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An argument that is often brought up when discussing the Objectivists' axioms is that any attempt to refute them depends on their existence. But what if a man in my dreams denies that he exists, does this imaginary man then exist because his refutation requires his existence? If not, wouldn't the same apply to the "real" refutations?

Also, in Galt's speech Ayn Rand says: "A consciousness conscious of nothing but itself is a contradiction in terms". Why is this? The definition of consciousness, according to Ayn Rand, is: that which identifies existents. If consciousness exists, and it identifies itself, why is this a contradiction in terms? One might argue that in order for us to be able to identify ourselves, we have to undergo a series of integrations based of observation of the outside world, but is it really necessarily so? What is essentially needed is stimuli, but that doesn't necessarily have to come from "reality" as we perceive it, the "Mad Scientist Argument" (which states that the things you perceive are simply the products of electrical impulses that a scientist sends to your brain) could account for this, as well as a scenario such as the one presented in "the Matrix" (the Universe is the creation of a computer program).

In OPAR, Peikoff addresses these issues claiming that they are "stolen concept" fallacies, that in order for us to arrive at the concepts of "scientist" and "computer programs", we have to undergo a series of complex integrations of sense data. But who is to say that these sense data have to come from our perceived reality, again these stimuli could be the products of electrical impulses sent by the scientist, or the computer program etc.

I understand that these axiomatic concepts cannot be proven as such, since in order for us to prove them we have to step out of existence as we know it, but wouldn't it be better to then say: to the extent that we can know anything, 1) Existence exists, 2) Consciousness is conscious, 3) Identity is existence

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Peikoff also points out that "[p]roof presupposes the principle that facts are not "malleable.""

Axiomatic concepts are grasped conceptually via sense perception by a process of validation. This usually requires more that a cursory familiarity with concept-formation, and the various aspects of reality which interact to make knowledge possible, including the validity of the senses and volition. After that, you may begin to grasp "to what extent we can know anything" and the degree of certainty which can be subsequently attained.

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An argument that is often brought up when discussing the Objectivists' axioms is that any attempt to refute them depends on their existence. But what if a man in my dreams denies that he exists, does this imaginary man then exist because his refutation requires his existence? If not, wouldn't the same apply to the "real" refutations?

The answer is yes, but only in your dream and for the duration of your dream and then further as a memory. It does not prove that there is an actual person outside of your head.

Also, in Galt's speech Ayn Rand says: "A consciousness conscious of nothing but itself is a contradiction in terms". Why is this? The definition of consciousness, according to Ayn Rand, is: that which identifies existents. If consciousness exists, and it identifies itself, why is this a contradiction in terms? One might argue that in order for us to be able to identify ourselves, we have to undergo a series of integrations based of observation of the outside world, but is it really necessarily so? What is essentially needed is stimuli, but that doesn't necessarily have to come from "reality" as we perceive it, the "Mad Scientist Argument" (which states that the things you perceive are simply the products of electrical impulses that a scientist sends to your brain) could account for this, as well as a scenario such as the one presented in "the Matrix" (the Universe is the creation of a computer program).

Actually to split hairs it is the process of differentiating that comes first, not integrating. Differentiating is just noticing differences. In the case you want to examine, a consciousness with nothing to differentiate itself from could not be self aware and since it is stipulated that there is nothing else to be aware of then it could not be aware at all. Continuing to even use the word consciousness as a noun is misleading, it is like saying red-ness actually exists when there is nothing that is red. The root word in the noun 'consciousness' is the verb 'conscious', and changing parts of speech makes it even more misleading than the case of 'red vs. redness'.

The solution to the "Mad Scientist Argument" and "The Matrix" is to understand that even this case still conforms to 'subject is aware of object', awareness is a verb and inherently relational, or in another word relative. So long as what you see comes from outside your own consciousness, then even if it is simulated it is real and the deeper reality is the memory states of some supercomputer. The expectation that things must really be just as we perceive them is unjustified, and has the name "naive realism." Here is a presentation and refutation of this and several other arguments against realism: Pierre Le Morvan, "Arguments Against Direct Realism and How to Counter Them", American Philosophical Quarterly 41, no. 3 (2004): 221-234. (pdf) Further information on the topic of "can I really believe my lying eyes?" see the book "The Evidence of the Senses" by David Kelley and my notes on that book here, (see esp. post #15 on chapter 4).

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Also, in Galt's speech Ayn Rand says: "A consciousness conscious of nothing but itself is a contradiction in terms". Why is this? The definition of consciousness, according to Ayn Rand, is: that which identifies existents. If consciousness exists, and it identifies itself, why is this a contradiction in terms?

Just to answer this specific thing, Galt is quoting Aristotle in that sentence. What is meant is that it would imply an infinite regress. So it would be "A consciousness conscious of consciousness... of what?" and so on. It has to have some content in order to be conscious. Thus, the rest of the sentence you quote from is: "before it could identify itself as consciousness, it had to be conscious of something."

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