To test hypothesis B [that it is morally acceptable to murder innocent children], a bunch of terrorist scum seize a school and murder 300 children. Observers note feelings of revulsion and a strong desire to punish the perpetrators. The experiment is reproducible (let's hope only in theory). Therefore hypothesis B must be rejected.
That is how objective moral truths are inferred from observation. We have a moral sense that we use to make moral observations, in the same way as we use our physical senses to make physical observations.
I don't find this sort of moral intuitionism very plausible. I've previously argued that value is best understood in terms of the relations between desires and states of affairs. But such complicated relations are of course not directly observable. So KP's view implies that value is instead some sort of property that is intrinsic to objects or actions.
Even supposing moral properties really were that superficial, and that we had some "moral sense" capable of perceiving those properties, it seems highly unlikely that we would interpret them appropriately. As Alonzo Fyfe (chapter 12) points out, any such faculty of perception would soon be hijacked by evolutionary forces.
We would evolve to be attracted to objects or actions that promote evolutionary success, regardless of their 'intended' (by who?) moral status. Moral properties would be just another bit of data (alongside photons and soundwaves) for us to work with and interpret for evolutionary ends. Again, there's no reason at all to think that we would learn to distinguish which properties were intended to mean 'good' and which 'bad'. If intrinsically 'bad' things increased our evolutionary fitness, then we would come to (mistakenly) interpret them as 'good'.
So unless one wants to conflate 'good for my genes' with 'morally good' (not a good idea!), we must deny that objective moral properties exist independently, 'out there' in the world, waiting to be perceived by humans.
But in that case, the intuitionist seems left with a hollow subjectivism. The intuitionist thinks that he can just 'see' moral properties, but in fact the object of his observation is merely his own psychological response. Instead of working out the real relations between various actions and human wellbeing, the intuitionist makes unsupported assertions based on his emotional reactions and prejudices. If something makes him feel 'icky', then he calls it 'wrong'. But this yuk factor is morally arbitrary.
Appealing to intersubjective corroboration will not help. That will merely tell us what prejudices and norms are prevalent within the given society (as opposed to what norms they should have). This then implies cultural relativism. After all, it wasn't that long ago that most people felt "feelings of revulsion and a strong desire to punish" homosexuals, interracial marriages, and so forth.
So quite apart from the evolutionary argument, the diversity (cultural and historical) of mainstream values also spells doom for intuitionists' hopes for objectivity. But given the obvious influence of socialisation on our moral intuitions, it was probably a hopeless project from the start.
It's actually a bit more complex than that because moral facts are partly dependent on physical facts, but the converse is not true. So I should qualify the above claim by saying that it applies only where the relevant physical facts are known to all observers.
As a naturalist, I would submit that moral facts are (ultimately) entirely dependent on physical facts - specifically, facts about states of affairs, human desires, and the relations between them.
The problem (for KP) here is that the relevant physical facts are not ones that can be learnt simply by observing the event in question. Rather, we requires all sorts of background knowledge, and even some basic predictive abilities (to see the likely consequences of the action). Then we need to reason well enough to draw accurate inferences from all these empirical facts. It's no simple matter.
Now, as I've said before, the method of consulting our emotional responses might be a useful shortcut, as they should at least correlate with the moral facts to some degree if we have been well socialised. But it's a very unreliable heuristic.
Most importantly, it doesn't really say anything about the event (since we can't directly perceive moral properties). Rather, it just tells us about ourselves. So the moral 'experiments' outlined by KP really wouldn't support any moral conclusions at all. Ethics isn't that easy.
Update: More here.