Not sure if I agree here. For example, how do we arrive at a moral axiom through science? I would prefer to say that we can arrive at some understanding of moral sentiments through science, but I disagree that we reach moral axioms.
I think if we start with a high level moral sentiment like "abortion is wrong" we can work backward with an individual or a group until we get to moral beliefs that aren't derived from other moral beliefs.
I think the role of science is in differentiating stated beliefs from sincerely held beliefs, identifying more base level beliefs in people who can only articulate higher level beliefs etc.
I can't think of a perfect real world example off the top of my head, but I can give you something in the ballpark.
A recent study investigated the question of how much charitable giving was motivated by altruism and how much by social pressure. Almost all respondents claimed their giving was altruism based, but by varying social pressure in a controlled way, they found that the large majority would not give if no one was watching them, nut would give when those social pressures existed.
Hmm...I think there could actually be a disagreement here nonetheless. While I agree with you that the question of "outcomes" could be, in principle, an empirical matter to the extent we can see whether a child will improve her spelling scores by being repeatedly rapped over the knuckles whenever she comes last, the question of corporal punishment may still rest on other moral axioms such as:
1) Pro-corporal punishment advocates may not be of the opinion that it will improve a child, but may think that the retribution is a matter of "justice". Of course, I am not advocating this, but merely showing the axiom may be different from what we imagine it is.
In the case of corporal punishment, it isn't simply that it doesn't improve behavior, but that it has clear negative effects, so the only people arguing for the value of punishment as justice will be those who value that justice more than the welfare of their child, and I think that's a rare group, at least in the west. But you're in general, correct that we can't say the moral problem of corporal punishment is universally sidestepped.
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2) Anti-corporal punishment advocates may not be of the opinion that it will fail to improve a child, but may find violently beating a child to be wrong in and of itself. In other words, they may have a rights-based intuition that is barbaric and inhumane to inflict suffering of this kind regardless of the improvement in a child's spelling tests. Such advocates may argue that it is dangerous to simply apply an empirical test such as whether beatings are efficacious or not as to do so would leave us hostages to fortune.
I'm not saying that efficacy or outcome supplants moral discussion in all cases, or even in this case. It isn't that an empirical test determines what is right. In many cases, there's a lack of clarity among people about where the moral and factual parts of the debate begin and end. Science and reason can remove the factual part from the tangle and leave us "simpler" moral problems.
In the case of corporal punishment, the moral problem becomes: Do you value something else (justice, tradition, whatever) more than your child's academic, social and emotional well-being. When the question is put like that, there still may be individuals or societies that support corporal punishment, but for a lot of people, at that level of clarity, it's less of a question.
If scientific findings had shown the opposite to be true, it would still leave the question of how much parents value the long term outcomes for their child vs their short term experience.
If people object to science removing the factual confusion from a moral issue, then they may be emotionally attached to the lack of clarity. If they value truth to any extent, this is probably a problem.