You write, “People often object to relying on intuitions. But I’m curious how they get their foundational beliefs. One’s most basic beliefs always seem justified by the fact that they seem right.”
No.
As I show in Chapter 1 of my 2016 book, Rightness as Fairness: A Moral And Political Theory, seeming true and being true are two very different things. Intuitions at most establish what seems true. Yet, most false beliefs of the past (e.g., “the Earth is flat”) were based merely on how things seemed—things that seemed true but weren’t.
We discovered these things to be false because ordinary standards of evidence—the same ones that stand as the foundation of empirical science—demand more than “X seems true; thus, X is true.” They demand that we can demonstrate that X is true to virtually any observer, and indeed, from any observational perspective: first-, second-, or third-personal. (This is the scientific standard of independent replication).
Next, as my new article in The Journal of Moral Philosophy, “Metaethical Lessons of a Failed Ontological Proof of Robust Moral Realism” shows, this standard is precisely what isn’t met by moral intuitions or by robust moral realism more generally—since whether it even seems to one that one has moral reason depends on contingent attitudes (beliefs, desires), contra robust moral realism.
But these are not stance-independent reasons (they depend on interests, which are stance-relative), and it is by no means obvious (and not establishable by intuition) that everyone has them.
You write, “People often object to relying on intuitions. But I’m curious how they get their foundational beliefs. One’s most basic beliefs always seem justified by the fact that they seem right.”
No.
As I show in Chapter 1 of my 2016 book, Rightness as Fairness: A Moral And Political Theory, seeming true and being true are two very different things. Intuitions at most establish what seems true. Yet, most false beliefs of the past (e.g., “the Earth is flat”) were based merely on how things seemed—things that seemed true but weren’t.
https://link.springer.com/chapter/10.1057/9781137541819_2
We discovered these things to be false because ordinary standards of evidence—the same ones that stand as the foundation of empirical science—demand more than “X seems true; thus, X is true.” They demand that we can demonstrate that X is true to virtually any observer, and indeed, from any observational perspective: first-, second-, or third-personal. (This is the scientific standard of independent replication).
Next, as my new article in The Journal of Moral Philosophy, “Metaethical Lessons of a Failed Ontological Proof of Robust Moral Realism” shows, this standard is precisely what isn’t met by moral intuitions or by robust moral realism more generally—since whether it even seems to one that one has moral reason depends on contingent attitudes (beliefs, desires), contra robust moral realism.
https://philpapers.org/rec/ARVMLO
Notice, furthermore, that all of your main examples (1-6) aren’t about moral reasons in any obvious way, but rather self-interest.
As I show in my 2023 article in Inquiry, From rational self-interest to liberalism: a hole in Cofnas’s debunking explanation of moral progress and my 2020 book Neurofunctional Prudence and Morality: A Philosophical Theory, most of us do have self-interested reasons to behave morally.
But these are not stance-independent reasons (they depend on interests, which are stance-relative), and it is by no means obvious (and not establishable by intuition) that everyone has them.