Austin Nedved is kind enough to be providing me with a useful foil with which i can work. His comments are reasonably well informed, well presented, and represent common forms if objections that I gave encountered. I hope that Austin does not mind my use of these conveniences.
AUSTIN: [W]hat we desire the most is justice.
ALONZO: Regardless of what your personal concept if justice may be, we can falsify this claim just by pointing out the massive differences in what different communities call "justice". This alone should disprove any claim that there is a thing called "justice["] that we desire the most.
AUSTIN: The conventionalist argument you are making here refutes itself. A great number of cultures subordinate experimental science to divine revelation. Some have even rejected the validity of experimental science outright. Others still have rejected anything that conflicted with what Aristotle had said. But surely this does not entail that there is no truth, or that there are no legitimate sources of knowledge. The Conventionalist's claim that the multiplicity of understandings of what constitutes truth prevents us from having an objectively true understanding of truth, is self-defeating.
I was not making a conventionalist claim. My argument was not, "Everybody disagrees, so there is no truth". Instead, my argument took Austin's claim as a claim that has implications in the observable world, and showed that the observations of the world falsify the hypothesis.
Let us assume that somebody were to make the claim that what we desire most is broccoli. If true, this would have implications for what we would expect to observe in the eating habits of different cultures. That is to say, we would expect to find people throughout the world eating a lot of broccoli if it were available, and putting a great deal if effort into making sure it is available.
Let's assume that we discover in places where broccoli is available that one group mostly eats potatoes, another mostly eats beef, and yet another mostly eats pasta, while a fourth mostly eats broccoli. In the light of these observations, it would be hard to maintain the thesis that what we desire the most is broccoli.
One way out of this would be to note that the first culture's word for potatoes us 'broccoli'. The second culture calls beef 'broccoli', while the third calls pasta 'broccoli'. If this us what we find, then thus too would refute the thesis that there is a single thing called 'broccoli' that we desire.
Neither horn of the dilemma makes use of "the conventionalist argument". That is to say, if one were to make these objections to the claim, "What we desire the most is broccoli," we would not expect the broccoli theorist to answer, "Your conventionalist argument refutes itself." The conventionalist argument is not in play.
There are those who argue that the fact of moral disagreement among individuals or cultures implies that there is no fact of the matter. One leading proponent of this argument was J.L. Mackie. He had two main arguments against 'objective value' - one of which is the Argument from Disagreement. People have different opinions on what has value, so objective value does not exist.
However, this us as problematic as saying that people have different opinions on the age of the earth, so there us no fact of the matter. Or, even more problematically, people have different opinions on objective value, so there is no objective value.
Now, we could interpret Mackie as saying that we gave no objective way to resolve these disputes. However, this is a mere assertion - not an argument. And it us a question-begging assertion at that.
Anyway, I am a moral realist. I hold that there are moral facts independent of the sentiments of the speaker. There is moral disagreement, but that simply implies that some people are wrong. It does not imply that there is no fact of the matter.
The types of things that people can be wrong about include beliefs in a god, or intrinsic values, or making inferences from false premises such as a social contract, impartial observers, or decisions made behind a veil of ignorance.
Certainly, one of the things we can know in this world of facts is that it is not the case that what we desire the most is broccoli. We know this by looking at the world and seeing people showing great interest in a number of things, many of which are not broccoli. This appeal to the fact that people have a number of different likes and dislikes is not a "conventionalist argument". It is an observation that falsified the hypothesis, "What we desire the most is broccoli." This same set if observations also falsified the thesis, "What we desire the most is stamp collecting," and, as it turns out, "What we desire the most is justice."
Instead, we have a range of desires - for sex, for pleasure, to eat, to drink, for companionship, to avoid pain. We have the capacity to learn desires based on our experience - cultural preferences, learned fears, and other likes and dislikes. In this, there is no evidence that what we desire most is broccoli or stamp collecting or justice.