I just read this essay on coherentism, and it resonated with a question I have about reconciling foundationalism, coherentism, and infinitism. The gist of the essay is that there are graph-theoretic structures that are isomorphic to epistemic support/justification structures, forms of epistemic regression.
One subsidiary discussion there, albeit brief, concerns serial vs. ramified regression. My "sense" was that there is "the" regress, that all given beliefs, when traced backwards, are integrated into a single pathway through sentential space. But something came up [see the motivation section below] that caused me to wonder about this, and in light of the serial/ramified distinction, I wonder: is there supposed to be a single integrated inferential pathway, or do our (justifiable) beliefs map to separate regresses (separate language games, non-overlapping magisteria)? For example, assume coherentism is true. Does this require that all coherentist justification involves integrating beliefs into a single (massive) circle, or could there be a plurality of circles? Moreover, in light of Susan Haack's analogy of the crossword puzzle modulo foundherentism, would these circles all intersect at some point or other, or would they be fully separate? (Assume, that is, that a node in some coherentist loop might be shared by two or more logical circles, forming part of the inferential loop in each case.) Taken literally, the crossword-puzzle analogy seems to involve a plurality of regresses, in my eyes.
Motivation: While working on my theory of mathematical justification, I noticed a seeming correspondence between three kinds of sets and the three non-empty solutions to the epistemic regress problem. That is, well-founded (WF) sets seem to correspond to foundationalism; looping sets like Quine atoms or sequences like, "x ∈ y, y ∈ z, z ∈ x," seem to correspond to coherentism; and infinite descending ∈-chains seem to correspond to infinitism.
What I got from this seeming correspondence was that the structure of the justification for an ∃-sentence in set theory depends on the structure of the set whose existence is in question. Viz., well-founded sets' ∃-sentences are justified in a foundationalist way, either in or from axioms; looping sets' ∃-sentences are justified in a coherentist way; and ∃-sentences for infinite descending ∈-chains are justified in an infinitist way. Consequently, it seems that axioms of antifoundation are not justifiable, though. This isn't to say that ∃-sentences for nonwell-founded sets are unjustifiable altogether, only that this justification ought not be pictured as axiomatic.
Now, I think of V as a very integrated "object." Moreover, modulo V = X sentences (like V = L, V = HOD, and so on), I have a proposal of my own: let J be the transet of all sets with sufficiently justifiable ∃-sentences to their names, and then say that V = J ("the justifiable universe"). So allowing all three non-empty solutions to the regress problem, here, we have that V would be the transet of the ∃-justified WF sets, the loop sets, and the descending sets.
But how does this transet collate these domains? Moreover, modulo the question of plural regresses, do we suppose one cumulative hierarchy, one absolute loop, and one (presumably ORD-long†) descending chain, or could we have a plurality of these? I think in paraconsistent set theory that there is room for a plurality of minimal elements (and ur-elements might also be part of such a picture), hence a plurality of WF-subdomains. I see no obstacle, there, to a plurality of the other subdomains, too.
Now, the major problem is that I have only come up with one actual infinitistic regress that goes through. That is, let axioms be not so much not inferred at all, but rather inferred from questions. The driving force in epistemic regression seems, to me, to be erotetic inference. My understanding, then, is that a given assertoric regress of inference terminates in axioms, and that what is traced backwards in the sequence of inference, relative to the point where the axioms appear, is a sequence of questions related by erotetic inference. For example, have there be a regress ... C?, B?, A?, A, B, C... and say that A is an axiom. What I am saying is that from A?, we reverse-infer B?, then C?, and from C? we reverse-infer D?, and so on, absolutely forever and ever.
As far as unifying V goes, then, this would have the descending domain logically attached to the cumulative hierarchy, at the point at which the axioms of the hierarchy appear. These two domains would not be inferentially closed off from each other.
Moreover, it seems that you could have points in a regress that ramified not into branches that went off in totally different directions, but into circles whose inferential halves converged at some further upward point. You could embed a coherentist loop into a foundationalist circuit, then. I'll admit, it's not clear to me how the corresponding V-structures would link up (this would require having a WF-set be part of a loop set at some point).
What strengths/weaknesses are there in having a single regress with all inference structures somehow integrated therein, vs. having a plurality of possibly separate (inferentially closed) such structures? I feel like the essay I mentioned at the beginning of this post is probably not the only analysis of this issue available. (I do remember Kant discussing the issue, if more or less implicitly, in the Transcendental Dialectic.)
†The argument for V = J is easy enough, then: we assume that the descending transet is ultimately equinumerous with the cumulative hierarchy, show that the only numerical value the descending transet can have is J, and then voila, we have that V = J.