In the following quote from Leonard Peikoff’s “Objectivism: The Philosophy of Ayn Rand”, we see the presuppositionalist argument (or transcendental argument) for proving three axiomatic concepts: existence, identity, and consciousness.
First, he appeals to our common sense perceptual judgments: things exist, things have definite identity, and we are consciously aware of them. We intuitively believe in these axioms because these judgments are implicit in every moment of conscious awareness:
One knows that the axioms are true, not by inference of any kind, but by sense perception. When one perceives a tomato, for example, there is no evidence that it exists, beyond the fact that one perceives it; there is no evidence that it is something, beyond the fact that one perceives it; and there is no evidence that one is aware, beyond the fact that one is perceiving it. Axioms are perceptual self-evidencies. There is nothing to be said in their behalf except: look at reality.
…
Objectivism: The Philosophy of Ayn Rand, by Leonard Peikoff, p.8
The above is the validation of the Objectivist axioms. “Validation” I take to be a broader term than “proof”, one that subsumes any process of establishing an idea’s relationship to reality, whether deductive reasoning, inductive reasoning, or perceptual self-evidence. In this sense, one can and must validate every item of knowledge, including axioms. The validation of axioms, however, is the simplest of all: sense perception.
Then, he proves that these axioms are inescapable – any argument which purports to deny them must concede them:
The three axioms I have been discussing have a built-in protection against all attacks: they must be used and accepted by everyone, including those who attack them and those who attack the concept of the self-evident. Let me illustrate this point by considering a typical charge leveled by opponents of philosophical axioms.
“People disagree about axioms,” we often hear. “What is self-evident to one may not be self-evident to another. How then can a man know that his axioms are objectively true? How can he ever be sure he is right?”
This argument starts by accepting the concept of “disagreement”, which it uses to challenge the objectivity of any axioms, including existence, consciousness, and identity. The following condensed dialogue suggests one strategy by which to reveal the argument’s contradictions. The strategy begins with A, the defender of axioms, purporting to reject outright the concept of “disagreement”.
A: “Your objection to the self-evident has no validity. There is no such thing as disagreement. People agree about everything.”
B: “That’s absurd. People disagree constantly, about all kinds of things.”
A: “How can they? There’s nothing to disagree about, no subject matter. After all, nothing exists.”
B: “Nonsense. All kinds of things exist. You know that as well as I do.”
A: “That’s one. You must accept the existence axiom even to utter the term ‘disagreement’. But, to continue, I still claim that disagreement is unreal. How can people disagree, since they are unconscious beings who are unable to hold ideas at all?”
B: “Of course people hold ideas. They are conscious beings – you know that.”
A: “There’s another axiom. But even so, why is disagreement about ideas a problem? Why should it suggest that one or more of the parties is mistaken? Perhaps all of the people who disagree about the very same point are equally, objectively right.”
B: “That’s impossible. If two ideas contradict each other, they can’t both be right. Contradictions can’t exist in reality. After all, things are what they are. A is A.”
Existence, consciousness, and identity are presupposed by every statement and by every concept, including that of “disagreement”. In the act of voicing his objection, therefore, the objector has conceded the case. In any act of challenging or denying the three axioms, a man reaffirms them, no matter what the particular content of his challenge. The axioms are invulnerable.
Objectivism: The Philosophy of Ayn Rand, by Leonard Peikoff, p.9-11
This position is not unique to Peikoff; he is faithfully fleshing out the arguments from Ayn Rand:
“Axioms are… propositions identifying a fundamental, self-evident truth.”
“An axiomatic concept is the identification of a primary fact of reality, which cannot be reduced to other facts or broken into component parts. It is implicit in all facts and in all knowledge. It is the fundamentally given… on which all proofs and explanations rest“
“Since axiomatic concepts refer to facts of reality and are not a matter of “faith” or of man’s arbitrary choice, there is a way to ascertain whether a given concept is axiomatic or not: one ascertains it by observing the fact that an axiomatic concept cannot be escaped, that it is implicit in all knowledge, that it has to be accepted and used even in the process of any attempt to deny it.”
Introduction to Objectivist Epistemology, by Ayn Rand
“You cannot prove that you exist or that you’re conscious,” they chatter, blanking out the fact that proof presupposes existence, consciousness and a complex chain of knowledge: the existence of something to know, of a consciousness able to know it, and of a knowledge that has learned to distinguish between such concepts as the proved and the unproved.
When a savage who has not learned to speak declares that existence must be proved, he is asking you to prove it by means of non-existence—when he declares that your consciousness must be proved, he is asking you to prove it by means of unconsciousness—he is asking you to step into a void outside of existence and consciousness to give him proof of both—he is asking you to become a zero gaining knowledge about a zero.
When he declares that an axiom is a matter of arbitrary choice and he doesn’t choose to accept the axiom that he exists, he blanks out the fact that he has accepted it by uttering that sentence, that the only way to reject it is to shut one’s mouth, expound no theories and die.
An axiom is a statement that identifies the base of knowledge and of any further statement pertaining to that knowledge, a statement necessarily contained in all others, whether any particular speaker chooses to identify it or not. An axiom is a proposition that defeats its opponents by the fact that they have to accept it and use it in the process of any attempt to deny it.
John Galt’s speech, Atlas Shrugged, by Ayn Rand