This topic has occurred to me as a result of a recent discussion about aesthetics and earlier discussions about morality. My own view is that we can make no case for any given moral system being objective. One always begs the question when trying to devise an argument in which one argues that a particular moral rule is objective. The same situation occurs in aesthetics.
However, one's feelings about such matters are often quite different--or at least my feelings are. I FEEL that moral judgements, or certain moral judgements, are quite real--not subjective at all. And even in aesthetic matters, I FEEL (though less strongly) that certain aesthetic judgements are quite real.
The question is whether these strong feelings we have matter--i.e., whether they are an indication that, though we cannot build a case, that perhaps some moral judgements and some aesthetic judgements are after all objective.