Moral relativism: Moral claims are of the form "[Subject] OUGHT [Action] [Object]" Moral relativism is the claim "[All moral claims] ARE [subjective]." Note that ARE is the plural for IS, so that moral relativism is not itself a moral claim. (Is-ought distinction.)

For something to be true or false, it has to state what was, is, or will be the case. Whereas moral claims of what should be the case are either right or wrong, which is completely separate from true or false.

Motion is relative, and so the motion of any object can be stated from any frame of reference. Likewise, the morality of any action is relative, so the right/wrongness of any moral claim can be judged from any moral code. The claim that "Moral system X is right" is correct under moral system X, but not under any other moral systems. To claim that "Moral system X is true" is the is/ought fallacy again.

Some frames of reference make more sense to use than others. We infer the motions and positions of distant celestial bodies with respect to our own planet Earth because it is central to us. A moral system that uses happiness as the highest good would make more sense than, say, a moral system that wants perfect peace or pure freedom. Happiness is desired by all, and each others' happiness is desired by most because of our altruistic nature. So probably the best moral axiom that most people would reasonably go for is one of happiness as an ultimate.

Perfect peace would not be a good ultimate. A lack of battling of any form would disallow us from reasonably resolving worldly problems. (This would involve more than just physical battling.) And pure freedom is not a good ultimate, because it means social anarchy where we are free to inflict suffering and pain upon each other - and not many people would desire that to be the case.

I don't understand the deep threat perceived in the non-objectivity of morality. The subjectivity of it does not lessen its strength. Similarly, that there is no objective purpose in life does not weaken the subjective convictions we create for ourselves. The appearance from moral relativism is that there is no accountability and no true righteousness. Of course there is accountability and righteousness, it's just that it is subjective.

That's it? What a disappointment.

You're confusing some sort of relativistic epistemology where our happy relativist thinks that beliefs about morals can be selected by individuals, rather than morals themselves. Totally wrong.

The simple fact is that we each pick our own morals, and while some may argue that their morals should be absolute or assert that they really are absolute, but they are mistaken. Moral relativists are under no obligation to accept all of these claims about morals.

The simple way to demonstrate that there are no objective morals is to ask how you test if a given moral rule is objective. You may appeal to some source: biology, evolution, social harmony, happiness, greatest good, whatever. But as you can see, this overriding goal is not objective. At best, you can say that one moral rule can be objectively derived from another general moral goal but you have no means of ruling if this general moral goal is the right one.

In your follow-up, you talk about happiness. So what's so great about happiness? More importantly, what your discussion lacks is any means of checking whether your conclusion is correct or not. How do we know whether maximizing happiness should be the goal of morality? What about justice, retribution, individual freedom, anarchy? Instead of maximizing happiness, what if the moral goal should be maximizing fluffy bunnies? It's bizarre and ludicrous to be sure, but there is no objective reason for deciding to maximize happiness instead of fluffy bunnies. Perhaps many people wish to maximize happiness, but this doesn't make it objective.

Consider this thought experiment.
Imagine two worlds otherwise quite like our own: one--as our host has described--in which morality is real and objective, and another in which "[moral] nihilism is a fact" but we are in no way obligated to believe it.
Could a person within either of these worlds determine which world he was in?

Unless either world is demonstrably impossible, I contend that he could not. Is a world with moral nihilism impossible? Perhaps not.

While Ebonmuse is entirely correct in claiming the following:

...if moral nihilism were true, there are no "shoulds" at all - it would not be true that we should do anything - and therefore we would be free to ignore the truth of moral nihilism and live as if it were false.

This alone is insufficient to discount the possibility of moral nihilism. Truth is indifferent of our beliefs and motivations. He treads dangerous ground when he claims:

If [moral nihilism] is true, it is not the case that we should believe it to be true.

There is a subtle distinction between saying that we should not believe it to be true and we should believe it to be not true. The key is that no moral statements follow from moral nihilism, not even that you should believe in moral nihilism. (Again, truth is indifferent to our beliefs and the motivations for those beliefs.) As such, moral nihilism is undoubtedly dissatisfying to those who seek to make moral claims and create moral systems, but this does not render it inconsistent.

Another challenge I anticipate will be made of moral nihilism is that it is subject to the same paradox of "meta-morality" that moral relativism was:

By definition, morality is precisely that system of thought which states how intelligent beings should act. Therefore, claims about how we should make moral decisions are themselves moral claims. Meta-morality is morality, and again the moral relativist cannot escape logical paradox.

I'm not entirely sure what you mean by meta-morality (as the term seems redundant), but meta-ethics--which deals with questions like "are there objective values?"--is strictly not morality as you have defined it. Moral nihilism makes no claims over how people should act or how they should make moral decisions but whether moral values exist.

So--in summary--I have claimed that the possibility of moral nihilism can only be discounted if it is demonstrably impossible, and the mere fact that no moral statements can be drawn from moral nihilism--including the obligation to believe it to be true--has no bearing on the truth of the matter.

(Also, is it too much to ask for you to address non-cognitivism as well before you assert that moral realism is the only remaining possibility?)

1. The comparison between science and morality is weak at best. The problem is that while is-claims can be verified empirically, as you noted, value-claims cannot. Thus, for science one can simply look at the world for guidance, but in the case of morality one cannot. The problem is that the ONLY source for moral knowledge is from personal opinion or that of others, period.

2. The comparison between math and morality is also weak. Math is deductively demonstrable while morality is not. It only this deductive demonstrability which gives math its objective nature. Morality is not deductive and therefore is not objective.

3. Post-modern criticisms would suggest that even math and science are not completely objective. Claims in these areas can only be considered to be true within the framework of various assumptions, axioms and/or values which are taken for granted. Since morality seems to be little more than assumptioins, axioms and/or values combined with a bit of inductive and inferential reasoning, it would seem to suggest that these post-modern criticisms apply ten-fold to moral claims.

4. I know this one has been said in a comment, but I don't think that you responded to it very well: Moral relativism is a fact-claim, and the moral relativist is not bound to total relativism. The idea that others should accept moral relativism is based in the moral claim that others should accept true fact statements. This moral claim can be completely relative without denying that moral relavitivism is objectively true.

Perhaps you could respond to these claims, or point me to where you have already done so.

Morality is a method that deals with how humans interact with each other (i.e. as some call it "human causality").

No my friend, that is sociology. Sociology describes how people interact, morality describes how people want others to interact. Science does not dictate how we should apply it or what devices or experiments we should perform, that is a matter of opinion.

Science is a "man-made" method that deals with how material objects interact with each other. Morality is no different. Morality is a method that deals with how humans interact with each other (i.e. as some call it "human causality"). You can use quite similar objective tests to reach rational conclusions with either method.

However, this does not mean one can freely choose to disbelieve nihilism any more than one can make onesself believe in geocentrism by pure force of will. Can you mould your own perceptions so readily? Certainly it is not an ability I possess - I think I would be much happier if I believed in a benevolent deity, pleasant afterlife etc; the lives of those around me would probably benefit too, as I would be more selfless and have greater concern for society. But the evidence does not convince me.

So, in your examination of morality, are you considering what is true, scientifically, or what you want people to believe is true? Because I for one would much rather everyone believed in your universal moral code, but again, the evidence does not convince...