PH's and gang's argument that Morality is not objective is because there are no Mind [brain, human]-Independent Moral Facts.
But such grounding on Mind [brain, human]-Independent Facts is fatuous, meaningless and illusory in the sense of ultimate reality.
Note: I will use the shorter term "Mind-Independent Facts" to represent the intended Mind [brain, human]-Independent Facts
Here is paper on why there are no mind-independent facts and it argues that there are only mind-interdependent facts conditioned by its specific Framework and System [FSK] of Knowledge or Reality.
WHERE ARE FACTS?
A Case For Internal Factual Realism
Xinli Wang
1. Introduction
2. Many Faces of Facts
3. External Factual Realism
3.1 Factual Naturalism [concrete]
A. Facts As (Locatable) Particulars
B. Facts As (Datable) Happenings
C. Facts As (Observable) Situations
3.2 Factual Ontologism [abstract]
3.3 A Misconception of the Nature of Facts
4. Facts as True Propositions
5. Internal Factual Realism
6. Conclusion
1. Introduction
What exists in the world?
Are there only simple entities like objects, properties, and relations,
or are there also Complex Entities like events, situations, and Facts?
Against the background of traditional Aristotelian metaphysics, which pictures Reality as consisting of the aggregate of mutually independent individual objects or things,
the outstanding innovation of Wittgenstein's ontology is his characterization of the world as an aggregate of Facts, not of things (Tractatus, I, 1.1).
Facts are generically different from things or objects.
Facts are usually regarded as Complex Entities consisting of objects, properties, and external relations which
either relate a property to an object (say, John is bald)
or connect several objects (say, John is taller than Joe).
Facts are not simply the aggregate of objects; they [Facts] are objects standing in relations to each other.
Facts are configurations of objects.
If the world is a totality of Facts, what sorts of Facts exist in the world?
Where are Facts?
For the details of the argument read Section 2, 3, 4 and 5 of the attached article.6. Conclusion
The traditional questions, ''What is a Fact?" or "When do we use the phrase 'is a Fact'?" is misleading.
The question seems to suggest that Facts have some intrinsic nature awaiting discovery.
External Factual Realism arises as an attempt to answer such a question.
Such an attempt is doomed to failure for there are no mind-independent Facts.
Most (if not all) Facts are language-dependent.
Our task is not to make sense of sense unfathomable entity, but in so far as there is a problem, it is one of knowing when we are entitled to proclaim something a Fact.
So the more appropriate question to ask about Facts should be:
"How do we use the phrase 'is a Fact'?" or
''When are we entitled to proclaim some complex entity a Fact?"
Such a switch in the way of asking questions about Facts is necessary and illuminating.
The answer to the question has to be language dependent.