It's no use asking me to supply a definition for you. The main reason I am a moral anti-realist (essentially a non-cognitivist) is because it seems impossible to define moral terms (without using other moral terms).
If you refuse to choose a definition then how are we supposed to engage in any type of discussion? We can't even reduce ourselves to grunts and whistles without, first, assigning meaning to grunts and whistles.
As it turns out, I am an anti-realist too. And I am a non-cognitivist. I am also a realist and a cognitivist.
How is it the case that I can be all of these things? Don't they contradict each other?
No, they do not. What people take to be different moral theories is, in fact, different moral languages. Realism and anti-realism no more contradict each other than Einstein’s theory of relativity in German contradicts Einstein’s theory of relativity in Chinese
They appear to contradict each other because both theories use the same terms. So, "Moral properties do not exist" in one language appears to contradict “Moral properties do exist” in another language. However, they are different language (as opposed to different theories) precisely because they use two different meanings of the term "Moral properties".
Here's the argument as I see it.
Person 1: Choose your definition of 'morality'.
Person 2: (Chooses a definition).
Person 1: Now, defend that definition as the correct definition.
Person 2: I cannot.
Person 1: Then we can throw out that definition of morality.
I take this to be logically equivalent to the following.
Person 1: Choose your language for expressing Einstein’s theory of relativity.
Person 2: I choose German
Person 1: Now, defend that language as the correct language for Einstein's theory of relativity.
Person 2: I cannot.
Person 1: Then we can throw out Einstein's theory of relativity.
Throw out moral terms, if that is what pleases you. I do not need it. Throw out every moral term – just cross it out of the dictionary and resolve never to use any of them ever again. Let us speak the language of moral anti-realist non-cognitivism.
Desires exist. Desires are reasons for action. People still act so as to fulfill the most and strongest of their own desires. Some desires are still malleable – they can be created or destroyed, strengthened or weakened, by social forces such as praise, blame, reward, and punishment. People still have “reasons for action that exist” (desires that they seek to fulfill) for promoting desires that tend to fulfill other desires. We have a whole family of propositions here that are objectively true or false.
Only, because we have adopted the language of moral anti-realist non-cognitivism we are not reducing any of these statements to moral terms.
Or, we can choose a different language. Here's a suggestion:
"Good1" means "Is such as to fulfill the desires in question." That is to say, to call a state of affairs S "Good" is to say that there is a desire that P and P is true in S. It follows from this that those people who have a desire that P also have a "reason for action" to realize S.
They may also have reasons for action that realize not-S. That is, they may have a desire that Q, where Q is false in S. That desire that Q may be stronger than the desire that P. In this case, the agent has more and stronger reason to realize not-S than S. Let us use the term "Good2" to refer to states of affairs according to how well they fulfill all of an agent's desires. "Good2" means "Is such as to fulfill the most and strongest of an agent's desires." Notice that Good2 is just a species of Good1 where "The desires in question" are all of an agent's desires.
Now, we have an agent with a desire that P and a desire that Q. Now, if there is a state of affairs S' where P is true and S and Q is true in S, then the agent has reason to realize S'. He has more and stronger reason to realize S' than to realize S. So, in this case, let us say that S’ is "better1 than" S. "Better1 than 1" means "Fulfills the more and stronger of the desires of the agent."
Now, let us introduce a second agent. Agent2's desires are malleable, meaning that Agent1 has the power to choose Agent2's desires. Agent1, recall, has the most and strongest reasons for action to realize S' (since it fulfills both his desire that P and desire that Q). Agent1 can give Agent2 a desire that R1. R is true in S', so this means that Agent2 will have reason to realize S'. Or Agent1 can give Agent2 a desire that R2. R2 is true in not-S', so this means Agent2 will have reason to realize not-S'. Agent1 has more and stronger reason to give Agent2 a desire that R1.
Now, let us define Good3 in such a way that we will only use this definition of “Good” when we are evaluating desires. A desire is Good3 if it is such that it tends to fulfill the most and strongest of other desires. That is, it will tend to motivate the agent to act in such a way so as to make or keep true the propositions that are the objects of other desires. In this case, R1 is good3.
We can also add Good4 by the way. Good4 will only be used to evaluate intentional actions. A Good4 action is an action that a person with Good3 desires would have performed. That is to say, a Good4 action is an action that would fulfill the most and strongest desires of a person with Good3 desires.
Since all of these numbers are confusing, let us say that Good1 = Good in the generic sense. Good2 = Practical goodness. Good3 = Virtue. Good4 = Obligation,
What I have done here is invented a language. Furthermore, I would claim that this language is so close to common English that most native English speakers (particularly those who have not committed themselves to speaking a different moral language but who take English moral terms "as is") would be hard pressed to tell the difference.
Now, the challenge is, "Can I defend this language as being the correct language?"
Answer: I cannot.
But, see the previous section. Choosing between moral languages is no different than choosing between Einstein’s theory of relativity in German or Einstein’s theory of relativity in Chinese.
It would be useful if we all spoke the same language. It would certainly make communication more efficient. However, whether that language should be English, Spanish, Chinese, or some other language is not a subject that interests me. As for me, I speak English, and that is the language I will continue to write in. If you prefer a different language, I will leave it to you to do the translation.
It would be useful if we all spoke the same moral language. It would certainly make communication more efficient. However, whether that language should be cognitivist, non-cognitivist, realist, or anti-realist is not a subject that interests me. I speak cognitivist realist. If you read a different language, I will leave it to you to do the translation.
Though, actually, I do speak non-cognitivist anti-realist and can do the translation myself if I have enough reason to do so. I can't say the same about Chinese.