Non-cognitivism
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Non-cognitivism is the meta-ethical view that ethical statements (such as 'Killing is wrong') do not assert propositions; that is to say, they do not express factual claims or beliefs and therefore are neither true nor false (they are not truth-apt). This distinguishes it from moral realism, which holds that ethical statements are objectively and consistently true or false; ethical subjectivism, which proposes that ethical statements assert propositions about the speaker's own attitudes; moral skepticism, which proposes that all ethical statements are false; and cognitivist irrealism, which asserts that ethical statements are true or false (this is ethical cognitivism), although there are no worldly facts to make them true or false.
There are three major schools of thought among non-cognitivists as to what meaning ethical statements do have. Emotivism, while not necessarily non-cognitive, is generally defended as a non-cognitive theory. Emotivists suggest that they are expressions of emotional response, desire and aversion, approval and disapproval: under this view, the statement "Killing is wrong," for example, can be paraphrased as "Boo to killing!" or "Down with killing and killers!". Statements such as "helping people is good" can be paraphrased as "yay helping people!" A close cousin of Emotivism, developed by R. M. Hare, is called Prescriptivism. Prescriptivists interpret ethical statements as commands or prescriptions: "Killing is wrong," in this model, is equivalent to "Do not kill!" Expressivism, including Simon Blackburn's quasi-realism and Allan Gibbard's norm-expressivism, defends that non cognitive attitudes underlie moral discourse and this discourse therefore consists of non-declarative speech acts, although accepting that its surface features may consistently and efficiently work as if moral discourse were cognitive. The point of interpreting moral claims as non-declarative speech acts is to explain why moral claims are neither true nor false. "Boo to killing!" and "Don't kill!" are sentences that aren't candidates for truth or falsity.
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Arguments in favor of non-cognitivism
The principal argument in favor of non-cognitivism is that ethical properties, if they exist, have no observable effect on the world. The amount of killing, for example, is not altered by its being wrong or right, but only by individual distaste for killing. Thus there is no way of discerning which, if any, ethical properties exist; by Occam's Razor, the simplest assumption is that none do. Non-cognitivists then assert that, since a proposition about an ethical property would have no referent, ethical statements must be something else. (Note that moral skeptics would say ethical statements are indeed propositions with no referent, and therefore false.)
A person stating that killing is wrong presumably has some negative emotional reaction when confronted with the concept of killing; the statement of its wrongness can thus be construed as a direct consequence of this reaction. From this observation, emotivists claim the statement is not distinct from the reaction itself. Likewise, a person telling another that killing is wrong probably does not want this other person to then go off and kill someone, and may be explicitly attempting to stop him from doing so. Thus the statement "Killing is wrong," calculated to prevent someone from killing, can be described as an exhortation not to do so.
Arguments against non-cognitivism
One argument against non-cognitivism is that it ignores the external causes of emotional and prescriptive reactions. If someone says, "John is a good person," something about John must have inspired that reaction. If John gives to the poor, takes care of his sick grandmother, and is friendly to others, and these are what inspire the speaker to think well of him, it is plausible to say, "John is a good person (i.e. well thought of) because he gives to the poor, takes care of his sick grandmother, and is friendly to others." If, in turn, the speaker responds positively to the idea of giving to the poor, then some aspect of that idea must have inspired a positive response; one could argue that that aspect is also the basis of its goodness.
Another argument is the "embedding problem." Consider the following statements:
Eating meat is not wrong. Is eating meat wrong? I think that eating meat is wrong. Mike doesn't think that eating meat is wrong. I once thought that eating meat was wrong. She does not realize that eating meat is wrong.
Attempts to translate these sentences in an emotivist framework seem to fail ("She does not realize, 'Boo on eating meat!'") Prescriptivist translations fare only slightly better ("She does not realize that she is not to eat meat"). Even the act of forming such a construction indicates some sort of cognition in the process. In some non-cognitivist points of view, these sentences simply assume the false premise that ethical statements are either true or false. They might be literally translated as:
"Eating meat is wrong" is a false statement. Is "eating meat is wrong" a true statement? I think that "eating meat is wrong" is a true statement. Mike doesn't think that "eating meat is wrong" is a true statement. I once thought that "eating meat is wrong" was a false statement. She does not realize that "eating meat is wrong" is a true statement.
These translations, however, seem divorced from the way people actually use language. A non-cognitivist would immediately disagree with someone saying, "'Eating meat is wrong' is a false statement" (since "Eating meat is wrong" is not truth-apt at all), but may be tempted to agree with a person saying, "Eating meat is not wrong."
One might more constructively interpret these statements to describe the underlying emotional statement that they express, i.e: I disapprove/do not disapprove of eating meat, I used to, he doesn't, I do and she doesn't, etc.; however, this interpretation is closer to subjectivism than to non-cognitivism proper.
A similar argument against non-cognitivism is that of ethical argument. A common argument might be, "If killing an innocent human is always wrong, and all fetuses are innocent humans, then killing a fetus is always wrong." Most people would consider such an utterance to represent an analytic proposition which is true a priori. However, if ethical statements do not represent cognitions, it seems odd to use them as premises in an argument, and even odder to assume they follow the same rules of syllogism as true propositions.