Someone could believe in other errors or assumptions while still knowing that logical axioms are inherently true and that their own mind exists, that they are thus absolutely certain, and that no matter what else is true, it cannot contradict axioms. They could wrongly believe in the knowability of scientific laws beyond mere perceptions and still know that logical axioms are true separately, intrinsically, and universally. They could mistake evidence for the presence of other minds or for Yahweh being the same as the logically necessary uncaused cause for proof, and still they would know that axioms are epistemologically self-evident because they are true in themselves if only they at least were willing to not make assumptions about the very foundations of all things--that which is true independent of all else and yet that everything else depends on.
There is no ultimate irrationality other than denying logical axioms. It is not thinking Earth is flat, or that something one is looking at does not exist (not that one could know these things either way), that God does not exist (and an uncaused cause is provable), or that emotion is relevant to whether morality exists, to give just some examples, that is the supreme type of irrationality. No, this could only be the denial of the specific truths that could not be false without still already being true. That something logically following from another thing is true, that something is true about reality, that something is either true or false, and a small handful of other truths, like what might be termed the law of identity, cannot be or have been any other way--not because of God, nature, or human beliefs or endeavors.
Whoever denies logical axioms is an irrationalist. Whoever recognizes logical axioms at least on a basic level, even if they believe other things that contradict what follows from them or make assumptions elsewhere, has to a partial extent avoided assumptions. As long as they truly do see that logical axioms cannot be anything but inherently true, they have genuine knowledge of at least this most important of subjects. One must flee from assumptions to have knowledge, and no one can have knowledge apart from knowing the axioms that have self-evidence and intrinsic veracity. Partial avoidance of assumptions is possible. It is just not enough to truly know or honor logical axioms beyond the most cursory acknowledgement.
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