I define a an objective claim as one whose truth does not depend on what persons think its truth ought to be. Previously, I wondered whether this definition precluded the possibility of an objective morality, but I've just realized that this isn't the case.
In physics, we correlate initial states with final states. Based on observations of the initial state, we predict attributes of the final state. Physics is objective because we can test physical laws independently of how we feel about those laws. We can even control our experiments by limiting the knowledge of the experimentalists performing the work. We can look at the radioactive decay rates of different isotopes, and it doesn't matter whether I think Carbon 14 ought to decay faster than Plutonium 238, it simply doesn't.
So, what would be the equivalent for moral objectivity?
Two different moral histories can result in the same physical intermediate state. If morality were objective, then observers could see the intermediate physical state evolve into different physical final states with different moral values.
For example, suppose we obtain two apparently identical tanks of heating oil. The tanks will be used to heat the ACME Corp warehouse. One tank is paid for by monies secretly stolen from the local orphanage. The other tank is paid for from the ACME coffers. If morality is objective, then we might expect that the two tanks of heating oil will have different caloric outputs. Maybe, the tank funded by ill-gotten money will generate less energy than the tank paid for out of the ACME bank account.
One can argue that the experiment won't work because we arranged to have the money stolen from the orphanage as part of the experiment. However, we can design other experiments that will bypass this problem.
Maybe stolen appliances will be more vulnerable to malfunction and failure than appliances purchased at a fair price. Or, what if guns and ammunition used by terrorists will be less effective, less accurate than weapons used by security forces? Does fair trade coffee always taste better than unfair trade coffee? Would medical knowledge obtained through cruel experimentation on humans work less effectively than the same knowledge obtained through ethical practices?
A priori, the world might have demonstrated such objective morality. In reality, this is simply not the case. Bad things happen to good people, and good things happen to bad people. Physics turns out to be independent of the moral history of the initial state. There is no universal justice as far as we can determine, and so morality is a property of our personal feelings, not a property of actions or things in and of themselves.
Christians will claim that everything gets balanced out in the hereafter, but this is just irrational. Claiming that universal justice exists is like claiming that cucumber sandwiches are conserved and that eating a cucumber sandwich on Earth results in the appearance of an identical sandwich on the other side of the galaxy. If we are to believe in objective morality, we ought also to believe in sandwich conservation laws, too.