Even if we were to accept that logic exists (in form of law of identity only), these premises follow:
P1. In each closed system of limited claims, each claim is dependent on "something" to prove itself true.
P2. that "something" can only be law of identity itself
P3. If it is not law of identity, then claims are essentially unproved by law of identity
P4. It is impossible in a closed system of limited number of claims to prove that all claims actually follow law of identity
P5. Premise 4 is true due to forced infinity regress or circularity any time when all claims in a system are questioned.
For each claim, someone can ask "Why is that true?", forcing any closed system to either run out of proved claims, or force circularity. And this is true even when we accept law of identity as valid.
"bachelor is unmarried man"
Why is that claim true?
"Because it follows law of identity"
Why is that claim true?
"Because bachelor is defined as unmarried man"
Why is that claim true?
"Because society assigned such definition"
Why is that claim true?
"Because society exists and we observe it"
Why is that claim true?
"You are observing it too"
Why is that claim true?
"You are doing infinite regress fallacy"
Why is that claim true?
"I see you do it"
Why is that claim true?
the question "Why is that claim true?" can be repeated endlessly for all claims made by any closed system of claims.
Godel noticed this in math and made few incompleteness theorems, however, math isnt the only thing suffering from the very same problem.
If all claims need proof, no system of closed claims can prove all claims true in the system.
"And why is that true?"