"The fundament upon which all our knowledge and learning rests is the inexplicable." ~ Schopenhauer
The methodological approach of the philosophical-skeptic must be the methodology of all philosophical theory which wishes to remove errant assumptions, as the methodological approach of the philosophical-skeptic denies all assumptions. This is not to say that the conclusions of the philosophical-skeptic are necessarily the conclusions that the philosopher would necessarily wish to emulate, but the methodology itself. Philosophic questioning is efficacious to reveal incredulous assumptions as the absurdities that they are. When we proceed from that which is most fundamental, we are less likely to deceive ourselves or assume certain, that which is inherently uncertain/unverifiable.
Now, removing all assumptions allow me to examine what must examine what my axioms/premises must be.
Axiom-1. I am conscious. If this were not so, I could not be conscious of my own consciousness. A study of epistemology, or the theory of knowledge, or the study of "How do we know, what we know?" must begin with an assumption of a know-er, a subject, a thinking/conscious person. Peering further, we cannot examine knowledge without a know-er, without a subject. Consciousness of ourselves is properly, the first axiom. It is philosophically inescapable. I have an inclination that a philosophy that begins from any other assumption is doomed to error.
Premise-1. I am have inescapable patterns of consciousness. This premise might be considered as the premise of logic, or as some others have deemed it, the axiom of identity: that "'A' is 'A'", from which the law of non-contradiction is ascertained. Is this an indisputable premise? I am not entirely certain that any premise can be epistemically prior or more fundamental than the first axiom, yet, I must reject the denial of "'A' is 'A'" merely because I cannot conceive of it to be otherwise. This is no rigorously epistemological-certainty to be sure; it is rather an admission of limitation of the patterns/categories of thought that I can conceive, perhaps some person, alien or entity may be able to conceive differently but for my consciousness, I cannot conceive differently from that which is denoted by what is called 'logic'. Quantum-physics might be able to demonstrate some denial of this "'A' is 'A'" premise; those knowledgeable in such things might respond that we may only say that "'A-ing' is 'A-ing'" but even this modification indicating constant action, indicates that there are patterns of thought which are constant (because the logical form is the same: the 'is' is still the operator of thought) at least in my modest understanding. We may theorize why this is, perhaps our minds have some limitation connected with our biological brains; or perhaps our minds are in complete accordance with 'reality', or any number of additional theories; but whatever the case I cannot examine this premise further, as to probe further past the premise of logic reveals only absurdity to my consciousness. This 'absurdity' is not an argument in and of itself, as logic, itself would dictate that certain premises are necessary for the analysis and synthesis of ideas to take place, and that there are certain premises which are necessarily indisputable and these indisputable premises are known as 'axioms'; these 'axioms' are the fundamental ideas, without which, no logic could take place; a logic without axioms or premises is necessarily void.
Premise-2. I have the apprehension of sensory-experience of phenomenon. Currently, I am
apprehending many sensory experiences of various phenomenon, I see this 'computer', I feel the
touch of the 'keyboard', I hear the clacking of 'computer keys' on a
'keyboard' along with the 'music' of 'Johnny Cash' in the 'background', I have the slight taste of 'unsweetened iced-tea' and the smell of 'sauerkraut' is faint but noticeable;
you are reading the words on the 'screen' or 'paper' from which you are
(P2-A&2-B). My apprehension of sensory-experience of phenomenon leads me to two sub-premises, the first (P2-A) that my experience occurs through 'time', and the second (P2-B), that there is 'substance' outside/independent of my
consciousness. These two sub-premises of sensory experience together, led me to yet a further sub-sub-premise (P2-A&B-a), that the phenomenon of
my sensory experience relate in a way that I apprehend as 'causation' (P2-A&B-a); that there are causes which have effects.
(P2-A&B-a-1&2) I have had many experiences, but an important distinction between those experiences which must be acknowledged because of Axiom-1, are those experiences which are satisfactory/preferable/pleasing (P2A&B-a-1) and those which are unsatisfactory/unpreferable/painful (P2-A&B-a-2); there may be between these two important distinctions of sensory-experience , a continuum of experiences of which may be described as ambivalent (P2-A&B-a-3) but for the moment, I will put these aside. Because of an apprehension of this 'causation' that 'substance' through 'time' are related and that this can relate to my satisfaction (P2A&B-a-1) or unsatisfaction (P2-A&B-a-2), I am (highly) motivated to create causes that will result in effects that will lead to my satisfaction and not to my unsatifaction (P2-A&B-a-1&2-A: premise of 'action' = praxeology).
Premise-3. I have symbols for communication. Of course this premise has been assumed this entire time, but without the "I" that speaks and without which the logical ideas for which those symbols may represent and without the "you" that receives these symbols, the premise of symbols for communication (otherwise known as "language") is itself void. So I place this premise fourth, though it must have been assumed this entire time. I presume that I have available to myself, symbols which represent ideas, which may trigger/initiate similar ideas in your mind. Is this premise necessary? Like the second, it is by no means epistemologically-certain; but I assume this nonetheless as a theory of solipsism (that I am the only mind that exists and that nothing else but my mind exists) is unsatisfactory to my preferences. If I would deny this premise, then there would be little justification for constructing this theory at all, other than my own personal satisfaction.