Joy, who likes to go by the uglier name of Hulga, has an artificial leg. She is both proud of it and sensitive about it, letting no one but herself touch it until Pointer demands it. The text says she was
as sensitive about her artificial leg as a peacock is about his tail.
Hulga is also very proud of her PhD in philosophy, which has led her to embrace atheism. She thinks she is smarter and more world-wise than other people, especially the simple country people all around her, but finds out by the end of the story she is wrong.
O'Connor said of the wooden leg:
We're presented with the fact that the PhD [Hulga] is spiritually as well as physically crippled . . . and we perceive that there is a wooden part of her soul that corresponds to her wooden leg.
By the end of the story, Pointer has both stolen her wooden leg out of spite and demolished her idea that she is the worldly one by revealing himself as a hard-hearted con artist and atheist, far more evil than anything she has ever encountered. She learns she is not as worldly as she thinks, because she has never confronted pure evil.
Hulga cannot rely on her wooden leg to make her whole, because it can be stolen from her. Likewise, her rationalist philosophy is a disability, a "wooden part of her soul."