What is the folktale about two people who both claim to posess an object.?

That's from the Bible. It is a story to show the wisdom of King Solomon. Two women were fighting over a baby. Both claimed to be its mother. So, Solomon ordered the baby cut in half, and half given to each woman. One woman was satisfied, but the other cried out to let the other woman have the baby instead of cutting him in half. Solomon said to give the child to the woman who was willing to give up the baby to save its life, because she must be the true mother of the child.

Oh you have this SO messed up -

The tale is about two women who came before Solomon, King of Israel.

They each said a certain baby was thier's, so Solomon decreed to have the baby cut in half and shared that way, knowing full well that the real mother would give up her baby rather than see it cut in half..

It's in the Bible.

Are you sure you're not referring to the story of the biblical King Solomon and the two women who bring him a child, both claiming to be the mother? Solomon suggests that they cut the child in half. One of them says, basically, "Fine, go to it!" The other says, "No! Let her have the baby." Thus Solomon knew who the baby's true mother was.

I think it was the Wisdom of Solomon. Both men claimed the same child. Solomon ordered the child killed if the two men could not reach agreement. One man immediately gave his claim to the child to the other man. It was then that Solomon knew that the man who would rather part with his child than to see him die was the man the child belonged to.

Supposedly Solomon was the judge and the "object" was a baby