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If a culture develops a belief that killing the weakest of its citizens is morally preferable in order to protect itself from other cultures (you're only as strong as your weakest link, right?), then is it still wrong? Or  is it ok because that culture in particular has deemed it morally right?

No, I don't actually believe that murder is ok in any culture. And neither do you (I hope). Quite a few people in this day and age, however, claim beliefs about reality which actually conflict with their moral beliefs, to the point that if they truly lived consistently with their beliefs, the above society would be totally acceptable. Many, many people claim moral beliefs with their intellect which conflict with the moral beliefs they actually demonstrate in the way they live their lives. This is an interesting paradox, and one which very much gets taken for granted these days. Let us not grant it so easily here.