(The following is a revised version of my “Arguing from Moral Ontology by Eliminating Nontheistic Alternatives.”)
Moral ontology is the branch of meta-ethics concerned with the ontology or metaphysics of moral facts and properties. Within the field of moral ontology, it is a commonplace that the nature of moral facts and properties fall into one of three categories: natural, nonnatural, or supernatural.
In order to define these three categories, allow me to first define some other terms. A physical entity
is an entity which is either (1) the kind of entity studied by physicists or chemists today (e.g., atoms, molecules, gravitational fields, electromagnetic fields, etc.); or (2) the kind of entity studied by physicists or chemists in the future, which has some sort of nomological or historical connection to the kinds of entities studied by physicists or chemists today.
A natural entity is an entity which is either a physical entity or an entity that is ontologically or causally reducible to a physical entity. A natural entity is causally reducible to a physical entity just in case the natural entity’s causal powers are entirely explainable in terms of the causal powers of the physical entity. A natural entity is ontologically reducible to a physical entity just in case the natural entity is nothing but a collection of natural entities organized in a certain way.  Nature is the spatio-temporal universe of natural entities.
A supernatural person is a person that is not part of nature but can affect nature. Examples of supernatural persons include God, angels, Satan, demons, ghosts, etc.
Like a supernatural person, a non-natural entity is not part of nature. Unlike a supernatural person, however, a non-natural entity is unable to affect nature. Abstract objects and Platonic Forms are examples of impersonal non-natural entities.
Corresponding to these three kinds of entities (natural, supernatural, and non-natural) are three views about moral ontology. Ethical naturalism is the view that moral properties are ultimately reducible to natural properties. For example, moral goodness is ultimately reducible to human flourishing. Ethical supernaturalism is the view that moral properties are ultimately reducible to supernatural properties. For example, moral obligations are ultimately reducible to God’s commands. Ethical nonnaturalism is the view that moral properties are irreducible, that is, that moral properties cannot be reduced to any non-moral properties. For example, moral values like goodness and badness exist as Platonic Forms or abstract objects.
Unfortunately, many writers assume that moral objectivism entails ethical nonnaturalism. Geoffrey Sayre-McCord observes, “Objectivism in ethics has suffered from the mistaken assumption that objectivists must hold that moral properties are nonnatural …”. The metaphysical nature of moral properties can only be determined by a thesis about moral semantics,not by the “metaphysically neutral” theory of moral objectivism. Mere (moral) objectivism neither logically entails nor makes probable the view that moral properties are nonnatural properties. As John Post writes, moral objectivism does not require “some shadowy Platonic realm ‘out there,’ perhaps beyond space and time.” Rather, moral objectivism is simply the view that there is an objective “truth of the matter as regards the correctness or incorrectness of our value judgments.”
2. The Argument from First-Order Ethical Beliefs for Objective Moral Properties
Ordinary ethical sentences and commonsense first-level moral beliefs imply objective moral properties.
(2) There are no empirical or a priori reasons to believe that first-level moral beliefs are all false.
(3) It is more reasonable to believe there are objective moral properties than not to believe this.
(4) There is no reason to believe that the conjunction of (1) and (2) is a defective reason to believe objective moral properties.
(5) Therefore, the belief in objective moral properties is indefeasibly justified.
(6) Therefore, there are objective moral properties.
(See here for more about this argument.)
3. The Evidential Argument from Unknown Identity against Ethical Naturalism
Let E be the failure of ethical naturalists to locate the exact identity of natural moral properties.
(7) E is known to be true.
(8) E is more probable on the assumption that ethical naturalism is false than on the assumption that ethical naturalism is true.
(9) The prior probability of ethical naturalism is not significantly higher than its denial.
(10) Therefore, other evidence held equal, ethical naturalism is probably false.
4. The Unintelligibility Argument against Ethical Non-Naturalism
The following argument was inspired by William Lane Craig. Craig writes
First, it’s difficult even to comprehend this view. What does it mean to say, for example, that the moral value Justice just exists? It’s hard to know what to make of this. It is clear what is meant when it is said that a person is just; but it is bewildering when it is said that in the absence of any people, Justice itself exists. Moral values seem to exist as properties of persons, not as mere abstractions—or at any rate, it is hard to know what it is for a moral value to exist as a mere abstraction. Curiously, since the abstract object Justice is not itself just (just as Quickness is not quick or Laziness lazy), it would seem to follow that in the absence of any people justice does not exist—which seems to contradict the hypothesis! Atheistic moral Platonists seem to lack any adequate foundation in reality for moral values but just leave them floating in an unintelligible way.
This suggests the following argument. (Not that Craig would agree with it in its present form.)
(11) If ethical non-naturalism is true, then moral properties can exist as mere abstractions, i.e., even in the absence of any people.
(12) The hypothesis that moral properties can exist as mere abstractions is unintelligible.
(13) If a hypothesis is unintelligible, it is probably false.
(14) Therefore, ethical non-naturalism is probably false.
5. The Moral Ontological Argument against Metaphysical Naturalism
(6) There are objective moral properties.
(15) If there are any objective moral properties, they must be natural, nonnatural, or supernatural properties.
(16) Objective moral properties are not natural properties. [from (10)]
(17) Objective moral properties are not nonnatural properties. [from (14)]
(18) Therefore, objective moral properties must be supernatural properties. [from (6), (15), (16), and (17)]
(19) If there are supernatural properties, there exists at least one supernatural being. [by definition]
(20) Metaphysical naturalism entails the nonexistence of supernatural beings.[by definition]
(21) Therefore, metaphysical naturalism is false. [from (18)-(20)]