But, all kidding aside, when most academic philosophers (who aren't deeply acquainted with Objectivist literature and lecture courses) read about Ayn Rand's axioms, they tend to deride them as tautologies. They can't possibly imagine how you could deduce a whole philosophical system from such tautologies. Well, they have something in common with my interlocutor (mirabileamavi) in the comments of a YouTube video. Hopefully, my concise answers to him (her?) should be clarifying:
Interlocutor: ...How do you derive causation from tautologies?
Me: The "tautologies" you speak of are axioms. If something is truly an axiom, it is too fundamental to be conceptually analyzed, but is perceptually self-evident. You need only observe reality to see that it is true.
An entity is itself, therefore it acts as itself. This mode of action consistent with its nature is causality. See: objectivismfordeepthinkers.blogspot.com/2012/06/axioms-of-objectivism.html
Int: i still dont understand how you can derive causation from tautologies.
'john', from that i can infer 'john' is 'john' but i can't infer that 'john is a fireman' can i? the predicate 'is a fireman' is not contained in 'john'. while 'john' is 'john' is necessarily true and tautological, 'john is a fireman' certainly is not. from a=a we cannot infer that a=b. heres an example: 'frank ramsey', who is his father, what is his occupation? obvious all you can infer is that F.R. is F.R., nothing else.
Me: At the level of bare axioms, all we can say is that, because John is John, John must act as John. That's it: causality is a corollary of identity. But to identify John as a fireman, we cannot simply deduce from the axiom. We must specifically observe firemen in order to form the concept "fireman." We must then observe John and see that he fits the concept. (Intro. to Objectivist Epistemology) Once we have observed he is a fireman, causality tells us he can't swim and lay eggs as a female squid.
Int: 'is' is not equivalent to 'act'.
okie, look at this from another angle. since identity is universally necessary, 2 is 2 is also an identity statement. but what does it mean to say that 2 act as 2? or for that matter, john act as john? if not just 'john is john'
from john is john nothing else follows. not causality, not anything. let me ask you again, what casual anything can you deduce from 'frank ramsey'. thats right, nothing.
Me: An entity's identity includes its qualities and capacities for action/reaction. We can isolate and focus on them in our thinking, by abstraction, but they cannot be separated in concrete reality. Causality is a corollary of identity, not a separately deduced fact. As a corollary, it is simply another way of looking at the same fundamental fact: an entity is itself. It's self-evident: look at reality.
2 is a quantitative abstraction. Whatever 2 entities you are focusing on will act as themselves.
Int: take our friend 'fr' as an example, obviously we can infer 'fr' is 'fr' via any standard of formal logic. but we can't infer 'fr' is also p. why? because additional information is needed to establish the new inference.
to say something is corollary is to say that something follows from another. but how do you infer from 'fr' without the additional info. that 'fr' is also p? can i seriously validly infer that A, therefore B, C, D, X...?
i.e. how am i justified in seeing arsenic for the first time to infer that it can or cannot kill? surely none of its properties follows from my visual perception of it or the mere knowing of its name. yes we can know its effects/properties through observation, but thats an additional step, not something that merely follows from our acquaintance with it.
Me: For the last time, Objectivism doesn't say you can infer any specific properties/actions of entities from "A=A." To see that arsenic is deadly, you make specific observations of its effects. Once you have induced that arsenic is deadly, you know that once you have identified a specific sample as arsenic, it will be deadly when taken. Without causality, arsenic wouldn't have to behave as arsenic, and there's no way to know what will happen if you ingest it; it could make you live 1000 years.So the basic point here is that, in Objectivism, proceeding from the axioms does not mean deduction, but induction. The truth of the axioms (including the validity of the senses) makes induction from observation (generalization) possible (including concept formation as a certain type of induction.)
The major model of system building in modern Western philosophy has been that of the rationalists, who deduce consequences from "a priori postulates," "intuitive" starting points, or mathematical axioms. Thus, when confronted with a philosophic system like Objectivism that claims axioms, most contemporary philosophers simply assume that the axioms are intended as a deductive starting point. They then rightly observe that nothing can be deduced from the axioms alone, and claim that Objectivism is a failure, or is not "serious" philosophy.
This is what I was referring to by "the impoverished reasoning methods employed by academic philosophy": Real induction, which is a method of generating principles, has been largely supplanted by probabilistic reasoning, which most contemporary philosophers call "induction."
The details of how induction works in various fields of knowledge is an active area of research among Objectivist philosophers. But we have cases of induction and general guidelines for how to form valid inductions left by Ayn Rand, and explicated by Leonard Peikoff. I recommend Understanding Objectivism: A Guide to Learning Ayn Rand's Philosophy by Leonard Peikoff.
The video below is not directly relevant to the above, but is an excerpt from one of Rand's essays that makes general points about philosophy, reason and emotions.
[Edited: 9-1-12]
-----
Related Articles:
The Structure of Objectivism
The Axioms of Objectivism
.)
by Ayn Rand and 
by Craig Biddle. For those who are more philosophically oriented, I also recommend 
and 
by Dr. Tara Smith.