Ripheus23

In setting forth the new picture of set-theoretic axioms, we would be called to justify our appeal to the Law of the Excluded Middle, and double-negation elimination, so that constructive intuitionism is invalidated. In fact, the construct intuitionist's picture of logical knowledge is deeply flawed: we do not start with an interpretation of

A ^ X = X ^ A [where "^" = "and"]

... and proceed to derive the Law of Noncontradiction from this, and then LEM/DNE if we are "lucky," so to speak. Rather, the LEM is itself the foundation of the system, not in the sense that we deduce the Laws of Identity and Noncontradiction from this, but rather in the sense that our symbolic representation or indication of LEM is prior so that otherwise, we know all three logical laws simultaneously in the same act of cognition. There is, therefore, no actually possible logic that involves identity or noncontradiction but adverts away from LEM/DNE.

But as has been explained, the LEM admits of a proof by erotetic adduction. That is, adduction of the very form of questions themselves gives us the schematic of LEM, which by Q --> A [every form of question implies a possible form of answer] is equivalent to an assertion of the LEM.