By the definitions above, I'm a unitary but not an absolutism theorist. I would describe rationally binding constraints as those that govern prudence, not morality; one can be perfectly prudent without being moral (indeed, if one does not have morality among one's priorities, perfect prudence could require immorality). A brief sketch of my moral theory can be found here.
Why is there only one particular morality?
What would it mean for there to be several? I think morality drops out of personhood. It's possible that other things drop out of personhood, too, or that categories other than persons produce their own special results (although I don't know what any of that might look like), but I wouldn't refer to such things by the same name; that would just be confusing. If there were several moralities it's unclear which would bind actors or how they'd interact. Of course people have all kinds of preferences, but these govern what it's prudent for those actors to do and what axiology is likely to inform their attempts at world-steering, not what is moral.
Where does morality come from?
People. Only people are morally obliged to do or not do things. Only people have rights that makes it particularly moral or immoral to do or not do things with them. (I have a secondary feature to my system that still only constrains people but doesn't refer so specially to acting on them, of which I am less confident; it's a patch for incompleteness, not a grounding principle.) Rights and the obligation to respect them are just a thing that happens when something complicated and persony exists.
Are moral facts contingent; could morality have been different?
Only cosmetically. There could have failed to be any people, or there could be only one person in the world who could find it a practical impossibility to violate their own rights, or such far-flung people that they couldn't interact in any potentially immoral way. But given the existence of people who can interact with each other, I think morality is a necessity.
Is it possible to make it different in the future?
Only cosmetically. If there were no people - or if everyone's preferences changed so they always waived all their rights - or something, then morality could cease to be an interesting feature of the world, but it would still be there.
Why should we care about (your) morality?
Caring is not even morally obligatory (although compliance is), let alone rationally required.
Interesting thoughts. Definitely agree that morality comes from people, and specifically their interactions with each other. Although I would additionally clarify that in my case I consider morality (as opposed to a simple action decided by personal gain or benefit) comes from the interaction between sentients where one or more can act on another based on knowledge not only of their own state but the state of that other. This is because I consider any sentient to have some nonzero moral value to me, but am not sure if I would consider all of them person...
My meta-ethics are basically that of Luke's Pluralistic Moral Reductionism. (UPDATE: Elaborated in my Meta-ethics FAQ.)
However, I was curious as to whether this "Pluralistic Moral Reductionism" counts as moral realism or anti-realism. Luke's essay says it depends on what I mean by "moral realism". I see moral realism as broken down into three separate axes:
There's success theory, the part that I accept, which states that moral statements like "murder is wrong" do successfully refer to something real (in this case, a particular moral standard, like utilitarianism -- "murder is wrong" refers to "murder does not maximize happiness").
There's unitary theory, which I reject, that states there is only one "true" moral standard rather than hundreds of possible ones.
And then there's absolutism theory, which I reject, that states that the one true morality is rationally binding.
I don't know how many moral realists are on LessWrong, but I have a few questions for people who accept moral realism, especially unitary theory or absolutism theory. These are "generally seeking understanding and opposing points of view" kind of questions, not stumper questions designed to disprove or anything. While I'm doing some more reading on the topic, if you're into moral realism, you could help me out by sharing your perspective.
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Why is there only one particular morality?
This goes right to the core of unitary theory -- that there is only one true theory of morality. But I must admit I'm dumbfounded at how any one particular theory of morality could be "the one true one", except in so far as someone personally chooses that theory over others based on preferences and desires.
So why is there only one particular morality? And what is the one true theory of morality? What makes this theory the one true one rather than others? How do we know there is only one particular theory? What's inadequate about all the other candidates?
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Where does morality come from?
This gets me a bit more background knowledge, but what is the ontology of morality? Some concepts of moral realism have an idea of a "moral realm", while others reject this as needlessly queer and spooky. But essentially, what is grounding morality? Are moral facts contingent; could morality have been different? Is it possible to make it different in the future?
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Why should we care about (your) morality?
I see rationality as talking about what best satisfies your pre-existing desires. But it's entirely possible that morality isn't desirable by someone at all. While I hope that society is prepared to coerce them into moral behavior (either through social or legal force), I don't think that their immoral behavior is necessarily irrational. And on some accounts, morality is independent of desire but still has rational force.
How does morality get it's ability to be rationally binding? If the very definition of "rationality" includes being moral, is that mere wordplay? Why should we accept this definition of rationality and not a different one?
I look forward to engaging in diologue with some moral realists. Same with moral anti-realists, I guess. After all, if moral realism is true, I want to know.