You are asking the right question. Your intuition that something is missing is accurate. Gödel’s theorem essentially predicts it. Transitive operational level logic is not fundamental.
But are you prepared for the actual answer?
There is in fact an escape from Gödel’s theorem. I’ve spent 20 years peering into the matter myself, though not in search of getting behind the theorem. That was a only consequence of the discovery.
The 3 fundamental laws of logic are not independent axioms that are merely related in a transitive operational sense. In fact, they make no logical sense as a transitive relationship. Their relationship is coherent on a more fundamental level. And that level makes transitive operations like Gödel’s theorem possible.
Each of the 3 laws can only be defined in the presence of all 3 laws. So each law is a distinct expression of all 3.
Take the law of non-contradiction for example:
A thing cannot be A and -A at the same time or in the same sense.
Formulate it however you like, the result is the same. Formulation requires Identity, Distinction, and Exclusion.
The same structure applies to all laws. They are not individual axioms, but mutually dependent statements.
This fundamental level of logical structure makes every other level of logical and mathematical systems possible. It is the level of holism.
Furthermore, the 3 fundamental laws can be mathematically described and is fully coherent as a holism. I’ve included the python script below.
This hollistic structure is not caught in or bound by the transitive axiomatic logical loop that Gödel’s theorem describes. It is grounded in the objectivity of 3 necessary perspectives as expressed in the distinct laws.
In other words, the logical holism contains the minimum number of variables/witnesses required for objective verity.
I asked if you really wanted the answer to your valid and solid intuition, because the implications are staggering. It is not an answer many people will be comfortable with. We have encountered this 3 in 1 anomoly before in the Trinitarian theology stubbornly articulated by Jesus of Nazareth.
We couldn’t make transitive coherent sense of the relationship structure He claimed to embody and didn’t discover the mathematical category until 1900 years later.
Well, now we can make sense of it as a holism. Implications be damned. Python script below:
import numpy as np
Define the shared Relational Matrix
A = np.array([
[0, 1, 1, 1],
[1, 0, -1, -1],
[1, -1, 0, -1],
[1, -1, -1, 0]
], dtype=float)
Extract the isolated peripheral submatrix (Center removed)
A_isolated = A[1:, 1:]
print(f"1:1 Structural Isomorphism Verified: {np.array_equal(A, A.T)}“)
print(f"Full System Spectrum: {np.sort(np.linalg.eigvalsh(A))}”)
print(f"Isolated Submatrix Spectrum: {np.sort(np.linalg.eigvalsh(A_isolated))}“)
print(f"Full System Determinant: {round(np.linalg.det(A))}”)
print(f"Isolated Submatrix Determinant: {round(np.linalg.det(A_isolated))}")