“Someone might say…”Muslims should go to Mecca, Catholics to Mass.” If you’re not a Muslim, though, you don’t really think Muslims should go to Mecca, and if you are a Muslim, you don’t think that anyone, not even a Catholic, has a duty to go to Mass…Obviously, Muslims believe that they ought to make the hajj and Catholics that they ought to go to Mass. But if you don’t have the beliefs that give those acts their meanings, you presumably think that the people who do think so are mistaken. Either Muhammad was the Prophet or he wasn’t. Either the Koran is the definitive Holy Writ or it isn’t. And if he wasn’t and it isn’t, then Muslims are mistaken.”
One might argue, in response, that some beliefs are not about facts–they’re not about being mistaken or not. Rather they are about values. When a muslim says that he ought to go to Mecca he is saying that according to a framework that he accepts for personal or practical reasons, he ought to go to Mecca. To not believe this is not necessary to believe that the muslim belief is mistaken. It is rather to simply adopt a different framework about certain actions in which going to Mecca is not present.
Appiah would point out, I think rightly, that there is tremendous importance in why a muslim goes to Mecca on the hajj. One might mistakenly make the point that a Muslim ought to go to Mecca because that is what she believes she ought to do. She, as a Muslim, however, would not agree with you. She would aver that she ought to go to Mecca because God commanded it.
The question, however, is what we think justifies belief. A Muslim may be justified in believing she ought to go to Mecca because she thinks God commanded it and I may be justified in believing that a Muslim ought to go to Mecca because according to a theory that she accepts, it is what she ought to do. If both of these are justified beliefs then even if we believe what we believe for different reasons it does not mean that I necessarily think she is mistaken in her belief simply because I do not hold it.