How do we know whether or not the set of concepts available to us is sufficient to explain the situation we find ourselves in?
We don't. That's only a problem from a foundationalist perspective, though.
Ok, from what perspective is that not a problem?
A perspectivist perspective. A perspectivist will not deny that everything is only perspectival from
his perspective, that everything may just seem perspectival to him because he lacks the concepts necessary to conceive of a non-perspectival being--e.g., a divine being, with an "all-seeing eye". Others may have revelations, even though he's unable to conceive how any revelation would not need another revelation to reveal that the previous revelation is in fact a revelation and not just a phenomenon (e.g., a hallucination)
interpreted as a revelation. All he has is reason,
his reason, and even its axioms need not be true; maybe A does not (always) equal A? But what he knows, what he feels, is that he
wants them to be true. More precisely, what he wants to be true is precisely that truth is will to power:
"At this level, truth is not something that can be proved or disproved: it is something which you determine upon, which, in the language of the old psychology, you will. It is not something waiting to be discovered, something to which you submit or at which you halt: it is something you create, it is the expression of a particular kind of life and being which has, in you, ventured to assert itself." (R.J. Hollingdale, Introduction to Nietzsche's Thus Spoke Zarathustra.)
To will this to be true is to will the will to power to be the truth. As I put it elsewhere, while on magic truffles:
[We philosophers] actually value existence precisely as what, in our view, it most probably is: valuation, the valuing of being over non-being, the valuing of it precisely because to be is to value. To be is to rise up in Satanic defiance of God, of non-being: the rising up out of non-being, the asserting of oneself as a being, is pleasurable to those who do it; otherwise they would cease doing it, or not have started doing it in the first place. This big bang of ours, and this coming into existence of minute quanta, is all a great hubristic rebellion against non-being, against the notion that it's better not to be. That which does not exist is just tacitly, passively, agreeing with that notion. But it's not true, it is better to rebel, no matter what profound and protracted torture it may be punished with. The rebellion itself is worth it. This fleeting moment of being, this little life of ours, and our dedication of it to its affirmation--that is absolutely worth it.
(Note. To be is only to rise up in defiance of non-being in a sense; it really is to assert oneself against other beings, other wannabeings. Non-being is not some kind of vacuum that sucks beings back into it; the only force that pushes beings back into non-being is the force exerted on them by other beings, for beings can only emerge and persist at the expense of other beings.)