What does Paul’s shoe have to do with the belief he has that Coffey is innocent?

When Paul gave Coffey his shoe, he discovered that Coffey can’t tie a shoe or anything else. The day he was caught, his lunch was tied with string. If he had undone his lunch and given sausages from his lunch to the dog, he wouldn’t have been able to tie the lunch up again. Therefore, Coffey couldn’t have killed the little girls.