You pursue your arguments as far as they go, and eventually reach your bedrock assumptions: foundational premises that you accept (and might describe as seeming 'intuitive' to you), but that you can give no further argument for. Further, you realize that coherent philosophical diversity is possible: others could, coherently, accept (and find 'intuitive') different starting points from yours. What should you do about it?
One might deny the stipulated set-up, and insist that there is ultimately only one internally coherent philosophical world view. But that seems unlikely, so I will put aside that ambitious view for now. One might also reject the "foundationalist" structure I've assumed above, defending instead the idea that our beliefs might form an interlocking (ultimately circular, but "mutually supporting") web, with no privileged starting points. But I think what I say below will be easily translatable into coherentist idiom: the question just becomes what to think of our web of beliefs as a whole, given that coherent alternatives are possible.
Option 1: Skepticism
So, one obvious option is to throw up our hands in defeat. One might appeal to principles to the effect that, if we can't give a non-question-begging defense of our foundational assumptions (or of our web of beliefs as a whole) then we can't reasonably regard our beliefs as more likely to be true than the alternatives. And if that's so, then we can't coherently maintain them as beliefs (representing our best judgment of what's true) at all.
On the other hand, such radical skepticism is self-undermining: once you accept it, you can no longer believe it, nor the putative reasons that originally led you there. So it's worth exploring more doxastically stable alternatives...
Option 2: Epistemic Conservatism
We could hold that our beliefs are justified by default. Or, in the variant that is phenomenal conservatism: we are justified by default in believing whatever seems intuitive to us (and isn't ultimately defeated by other considerations that are even more intuitively plausible). This makes it easy to come by justification. Too easy, we might worry:
* It renders the ideally coherent Caligula, and other bad characters, epistemically justified in their acceptance of such (I would say) self-evident falsehoods as that it's good to go around torturing people for fun.
* It creates an awfully large gap between justification and truth. If we are committed to calling "justified" all sorts of philosophical beliefs that aren't even remotely on the right track, how great an epistemic merit can this really be?
Option 3: Objective Warrant
A more objective alternative looks for justification not in us, and our subjective status of finding something plausible-seeming, but rather looks to the contents of the propositions that are candidates for belief, and determines whether they are (or are not) intrinsically credible, or warrant belief. Just as moral objectivists are apt to hold that we should desire what's truly desirable, and not just whatever (possibly crazy/horrid) things seem so to us, so the epistemic objectivist can hold that we should believe the a priori propositions that are truly credible, and not just whatever (possible crazy) things happen to seem so to us.
Presumably any view of this form will hold that the a priori truths are self-evident, inherently credible, or warrant belief. And there will be some outrageous philosophical views (perhaps our imagined Caligula's views on the intrinsic value of torture, and grue-some views of induction / projectible predicates, etc.) that are ruled out of bounds, or self-evidently false, again no matter how subjectively plausible a crazy agent might find them. It remains very much an open question how much epistemic permissiveness we should expect: Are views that are ultimately misguided, but not totally crazy, a priori warranted? Variations on this approach might be more or less restrictive on such matters.
This overcomes the problems with unrestricted epistemic conservatism, but raises its own worries. In particular:
* Warrant has now become so objective (and so closely tied with the truth of the matter) that we might worry that it has lost its ability to fill the roles associated with normative guidance. There's an obvious sense in which we can no longer tell what's warranted and what's not. So the guidance to "believe in line with the evidence" is not much more useful than the patently unfollowable guidance, "believe just what's objectively true."
Other options? I'm not really seeing any, so option 3 is looking to me the best way to go, despite its faults. But what do you think? Which is the best of the above options? and Can you think of any better alternative views? Suggestions welcome!