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Per Hume's is-ought problem, most people consider ethical questions to be independent of empirical facts.

The physicalist/materialist world-view is based on the idea that only thing that exists are physical facts about the world.

If someone were to establish beyond any reasonable doubt that ethical principles such as "Killing is bad","Discrimination based on gender, race or orientation is bad","Stealing is bad" are universally true, does this amount to dualism?

That these propositions are objectively true (independently of any subject) would make them true facts about the world, but they would be independent of any physical facts as well. Wouldn't this amount to a form of dualism or platonism? Is this what Kant meant by "metaphysics of morals"?

Or is it the case instead that these ethical propositions would acquire a status similar to the laws of mathematics and logic? Or maybe the same stauts as the laws of science and physics?

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From a kind of Nietzschean point of view, inspired by the Genealogy of Morals, there is at least one counterexample. So you cannot presume this kind of idea would have either a dualist status, or become aspects of logic, or theories of science.

Consider this option: one might base objective moral stances in evolution and a form of human psychology answerable to animal behaviorism.

(I think this is the approach to an objective ethics that holds the most hope for eventually becoming real. It would constitute a physicalist grounding of something like Hegelian dialectical evolution. The internal approach, a la Kant is compelling, but lacks real proof you have considered an adequate range of potential sorts of minds -- healthy sociopaths exist, as do functional autistics -- and both seem to have a rather different spin on 'rightness' from Kant.)

Even though the standards would be real facts about the world without a physical manifestation, they would be the result of 'standing patterns' that are stable states of historical processes. We could see them coming from a physicalist point of view, and they can then be explained physically without being instantiated physically.

At the same time, if they were truly objective, only their source would be in theory. If we genuinely all shared them in a natural way, the moral principles themselves would not be science, because they would not be falsifiable or open to revolutionary restructuring.

That gives them something in common with mathematics, but it would still be possible to violate them without being mistaken. So they would have a form different from idealized logical facts, as well, in that they would be possible and not necessary forms.

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I think you may be trapping yourself. You start with "most people consider ethical questions to be independent of empirical facts," but then focus on a very extreme case that does not qualify as "most people's" way of thinking . You then "establish beyond any reasonable doubt" ethical positions using "most people's" way of thinking (ethical questions are independent of empirical facts), and apply it to the extreme physicalist's perspective. The physicalist would have derived those ethical principles from physical facts, not out of thin air, thus avoiding the metaphysics issue.

Whether such a system qualifies as a "system of ethics" is more of an academic exercise in linguistics. Some may choose to exclude them from the word "ethics" because they define ethics to be something which excludes such empirically derived facts. I think the real issue there is not that you can't have a system of ethics from a purely physicalist perspective, but rather that the vernacular gets stretched to an extreme when you do so. The issue reminds me a lot of projection in cartography. It's not that there isn't a point that is "the north pole," its that the Mercator projection, which is so effective in other scenarios, doesn't handle a pole well and ends up stretching that point out into a line, with all sorts of distortions.

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I consider dualism a position in ontology which assumes two different kinds of real objects.

1) In case certain moral norms could be detected and investigated as part of our world we would have indeed an ontological dualism: on one hand the laws of nature on the other hand the moral laws. These are two independent and different types of laws: A moral law does not enforce an actor to act according to the law.

Hence these moral laws are different from the laws of science.

2) These objective moral laws were also different from mathematical propositions or from the rules of logic:

Mathematical propositions derive from definitions and axioms. The latter have a high degree of arbitrariness. Also logical rules only hold within a certain calculus of logic, similar to a mathematical theory. Hence mathematical propositions and logical rules are valid only relatively.

3) Kant’s Groundwork for the Metaphysics of Morals claims that the Categoric Imperative can be detected as a basic moral principle by philosophical reasoning. Hence it is an example of an objectively valid moral principle – according to Kant.

In any case, the existence of objectively valid moral norms would introduce a new kind of laws besides the two existing types, the laws of science and mathematical/logical rules.

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