It's not a true fact that "anger in that situation is justified". That's a value judgment that cannot be true or false.
You are saying "anger in that situation is justified" is neither true nor false, but that "it is true that anger in that situation is justified" is false. That is, you are drawing a distinction between claiming something and claiming that that same thing is true. That is bizarre and, in my opinion, borderline incoherent.
To anticipate your response, a person who says "going to the movies is fun" is not lying. But saying, "it is true that going to the movies is fun" is not a lie either. To argue so is to misunderstand what "going to the movies is fun" means. As all the things you call value judgments are, it is a relationship claim. This claim is that going to the movies and the speaker are both so constructed as to have a particular relationship, namely that the speaker (or other unspecified people -- the claim is a bit vague) does in fact experience enjoyment when they go to the movies.
It is a claim that going to the movies has an inherent property that is demonstrated by a relationship. Just like "hydrogen is flammable" claims hydrogen has an inherent property that can only be demonstrated by exposing it to oxygen.
The basis for allocating public health spending in Australia, the UK and every other civilized nation I am aware of (the USA doesn't count as civilized in this respect) is "if extending human life and improving human quality of life is good, and every person's life and quality of life count equally, then the most effective way to spend the health budget is thus".
This is purely descriptive. It says, "if X, then Y". It is not a moral claim any more than "to make sodium explode, you add water" is a moral claim. "If you want X, then do Y" is another way of saying "Y causes more of X than the other possible things we could do". It's purely objective.
Is it true that the most moral way to spend the Australian health budget is, broadly speaking, to pay for treatments that provide at least one Quality-Adjusted Life Year per AU$40,000? Well, no. It's a consequence of the axiomatic assumption previously stated. If you discard that axiom then the policy has no basis.
Right, so anyone who claimed it was true would be wrong. It is, as you say, *not* true.
This isn't a problem. Well, maybe it's a serious problem for you. I don't see it as one. I recognise that there is simply no way of doing morality at all, that works, except adopting some axiom or axioms as a starting point. So I adopt some axioms and get on with it.
That's fine. But then all of your conclusions are dependent on the truth of your axioms. You are practicing descriptive morality, not proscriptive.
As you claim, an ought can only be based on another ought. So your root ought must forever be an axiom and any conclusions you draw from it are forever contingent on the truth of that axiom.
If that axiom is not true, as you conceded it is not, then the conclusions are not shown. Anyone who claimed they were shown to be true would be wrong, and if you make such claims, you are intentionally misrepresenting them.
There's one other point I need to make. If you were right about this, then moral claims would be as amenable to scientific claims as anything else. All it would mean is that you need one more axiom. But science has, and needs, many axioms. For example, you cannot prove that a person
should reject experimental results that cannot be repeated. This is an axiom that you must accept if you want to do science.
If you say it's not an axiom and you can prove it, it's only because you have accepted some other axiom. For example, if you show that rejecting unrepeatable results leads to more accurate predictions, you still cannot say that one should do that unless you accept that accurate predictions are
good. So even if it does require one root axiom that cannot be proven within science, that does not make it unscientific or else no conclusions are scientific.
So this is also a case of special pleading. Somehow, this one axiom makes it not science while all the other axioms of science are fine.
I actually believe that science, in the broadest sense, can in fact internally validate its own axioms, including a 'moral axiom'. (Though we can't do that for a moral axiom *yet*, but we have done it for many other axioms such as about repeatability, objectivity, and all the other shoulds in science.) It's only 'formal science' -- science that you can state as a set of rules or procedures -- that has to have unvalidated axioms. But that's a whole other issue that's probably not worth getting into here and I'm sure you and I have too many disagreements at lower levels, like this one about what values and judgments actually are, to make any headway on it. As a consequence, I believe that value judgments are relationship facts, like most facts are, and not in a different category at all.
We just speak in a form of shorthand. For example, if I say "Your wife is cheating on you", what I really mean is something like, "I have reasons to believe that your wife is cheating on you that justify you sharing my belief". This difference matters because the former claim appears to be false if I have good reasons that just happen to be wrong through no fault of mine. The latter statement can still be true even if your wife is not in fact cheating on you. So it matters to the claim whether when you say the first you are "really saying" the second.
Value judgments are also similar shorthands. They appears to be unverifiable, subjective assessments. But they are actually claims about objective relationship properties, just using a convenient verbal shorthand. For example, we all understand that a person who says "that dress looks great on you" can in fact be lying. (And, of course, we all understand 'that dress looks great on you' to really be saying something like 'I think that dress looks great on you' or 'From what I know about how people judge beauty, that dress improves the way most would assess you' or the like.)