So, if you attack the problem of reliable knowledge through the lens of the Agrippan trilemma creating an extended metaphor with a computational basis, then the we can provide a quick translation between your functional notation and current theory and praxis.
- S - Intuition; this presupposes of course that S is more than linguistic. While Ryle's knowledge-how and knowledge-that are the base, knowledge might be characterized in a variety of ways some of which is not linguistic.
- kS - Skepticism and the Demand for Proof; here, we see humans have doubts, and two opposite positions on a spectrum are faith and disbelief. Skepticism strikes the middle ground; to have doubts is human, and is generally aligned as an enemy of faith. But there are traditional theological positions that embrace various forms of limited skepticism. The current Pope of the Catholic church is a man of science, for instance. This places him at odds with a local, fundamentalist cult leader. Here we use language to express skeptic intuitions in full and work on achieving a theory.
- k2S - Epistemology and the Examination of the Demand for Proof; now, if we begin to become skeptical of various positions on skepticism, then we are, I believe, in the heart of conventional epistemology, often entertaining argumentation about what constitutes an adequate position on skepticism. Should we embrace Academic skepticism in the modern age and in the face of the Gettier Problems? Should we allow some segment of faith, or rather, is faith such that we can't avoid it? Here we not only use language to express skeptical theory, but we also examine the various theories.
- k3S - Meta-epistemology and the Search for Good Method; now we are starting to get heady. When we begin to debate theory, we then turn theorization on itself; we do so intuitively at first, but like all things recursion, we then have the ability to fully harness our language faculty; but the moment we start asking questions about what generalized theories relate to building and evaluating epistemological theories, we have crossed another threshold. We can doubt whether we can doubt. We can doubt whether we can know. Radical skepticism or the belief that eudaimonia is the aim of knowledge-seeking are not just epistemological, they are normative in regards to the epistemological, and so are also meta-epistemological. One might object that such a meta-level is unnecessary, that the positions themselves are merely epistemological, but the skeptic mind attempts to justify the normative aspects of language use.
So, while one can say 'this skepticism is better than that skepticism', the astute thinker will immediately draw the former epistemological claim into doubt by questioning the knowledge-methods used to justify it. We can also attempt to put epistemological practice in a particular Weltanschauung. A feminist will have a different approach to epistemology, because feminism is likely to presuppose certain skeptical positions in relation to male-female relations. A theologian will try to craft a theory of doubt around their faith in God. An athiest presupposes certain skeptical positions about supernaturalism. It seems generally that there is a be a strong analog to formal theories such as proof theory and model theory when considering a range of issues important to skepticism and skepticism about skepticism, respectively.
- We have doubt (S, skeptical intuitions)
- We can express and argue doubt (kS, skeptical theory)
- We can express and argue expressing and arguing doubt (k2S, epistemology)
- We can express and argue 'We can express and argue expressing and arguing doubt' (k3S, meta-epistemology)
These all seem intuitively reasonable and comprehensible and seems consistent with epistemology and meta-epistemology as currently practiced. Your question is what lies beyond this. Hic sunt dracones.
I would argue that the next target of skepticism turning on itself would be the examination of worldviews such that if one were to create a category to deal with the general class of all skepticism expressible and the relationship of all worldviews to skepticism, that is, the concern of all matters of faith, doubt, modality, reason, etc., then it makes sense to create a sort of universal set of discourse related to all propositions, arguments, and positions:
So far, I have created what I believe to be a largely positive account of philosophy, as the facts as have been presented are, I submit, consistent with tertiary sources on philosophy, whose specific content can be disputed, but not denied to exist. From here out, I believe the next level of recursion in skepticism would necessarily be a step from positive to normative claims meta-epistemology.
I'd tentatively submit having exhausted what are essentially positive claims about skepticism, one is simply left with original claims as to one's personal experience. On the one level, this is my worldview. As someone who has evolved his skepticism to a naturalized epistemology placed within the context of a general awareness of meta-epistemological concerns, I have simply exhausted, as you put it, the problem-space, at least in principle. I have taken permutation explosion, and created a system for locating claims within it through analytical means. But I now have to accept that my claims are rooted in bias and preference, and then set about my own worldview to discuss my worldview. So...
- k4+S - The Meta-Epistemological Examination of One's Meta-Epistemological Theory. Here, is the catchall category for all claims that are simply too complex to fit nicely in the simpler categories presented. For instance, consider the claim: "How do my emotional biases towards faith and skepticism color my worldview that affects my attitudes towards meta-epistemological discourse?" Certainly such a claim can be seen to be, in terms of predication, more complex than k3S by one degree of predicate. Here then is a simple proposition. We can create any meaningful kNs expression by simply meeting two criteria:
- N matches degree of predication. In the example listed, "biases affect attitudes towards meta-epistemology" (Affect(biases,attitudes:meta-epistemological)) attitudes is the predicate with an epistemological property.
- Attribution of skepticism to predication. In the example listed, the subject is conceptually related to skepticism. "Biases towards skepticism affect attitudes" (Affect(biases:skepticism,meta-epistemological)).
Thus, there is no limit to the depth of the kNS as a subject, though, obviously as a single claim, there are certain psycholinguistic limits to comprehension.
Is this useful? I suppose from an academic perspective, kNS would allow for the classification of the depth of skeptical predication, but I don't know that other for those of us who delight in mathematical pedantry, such a practice has any value.