But when they argue from "the impossibility of the contrary" this is precisely what they do. I guess you're not familiar with this style of apologetics. They claim that only the Christian worldview can account for knowledge, truth, intelligibility, good, evil, laws of logic, etc., and that they know this because an infallible being has revealed it to them in a way that they can be certain of it and without a belief in this being, one can not have this certainty. They do hold their position beyond skepticism. If I were a Christian, I'd be appalled by this. Just go watch them twist and evade and refuse to answer questions about their own worldview.
I think you are missing the point, though I am not familiar with the guys you list in the OP. Presuppositional Apologetics doesn't hold their position beyond scepticism, but points out there are good grounds to doubt others - their beliefs are based on Faith, thus giving certainty, but that Faith remains a requirement; or as you put it, an infallible Being has revealed it. That is the presupposition they start with. By taking that Faith as basis, the rest falls into place; while views without it, remain contradictory in some way. The classic example would be the Naturalist considering the valence of an idea to be illusory, but then argues from moral or value judgements to ideas like the best for society, or the good of humanity or posterity.
Cornelius van Till, the 'founder' of this school of Apologetics, argued that unbelievers are both Rational and Irrational, as they reject the Faith based on a semblance of Rationality, but this is shored-up by unconscious irrational positions. In the same way, the Christian Apologist has unconscious positions too. The known positions are the ones rationally chosen, but the point is to deconstruct them until the unconscious presupposition is laid bare - and then to see if that passes muster. If the Apologist finds his presupposition ends at a need to affirm Divine Sovereignty, or Infallibility, or Biblical Revelation, then that is a non-issue; as consciously being an Apologist, he has already affirmed these implicitly in the Faith. If the unbeliever deconstructs to presuppositions, they also need to be affirmed to support the rationalist construct, but there is no concept like Faith to fall back on. They must be taken on that person's whim, or by force of Will, but these are then not reasoned positions, but just the prefered perception subconsciously of that person. Often such positions are implicitly irrational, such as giving primacy to sense-data while well-aware of the existence of illusion, delusion and hallucination, for instance.
I am sympathetic to it, but I don't consider it the best form of Apologetics. In the end, it remains possible to doubt the outcome, as reconstructing unconscious presuppositions however Biblical or amenable to Christianity, are more often than not a form of circular reasoning. Everything can be doubted, and men must choose to draw their line in the sand somewhere or choose which hill to die on. People often need to be reminded of this. That said, you are clearly erecting strawmen to tilt against that you term Presuppositional Apologetics; but I don't know if what you are saying are valid reflections of the individuals you named earlier. It certainly is a misrepresentation of the school of Apologetics, though.
The axiom of existence identifies the fact that existence exists. Existence is everything that exists.
This is a circular argument. Something exists by being differentiated from something else, which is what the word means - to step forth. Something exists by nature that there is something it is not.
The axiom of consciousness identifies the fact that consciousness is consciousness of something, an object. An object is anything we perceive and or consider.
This says nothing about consciousness, but what you believe consciousness is doing - being perceiving something. This is a relational value based on perception - like defining leg as that which walks.
You said:"If my perception is invalid, by what means am I conscious. Blank out." So it seems you are not differentiating those concepts well. But is not a delusion or an hallucination invalid perceptions? Are they then unconscious? Consciousness is a very protean term, but it certainly is not just perception. What of mystical experience, which is perception or consciousness if you will, of non-duality?
The axiom of identity identifies the fact that everything that exists has a nature or a specific set of attributes and is itself. A is A.
This is dependant on what 'that exists' means, that you earlier used in a circular way. For things act on one another, like light giving colour when perceived by an observer. This requires participation, between the observer, the object and the perceived light. Is this identity not then when it is identified? Differentiated or made consciously aware off? Further, you are asserting a specific set of attributes, and a sort of independance - hence you are right into the problem of Universals. For instance, whose attribute is colour? The one perceiving it, or the object itself? What of things like rainbows then? A tautology doesn't mean much, as why can't A's attributes change into something not that same A as before? Why must A always be that A? Does it not depend on the ascribed attributes placed upon A? How can attributes in common between identities then be squared, if they are independant, or perceived as in common?
Taken together these 3 axioms imply a fourth: That existence exists and is what it is independent of conscious activity.
They certainly do not. They imply the opposite, that existence only exists when consciously perceived.
How on earth do you suddenly jump to independance from conscious activity, after stating that things exist as themselves and are consciously perceived? The former can only follow from the latter, so underneath lies a whole slew of assumptions on the validity of sense-data, perception, empiricism, etc.
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.
None of these things are implicit. You are acting fast and loose with terms. A computer can disagree with a result without being conscious of it, it depends very much what import we apply to 'disagree'. Simple negation does not mean the original exists necessarily nor is valid, except perhaps conceptually. So are you arguing a form of Idealism then? By saying A, everything else is Not-A, are you not then just creating conscious category? How on earth can this imply independant identity at all? If I state A is A, why can't A be a protean thing that changes, like the I of Buddhism or the weather? There is no objective content to a tautology.
What you are saying sounds very muddled to me, so please feel free to explain. Recycling Rand is not really very fruitful.