I hope you have given some thought to the logic problems presented in the last essay. Many of these can be analyzed at different levels of specificity, as we have seen. There is no one right way to expose the flaws.
Here are my solutions to the unadressed problems.
6.From William F. O'Neill, With Charity Toward None: an Analysis of Ayn Rand's Philosophy (I leave it to your discretion to decide how charitable to be toward this critic):
How are we to interpret "a bad boy is one who is heedless of others"? The usual interpretation is that if John is not heedless of others he is excluded from the category of bad boys and not that everyone who is heedless of others is automatically a bad boy.
Mr. O'Neill's first argument could equally well be demonstrated formally false by a Venn diagram. We need the hidden assumption "John is a boy" [which may or may not be true] before John falls into the 'bad boy' side of the 'bad boy / not bad boy' partition.
In both his first and second arguments, Mr. O'Neill basis his analysis on the supposed 'meaning', consistently or inconsistently used, of the word 'bad'. Yet, 'bad boy' is never partitioned into the categories 'bad' and 'boy', so the question of the meaning of 'bad' never arises as the concept never appears in isolation.
7.An argument made by someone with whom I was debating: ``Your belief in the power of reason can be sustained only by means of faith. After all, to argue that it is reasonable is circular.'' Faith and reason are two mutually exclusive epistemological methods. 'Faith' refers to beliefs held without sensory evidence and logical deduction (or in contradiction to such evidence). 'Reason' refers to beliefs held on the basis of sensory and logical evidence. One can no more have faith in ideas derived by reason than one can have a reason to accept ideas on faith.
How ironic that the argument is an appeal to reason (to the circular argument fallacy) that explicitly denounces the validity of reason itself. How would the person posing this argument answer the claim that there is nothing wrong with circular reasoning? Would he say I was being unreasonable?
George Smith called the basis of this argument the ``toolbox theory.'' Reason and faith are both cognitive tools and man, supposedly, picks the most appropriate one for a particular job. But what would be the method for choosing the appropriate tool? On what basis would we determine when to use reason and when to use faith? We would need epistemology to tell us that, and epistemology is based on metaphysics which means observed reality which means ... reason.
Reason is the process for establishing which knowledge claims are, well, reasonable, that is consistent with perceived reality -- no other standard is possible. Reason is not one tool in the cognitive-epistemological toolbox; it is the toolbox.
Every argument in support of faith must face a key issue: As soon as someone has reasons to believe something, it ceases to be an article of faith. Why believe on the basis of faith that which you can prove? Reason, since it presents a criteria for the acceptance of truth claims, demands that we reject any claims not presented with the necessary supporting evidence.
This leaves faith in a bind. Every belief claimed to be held based on faith must either meet or not meet the requirements of reason. Any beliefs that do need not be held on faith; reason is sufficient to establish them. Any beliefs that do not meet the rational-epistemological criteria for acceptance, but which are held on faith, are done in explicit defiance of reason. There is no reconciliation between reason and faith, they are competitors.
The moral analog of the original argument is, ``but aren't all men selfish? After all, they wouldn't have done something if they didn't want to!''
If this were true, selfishness would cease being a virtue or a vice, any more than allowing the universe to continue to exist is a virtue or a vice.
The terms `faith' and `reason' are most useful in the areas in which they compete. After all, the original argument was an attempt to grant faith an equal footing (or superior status) to reason.
Similarly the terms `selfish' and `self-sacrificing' are most useful in the ethical arenas in which they compete. A person's motivation in performing an action can be to further his own goals or someone else's. The actual benefactor of an act can be the actor or someone else. Look around you, people do act self-destructively. No matter how much they want to self-destruct they still are self-destructing.
These are both attempts at conversion by definition. Redefine `faith' so that it means convinced reason is valid, redefine `selfish' so that it means motivated, and redefine `god' so that it means ultimate reality and the whole world instantly becomes faithful, selfish, and theistic, and no one need even change a single belief in the process. What could be more irrelevant?
9.From Dr. Albert Ellis, Is Objectivism a Religion?: ``The very concept 'exist' is man-made and might well be meaningless to some being from another planet.... Of course the concept `exist' is man-made, but existence itself is a fact of reality. If this ``being from another planet'' were sentient, it is difficult to believe that it could have no concept of `exist'.
To be sentient it must have thoughts and ideas. Every thought and idea carries implicitly with it reference to reality. As David Kelley pointed out, even a false claim (I am Napoleon), is attempting to claim that in actual reality -- as a statement of fact -- I am Napoleon. I cannot imagine any sentient being not having a concept of existence, for such a lack would prohibit it from developing any concepts, making any claims, or having any beliefs -- to what could they refer?
Further, consciousness is a primary and the very existence of thoughts and ideas itself subsumes existence. In a later essay, I expect to discuss in detail the relationship between consciousness and reality.
