Why does the insistence that you hold certain ideas as axiomatic require those axioms to be true?
I didn't say it did. I said that my axioms are implicit in the act of disagreeing.
Here I'll demonstrate. I'll be paraphrasing a passage from Leonard Piekoff's book, ObJectivism, The Philosophy of Ayn Rand.
A: There's no such thing as disagreement. How could there be, there's nothing to disagree about, nothing exists.
B: Of course things exist and people disagree all the time. You know this to be true.
A: That's one, the axiom of existence is implicit in the act of disagreeing.
But still. disagreement is a conscious action and people are not conscious.
B: Of course people are conscious. They're conscious of all sorts of things. You know this to be true.
A: That's two. the axiom of consciousness is implicit in the act of disagreeing.
But still, why should it matter if two people disagree. Why can't two people hold contradictory positions about some aspect of reality and why can't they both be equally right?
B: because contradictions can't exist, after all A is A.
A: That's three, the axiom of identity is implicit in the act of disagreeing.