Are there any arguments against moral nihilism?

Moral nihilism, also known as ethical nihilism, is the meta-ethical view that morality does not exist as something inherent to objective reality; therefore no action is necessarily preferable to any other. For example, a moral nihilist would say that killing someone, for whatever reason, is not inherently right or wrong. Other nihilists may argue not that there is no morality at all, but that if it does exist, it is a human and thus artificial construction, wherein any and all meaning is relative for different possible outcomes. As an example, if someone kills someone else, such a nihilist might argue that killing is not inherently a bad thing, bad independently from our moral beliefs, only that because of the way morality is constructed as some rudimentary dichotomy, what is said to be a bad thing is given a higher negative weighting than what is called good: as a result, killing the individual was bad because it did not let the individual live, which was arbitrarily given a positive weighting. In this way a moral nihilist believes that all moral claims are false.