why do you think twain has pudd`nhead tell the joke about "half dog"?

Puddn'head has newly arrived in Dawson's Creek and a stranger to the residents there. Shortly after arriving in town (he hopes to open a law practice there), he comments about a barking dog. In an attempt to be sarcastic, Wilson mentions that he'd like to be half owner of the dog. The people question why he would wish for such a thing, and he answers that is half the dog were his..... he'd be able to kill his half. Rather than think him funny, the townspeople thought him foolish. This event is where he earned the name "Puddn'head."

The allegory is that the dog represented a divided nation in race, politics, social class and economics that lead to the civil war. David’s symbolism to cut the dog into two, and then kill his half, as a joke, cutting the annoying barking part out while keeping the good parts alive is misunderstood which let to social justice and truth in the end of the novel.