Consider sophisticated moral relativism. Say Anne's idealized self would conclude that abortion is wrong, whereas Bob's would not. So 'Abortion is wrong' is true for Anne but not for Bob. Do they disagree? They would not if they meant different things by 'abortion is wrong' (e.g. 'abortion is wrong to me'). They would just be talking past each other, as if Anne were to say "I like icecream," and Bob replied "No I don't!"
So, to preserve genuine disagreement, they must be expressing one and the same proposition. That much is shared and universal. What's relative is the truth (not the meaning) of what's said. Anne and Bob are both talking about the proposition that abortion is wrong, but the truth of the matter differs between them. (This seems crazy if truth is meant to correspond to worldly facts - how could facts be relative? But it makes more sense if we see truth as an epistemic construct.)
How are we then to understand the truth predicate? Jack points out to me that problems arise when we ask Bob to assess Anne's assertion that "'Abortion is wrong' is true." If 'true' in Anne's mouth means true-for-Anne, then it hardly seems that Bob can dispute her claim. It really is true-for-Anne that abortion is wrong, after all. But note that the problem again lies in attributing merely semantic relativity. We should instead insist that Anne and Bob mean exactly the same thing by 'true'. They just assess it differently. Bob correctly judges that Anne spoke falsely. Anne correctly judges that she spoke truly. They're both right, and they also genuinely disagree with each other -- a disagreement that will persist even upon semantic ascent. True?