Are there any convincing arguments against cultural relativism for agnostics and atheists?
There are ancient Greek arguments for objective morality. Plato argues that the just man has a more harmonious soul (substitute “psychological state,” if you like) than the unjust man because he is not plagued by desire and ambition. Similarly, Aristotle argues from man’s nature as a “political animal” to the necessity of cooperation and justice for achieving “eudaimonia,” (“happiness” or “flourishing”), which is the result of man’s rational function. More recently, there is the theory of evolutionary morality, which asserts that morality is reducible to a set of behaviors that has evolved in order to increase cooperative behavior amongst communities of organisms, thereby increasing the chance of reproduction for every individual organism. Of course, there is also the strong possibility that your choice to be “good” may be completely arbitrary, logically speaking. That may simply be something that you have to live with when you reject the authority of a higher power.