Objections to the Axioms (Part 6)

This will probably be my last response to the metaphysical axioms for some time.

A commenter raises the following issue:

It's often said that to deny axiom[sic] is to engage in self contradiction - and that wouldn't be a valid objection because in order to classify contradiction as an error one has to assume axioms to be true. I see circular reasoning in this answer against axiom deniers.[1] 

Objections to the Axioms (Part 3)

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Question: “Are Axioms Proven or Merely Assumptions?”

“Are first principles or the axioms of logic (such as identity, non-contradiction) provable? If not, then isn't just an intuitive assumption that they are true?[...]”[1]

The axioms are neither “proven” nor “assumed.” 

(In the Objectivist view of axiomatic corollaries, Aristotle’s “Laws of Thought” are corollaries of the Existence axiom.  And more specifically, the Law or Principle of Non-contradiction and the Law of the Excluded Middle are restatements/corollaries of the Law of Identity, which is a corollary of “existence exists.”[2] So I’ll consider this question as broad enough to encompass any first principle, including the Objectivist axioms.)

I’ll make several points about why this can’t be the case when speaking of actual axioms.

Induction of "The Arbitrary as Neither True Nor False"

[Previous post: "Induction and Reduction of 'Values as Objective'"]

The aim of this essay is to induce the Objectivist principle that arbitrary claims are neither true nor false, but are in a third class: non-cognitive. Ayn Rand said in regard to arbitrary assertions that, “it is as if nothing had been said, because nothing of cognitive value or validity has been said.”

The outline of this essay consists of three inductions and two clarifications: