The generic knowability principle is that if t is some truth, then it is possible for t to be known: t → ◊Kt. If foundationalism, coherentism, infinitism, and their combinations are taken as epistemic types, how do they specifically interface with the knowability principle?
For example, add a subscript F to K in the preceding to express that all truths are knowable foundationalistically: t → ◊KFt,E etc. So now my intuition is telling me that if the foundationalistic knowability principle is assumed, and held to be itself a knowable or even known truth, however, then the set of such knowable truths would end up containing itself, and so a coherentistic moment would appear in the system even on the universal assumption of foundationalism (I assume that coherentism and set loops go hand-in-hand, ultimately, just as infinite descending elementhood chains are infinitistic(al)). Unfortunately, for now, my attempts to formalize a derivation of this are not working very well. Is it possible to derive (an example of) coherentism, or perhaps foundherentism rather, from the foundationalistic knowability principle when this principle is self-applied?
Guesses (updated): (1) here is the knowability principle in set-theoretic form, and assuming foundationalism (see below), along with some other guesses at lines of the hypothetical derivation:
Definitions: ◊𝕂 is the set of foundationalistically knowable truths. ◊𝕂�� is the set of coherentistically knowable truths. T(ψ) = the truth of ψ. Emphasized: none of these guesses is particularly good, I must admit, at least for reasons of not being sufficiently refined.
ENot that all truths are knowable as foundations, i.e. non-inferentially, but that all truths are knowable as either foundations or by inference therefrom.
