tl;dr: Relativism bottoms-out in realism by objectifying relations between subjective notions. This should be communicated using concrete examples that show its practical importance. It implies in particular that morality should think about science, and science should think about morality.

Sam Harris attacks moral uber-relativism when he asserts that "Science can answer moral questions". Countering the counterargument that morality is too imprecise to be treated by science, he makes an excellent comparison: "healthy" is not a precisely defined concept, but no one is crazy enough to utter that medicine cannot answer questions of health.

What needs adding to his presentation (which is worth seeing, though I don't entirely agree with it) is what I consider the strongest concise argument in favor of science's moral relevance: that morality is relative simply means that the task of science is to examine absolute relations between morals. For example, suppose you uphold the following two moral claims:

1. "Teachers should be allowed to physically punish their students."
2. "Children should be raised not to commit violence against others."

First of all, note that questions of causality are significantly more accessible to science than people before 2000 thought was possible. Now suppose a cleverly designed, non-invasive causal analysis found that physically punishing children, frequently or infrequently, causes them to be more likely to commit criminal violence as adults. Would you find this discovery irrelevant to your adherence to these morals? Absolutely not. You would reflect and realize that you needed to prioritize them in some way. Most would prioritize the second one, but in any case, science will have made a valid impact.

So although either of the two morals is purely subjective on its own, how these morals interrelate is a question of objective fact. Though perhaps obvious, this idea has some seriously persuasive consequences and is not be taken lightly. Why?

First of all, you might change your morals in response to them not relating to each other in the way you expected. Ideas parse differently when they relate differently. "Teachers should be allowed to physically punish their students" might never feel the same to you after you find out it causes adult violence. Even if it originally felt like a terminal (fundamental) value, your prioritization of (2) might make (1) slowly fade out of your mind over time. In hindsight, you might just see it as an old, misinformed instrumental value that was never in fact terminal.

Second, as we increase the number of morals under consideration, the number of relations for science to consider grows rapidly, as (n²-n)/2: we have many more moral relations than morals themselves. Suddenly the old disjointed list of untouchable maxims called "morals" fades into the background, and we see a throbbing circulatory system of moral relations, objective questions and answers without which no person can competently reflect on her own morality. A highly prevalent moral like "human suffering is undesirable" looks like a major organ: important on its own to a lot of people, and lots of connections in and out for science to examine.

Treating relativistic vertigo

To my best recollection, I have never heard the phrase "it's all relative" used to an effect that didn't involve stopping people from thinking. When the topic of conversation — morality, belief, success, rationality, or what have you — is suddenly revealed or claimed to depend on a context, people find it disorienting, often to the point of feeling the entire discourse has been and will continue to be "meaningless" or "arbitrary". Once this happens, it can be very difficult to persuade them to keep thinking, let alone thinking productively…