Here is an analogy I use to think about our recognition and use of axioms or basic truths.
In our experience, something is incorporated into our awareness through practice (and my use here is not limited to only the wood-shedding practice of shooting 100 3-pointers a day, but also the sort of practice as opposed to theory) of that something. Consider musical instruments (if you have no experience playing any sort of musical instruments, then you’ll get what I mean with the next few examples); when you first started playing, the instrument itself was foreign to you, performing the requisite actions to get the right result took conscious thought at first, and every note you attempted to play was a torturous process of remembering the right fingering, keeping beat, playing the right rhythm, keeping the right form (for wind players, this would be called embouchure), reading the music on the page, blowing the precise amount of air to control note, pitch, and volume, etc. However, as one practices more with their instrument, they get the music “under their fingers,” being able to bend their mind towards not just getting the right note and rhythm, but to playing the piece dynamically, employing vibrato, precisely tuning the note in key with the chord.
In the beginning, the instrument was something outside your experience, outside of who you were. But, with practice, the instrument starts to become ‘centered’ within you, a part of you just as your limbs are. The mark of a master of any sort of instrument, be it musical, aesthetic (drawing, handwriting), motor (weapons, tools, cutlery, sports, etc) is that their trade has become nothing more than an extension of themselves. An ad-libbing musician thinks the tune in their mind and their body performs all the requisite functions from memory, because the foreignness of the instrument has been removed. Rafael Mendez, a masterful trumpeter, has a few wonderful things to say about musical instruments becoming second nature here.
To bring the idea more in relation to your experience, consider these experiences you’re probably familiar with.
Baseball; at first, utilizing your arm to its full potential and power feels awkward, but through repetition muscle memory develops until catching and throwing the ball in an efficient and powerful manner can happen without thought, becoming part of one smooth aesthetic enterprise of coordination, as demonstrated here.
Driving; how much give there is between the road and your tires in different conditions takes experience to be able to garner an expectation of how the entire car will react to your steering, accelerating, and braking. To have the car at your command is to feel a unity between mind and machine, as is needed for this.
Soccer; the bounce and arc of the soccer ball, the feel of your legs and feet, the pacing and precision needed to kick, dribble, and juggle, are gained through experiential practice with the soccer ball. The ability to “feel” where the ball is just through repeatedly kicking a ball in the same way and kick it up again is in juggling is something that almost seems like it shouldn’t be able to be achieved, yet it quite obviously can.
Now, what is my point in all of this? I wish to draw a line between experience of reality and our fundamental connection with reality, because I believe the same sort of experience that we draw can be trusted due to our ability to precisely interact and gain more precision through more practical experience of this reality. There are certain axioms that we grasp through repeated experience of; for instance, the effect of gravity and motion together that is needed to be able to juggle a soccer ball, the effect of friction and resistance to be understood not mentally but practically to drive, the feeling of motion and coordination in baseball.
If there can be drawn a direct connection between ourselves and reality, then it should be obvious that the laws of logic as basic truths necessarily follow and cannot in any circumstance be circumvented or denied. While we do gain our ability to think by experience of reality, reality appears to have no intention of lying to us, and we seem suited to grasp reality. Thus, axioms can and should be drawn and utilized in thinking about any thing, because we cannot actually imagine any thing that is not under some sort of relation to these axioms (i.e. logicality, rationality, truthfulness, order, etc).
A square circle or a blue white are understood to be logical impossibilities by merely understanding them in relation to the basic truths of logic, and require no further argument to be disproved.
Interesting consequences occur for naturalism and theism, and I obviously believe that theism is suited towards the existence of logic whereas naturalism is unable to explain axioms in nature.