First, so we're all on the same page, let me quote the good professor:
In its most extreme form, speaker-relativism amounts to the view that right and wrong are in the eye of the beholder, since it says that each person is correct in making moral judgments according to what seems right or wrong to him. But speaker-relativism also has less extreme forms, according to which the correct moral standard is determined by the speaker's culture, for example. On this version of the view, each person is correct in making moral judgments according to what his own culture deems right or wrong. Thus, a speaker-relativist might say that whereas we are correct to judge slavery wrong, the ancient Romans were correct to judge it differently, because their cultural standards were different.
Imagine the following short conversation between present-day Bob and an ancient Roman:
Bob: "Slavery is wrong!"
Roman: "Slavery is okay!"
These guys are disagreeing, right? I certainly think so. Slavery can't be wrong and okay at the same time, at least if we're talking about the same instance of slavery. But look what happens when we apply the speaker-relativist's translation of moral terms:
Bob: "Slavery is forbidden by 21st Century Western culture!"
Roman: "Slavery is permitted by ancient Roman culture!"
Here we don't have a disagreement anymore. These guys are talking about two different things, and it's possible for both of them to be right. (In fact, both of them are right.) Let me make clear that the problem here isn't that we can't determine who is right, or that we can't resolve this disagreement. The problem is that speaker-relativism presents people as making compatible claims when they're actually disagreeing. As a result, it's a bad theory of what moral terms mean.