Aquinas argues that the first principles of moral law are axiomatic knowledge, known by what he calls “synderesis.” If they are axiomatic, they can’t necessarily be “proven,” but only known.
For example, let’s suppose I said, “A equals A.” Someone questions this, so I give five examples. Then this person says, “That’s only five examples. Maybe somewhere A does not equal A.” At that point I would say, “Look, this is an axiomatic feature of the theoretical intellect; if you don’t see it, your mind is perhaps damaged somewhere or somehow. Sorry, but I can’t give all the illustrations of why this is true. You either see it or you (somehow) don’t.”
In the practical (not theoretical) intellect, the same is true of axiomatic moral principles. If someone honestly does not see that murder is wrong, there is something wrong with that person; there is nothing wrong, however, with the statement “Murder is wrong.” This is what Budziszewski calls “What we can’t not know” (another title of another book by him).
If we ask, “Why not do bad things?” the answer is, “Because they’re bad.” Part of the description of badness is, “We should not do this.” If you run into someone who pulls out Hume’s "You cannot derive an ‘ought’ from an ‘is,’ " the answer to that is the above: Part of what is KNOWN AS FACT by the practical intellect is what should and should not be done. If we know something is wrong, we also know that we should not do it.
A way this is often explained is based on the idea of “nature” (which is why the idea of “human nature” is so often attacked). Things come in natural kinds with natural functions. Someone that goes contrary to its natural kind and function is known as a “bad” example of whatever it might be (a cup with a hole in the bottom, for example, is a bad cup). By rational analysis, we can see what is “natural” for humans and the human constitution; the laws based on this nature are called natural laws. This is not the “law of nature” or “kill or be killed” or something like that; this is the laws of rational human nature.
By the way, I forgot about Rice’s book, mentioned above. It’s also recommended.
