This will probably be my last response to the metaphysical axioms for some time.
A commenter raises the following issue:
The axioms are not assumed but validated by sense-perception or direct introspective experience. Existence is not assumed: we can perceive things that exist and thus can eventually understand it, and why it is “self-evident.” Identity is not assumed: we perceive that things have certain natures and are finite, limited. Consciousness is not assumed: we are directly aware of our consciousness of existence. Objectivists know that the axioms are true because we perceive that their claims hold.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.
A similar process applies to validating the Law of Non-contradiction/law of contradiction. Objectivism agrees with these words once written by philosopher Brand Blanshard:
[…]If we hold that a thing cannot at once have a property and not have it, it is because we see that it cannot. The law of contradiction is at once the statement of a logical requirement and the statement of an ontological truth.
We can perceive that a tree’s leaf is flimsy, that it has an identity. From that insight (and several other kinds of perceptions), we can realize that while it is flimsy, it can’t be flimsy and sturdy as an oak wood table at the same time. A piece of paper cannot be burning and not burning at the same time and respect. A building cannot be complete and demolished at the same time. We observe at every moment that contradictions do not exist.
Contradictions are impossible and so even the attempt to earnestly hold a contradiction in thought is an error and must fail. It is literally impossible to maintain a contradiction, because every thought is about something definite, every belief has a single meaning. To believe something and its opposite at the same time is to believe nothing and think of nothing. Peikoff notes that, "[a] contradiction is a negation of identity and therefore of reality; to be A and non-A at the same time and in the same respect is to be nothing."
To believe a contradiction or to believe in contradictions as such is to be in error because of the truth of the laws of existence and identity, not because they can be assumed. It is the truth of these axioms that makes holding contradictions an error. On the contrary, it would be the belief (and non-belief) in contradictions that would be a baseless, arbitrary assumption. Believing a contradiction logically means not believing in it at the same time, which means: to believe in nothing in particular.
To deny the axioms, the objector, objection, the opposition, and many other things must exist (existence). They must have certain identities, or they are nothing and there is no objection (identity). And there must be at least two conscious beings for this discussion to even happen (consciousness). There simply is no escape from the axioms.
The axioms cannot be proved: even my defense of them here rests on their validity. But they cannot be meaningfully denied: the belief in contradictions or in nothing (the only other alternative to the axioms) cannot succeed. The attempt to deny the axioms ends exactly where the objectors wanted—in a host of contradictions.
: A comment made by an anonymous commenter on 16 August 2015 on my blogpost, "Objections to the Axioms (Part 4). http://inductivequest.blogspot.com/2015/08/objections-to-axioms-part-4.html.
: Brand Blanshard, Reason & Analysis, p. 276
: Leonard Peikoff, OPAR, pp. 115-116