In so many areas of inquiry (though, perhaps, not all), philosophical argument ends up bottoming out in a mere clash of intuitions, of considered judgments. But what happens now? Because these considered judgments will help determine the content and structure of our philosophical theorizing, to determine (once and for all) what the good life is for a person (or whether we should be descriptivists or causal theorists of reference, or whether justified true belief counts as knowledge or not, or — if Lewis Carroll is to be believed — whether modus ponens is a valid rule of inference) we — or so it would appear — need to settle which of these intuitions are the right ones.
To put my cards on the table, this seems like an impossible task. Indeed, it’s a task that seems (almost by definition) outside the bounds of philosophical argument.
One of the things I have been pointing out on this blog since I was in graduate school is that 'intuitions' aren't unitary things, and I have always been baffled at any suggestion that they are somehow a ne plus ultra. If we compare what contemporary philosophers say about intuitions and look at approaches from other times that at least cover part of the same ground -- Aristotelian topics, the Nyaya account of pramanas, and Scottish Enlightenment accounts of common sense, for example, all of which are finer-grained than contemporary talk of 'intuitions' -- we see clearly enough that we could be, and should be, much more precise about what is going on when they come up. And when we do we may realize that different kinds of intuitions are involved, or that the intuitions are not all equally evident, or even that some of the intuitions are not even coherent.
Intuitions are beginnings, not bedrocks. Conflicts of intuitions are the kinds of thing reasonable people wonder about, and wondering is the beginning of philosophy. Philosophy doesn't bottom out on them; the rational person will precisely start to wonder about what goes into these intuitions that leads to the differences.
One of the great temptations for philosophers through the century is to confuse features of communication and features of reason -- we always present our philosophy in certain ways, and it can be tempting to assume that those formats are just the way reason works, as if reason could not work in some other way entirely. For instance, someone may do all their philosophy in debates, and if they did, they might be tempted to think of good reasoning as being entirely disputational, because that's the way they polish it up in their own context. It has been difficult over the past century to convince many academic philosophers that you could perfectly well do philosophy with poetry or fiction, despite the fact that it certainly has been done -- because these are not usually features of how academic philosophers in this day and age communicate. And there are other ways in which this can be manifested. I think we see one of them in this case of intuitions. Appeal to intuitions has spread fairly widely in academic texts, and I think a plausible reason why is that you can't fully develop the argument in the formats academic texts usually have. If you are writing a thirty page paper on some very complicated topic, there are going to be things that you just have to ask the reader to take for granted for the sake of the argument; and when an academic philosopher appeals to the intuitive, that's precisely what's happening -- he or she is summarizing a reason for thinking that it's reasonable to accept this. It's something inevitable in the nature of the format, and it's certainly better that it be flagged and insisted upon than that it slide by without comment.
But in real life, we are not wholly bounded by page limits and the fact that we are writing about topic A and can't also write about topic B.Faced with conflicts among these intuitive appeals, we don't have to just throw up our hands and say, "Irreconcilable difference!" Reasoning itself doesn't bottom out like that. We can perfectly well ask more questions, look into the matter more deeply, test for consistency, analyze into constituent parts, compare to other cases, and any of a very large variety of rational activities.
I am sympathetic to Dorsey's notion of an 'atlas-drawing' approach to philosophy; it is, I think, an important part of philosophy, more important than is sometimes realized. It's also the sort of thing I do as a historian of philosophy -- just map out the positions to determine what positions there are. But there is no "phenomenon of intuitive bedrock". There are merely points at which inquiry begins to form along different lines.