The Merchant's Wig
There was once a young merchant who was sailing on his ship from India to Europe. He had his entire fortune on board. Due to a terrible storm, he was forced to throw all of his merchandise overboard. He was so upset that, that very night, his wig turned gray.

Explanation:
This joke comes from the Critique of the Power of Judgment (1790), one of Kant's three Critiques. He contrasts this version with another version: the merchant's hair turns gray. But in that version, we listeners or readers become more concerned and involved. We empathize with the merchant and feel his pain more than in the first version. When it's just a wig, we are in a better position to find the story amusing. We hear the narrative as a joke rather than as descriptive speech, a real story about the world that can be either true or false.

Kant thinks that when we hear or read a joke as a joke, we need to be removed from the situation or story; we cannot have something at stake in it. He calls this disinterestedness, which turns out to be a key principle in his aesthetic theory and account of beauty. "Taste is the faculty for judging an object or a kind of representation through a satisfaction or dissatisfaction without any interest. The object of such a satisfaction is called beautiful." While the humorous is not the same as the beautiful, Kant thinks that our response to both of them requires a kind of disinterestedness. A notion of disinterestedness can be found in the writings of Shaftesbury (Anthony Ashley Cooper) (1651–1713) and other eighteenth-century aesthetic theorists. Today the notion of disinterestedness remains controversial. Some theorists think that the idea can be better captured by the concept of absorbed attention or focus.

Happy Funeral Mourners
A man's rich relative dies. Suddenly he is rich. To honor his relative, the man wants to arrange a solemn funeral service. But he keeps complaining that he can't get it quite right.
"What's the problem?" someone asks.
"I hired these mourners, but the more money I give them to look grieved, the happier they look."

Explanation:
This is a second joke from the Critique of the Power of Judgment. Kant is using it to illustrate his incongruity theory of humor. When we learn that the mourners are happy because they are getting paid, he says, our expectation is suddenly "transformed into nothing."
The philosophical underpinning is that there are at least two levels of satisfaction at work: we feel sadness, joy, etc. (first-order satisfaction) and then can approve or disapprove of it using reason (second-order). A man can be glad that he is receiving an inheritance from a deceased relative, yet disapprove of his gladness. "The object can be pleasant, but the enjoyment of it displeasing."
The joke turns on something similar happening with the mourners. They are so happy that they are getting paid (second-order) that they are no longer able to look sad (first-order).
There is a similar anecdote in Plato's Ion, a dialogue about a professional reciter (a "rhapsode") named Ion, who aimed at moving his audience. Ion says that when he looks out at the audience and sees them weeping, he knows he will laugh because it has made him richer, and that when they laugh, he will be weeping about losing the money.