One Reason I'm Not a Moral Relativist
According to moral relativism, ethical truths are determined by the attitudes, desires, and preferences of a person or group of people.
However, people tend to think ethics is consistent. We make ethical arguments with each other all the time, which doesn't make sense unless moral claims are subject to the laws of logic. Something can't both be vicious and virtuous, nothing can be both obligatory and forbidden. And people are perfectly capable of having inconsistent preferences. Sometimes Joe might love pickles, others he might hate them. I like carrots in some dishes and can't stand them in others, and there doesn't seem to be any rhyme or reason about it.
A relativist might say that I'm improperly assuming our moral preferences must share all characteristics with our taste preferences. But it's trivially easy to find cases where people hold contradictory beliefs and attitudes about morality. If it weren't, moral dilemmas wouldn't be nearly as common as they are.
A relativist can avoid this problem by saying ethical truths depend on someone who happens to have consistent moral attitudes, as is the case with Ideal Observer Theory and Divine Command Theory. But I don't think either of these views flow from most of the arguments for relativism, and some philosophers don't consider them forms of relativism at all.