This October, bishops meeting at the Vatican will consider the possibility of ordaining married men to serve as priests in remote parts of the Amazon region. If the Synod of Bishops recommends such a move to ease celibacy rules and Pope Francis approves, it will be the first time in a thousand years that the Roman Catholic Church has routinely ordained married men as priests.

Though the change is proposed as an exception for a region where the church has struggled to recruit clergy, progressive Catholics hope, and conservatives...