!
Sir, that's the least of your problems. You have AIDS.
So, time to examine this.
First, does this require any prior knowledge? Yes, yes it does. This requires one know realize that AIDS is bad. It also requires one to know that it is an "anti-joke", presumably, to prevent offense from the punchline, rather than a laugh.
Second, does this require a stereotype? Well, not exactly. Except maybe that one goes to doctors when one has a medical problem?
Third, is this based off a joke chain? That it is; there are multiple "Doctor, doctor" jokes.
Fourth, is surprise necessary? Yes. The entertainment value of the joke comes from the plain deviation from the normal response, or the normal array of responses.
So, taking this "anti-joke" as an example for all other such things, is this a joke? This "anti-joke" derives its humor from the fact that one expects some witty line about how carrots are supposed to be eaten. Instead, it surprises the audience (as stated above) by telling the patient that he has AIDS. Using this "anti-joke" as a basis for all other "anti-jokes", one has come to conclude that "anti-jokes" are really just jokes, though perhaps their own subtype, like chain jokes, or one-liners. But to call them anti-jokes would be a misnomer.
If "anti-jokes" are not really anti-jokes? What would a true anti-joke be like?
It would have to require no prior knowledge whatsoever (thus eliminating the first three criteria) and would have to be completely expected.
For example: Why hello there.
If that made you laugh, then I've failed at making a true anti-joke. Ah well.