... All our concepts and [our?] realities depend on our perceptual-cognitive apparatus and therefore and not independently [provable] or absolutely provable.I added those two words in brackets to expose Dr. Ellis' gambit here. By implication, he attaches concepts to other concepts to which they can never apply. With the bracketed implications specifically denied, nothing important is said. With them affirmed, we are talking nonsense.
Our concepts depend on our perceptual-cognitive apparatus, but reality itself certainly does not. To preface (by implication) the word ``reality'' with the adjective ``our'' simply shows that Dr. Ellis does not understand the meaning and scope of the concept `reality'.
By ``independently provable'' he clearly means proof independent of anyone doing the proving. Here Dr. Ellis demonstrates he doesn't know the meaning of the concept 'proof'. His epistemology in rejecting 'our perceptual-cognitive apparatus' seems to be that of knowledge by osmosis.
What could ``our perceptual-cognitive apparatus'' mean if not our means of conception and out means of cognition. But, Ellis clearly regards this ``apparatus'' as an obstacle to obtaining knowledge about reality. In other words, because we have brains we cannot think; because we have eyes we cannot see. How could the method we have of obtaining something simultaneously be an obstacle that prevents us from obtaining it
Ellis considers ``independently or absolutely provable'' some higher standard of knowledge inaccessible to man. Thus, because man cannot obtain knowledge without thinking and without looking, he cannot obtain knowledge at all. The most powerful rebuttal -- just ask, ``Why?''
Ellis believes that knowledge obtained without a means of obtaining it (were such a thing possible) would be better than knowledge obtained by some specific means. I don't know what it would mean to obtain an ``independent'' proof when he means independent of me, but I don't think I would find it better than reasoned knowledge.
The ideas that `existence exists', `existence is identity', and `A is A' are all man-made notions that could actually disappear from the universe if no man or man-like creatures were around to think them and `prove' them.''Well, the ideas would disappear if no one were around to think them and prove them, but if people can prove them now then they are true, period. This is what ``proof'' means. The ideas are man-made, the facts to which they refer are not.
Ellis simultaneously regards man as omnipotent -- the creator of existence and identity -- and epistemologically impotent -- unable to obtain knowledge without going through a process of obtaining it and thus unable to obtain `real' knowledge. Man is neither. He is a resident of reality and can obtain knowledge through the use of his perceptual-cognitive apparatus, that is, his brain and his body.
10.From James D. McCawley, ``The Dark Side of Reason,'' Critical Review, Summer 1990: ``In the version of logic that currently rules as `standard' among Western philosophers, one can prove many results that are widely regarded as counterintuitive, such as the theorem that from contradictory premises one can draw any conclusion at all, even one to which the original premises are irrelevant ... The author would like to see some type of internal connectivity between `A' and `B' as part of the meaning of `A implies B'; however, he overlooks how completely reasoning from contradictory premises abrogates all reasoning. The very essence of reasoning is the process of drawing non-contradictory inferences from consistent premises. If one's underlying premises are self-contradictory, all bets are off.
Further, formal logic is not equivalent to reason. In formal logic, one can derive any conclusion from a contradiction -- in reality-based reason, there are no contradictions. What better way to show that one cannot derive meaningful results from a contradiction than to show that one can formally derive any results from a contradiction?
Formal logic proceeds without meaning and without intentionality. Throw a wrench into the works (such as a contradiction) and the whole mechanism goes wild. Human minds reasoning are at no such risk. Formal logic is an abstraction of part of reason -- the other part is meaning.
Reason does not establish that a conclusion is true, but at most that it involves no errors beyond those that one is already committed to.''So if one is committed to no errors, reason does establish that a conclusion is true, right? In other words, reason allows people to derive new true claims from preceding true claims. Nothing else is asserted. No one would claim that a person can reason invalidly, or reason `validly' from false premises, and still reach exclusively correct results.
By the way, there is no dark side of reason, not even in one's emotional life. Consider all of the times another person did something that upset you. If your anger was rational, the action that upset you was not. Only irrationality has a dark side, and like a Moebius strip, that is its only side. Only reason provides light.
11.[Zeno's Paradox]
I have received a communication taking issue with my dismissal of any finite-sum-of-an-infinite-series resolution of Zeno's Paradox. The responder feels that the solution does lie in that direction. Perhaps I should address the issue further.
Such a resolution would say: It's true Achilles must complete an infinite number of steps, but since these steps sum to a finite distance, they can be traversed in a finite time. And since we know that Achilles does overtake the Tortoise, that should end the matter.
One can only appreciate the inadequacy of this response by keeping in mind that we deal with a problem in reality, not a solution to a mathematical exercise. In any mathematical exercise, we need not worry about actually completing an infinite number of operations. For example, we accept that mathematical induction establishes the truth of a statement ``for all integral values of `X''' understanding that what we really mean is for any finite integral value of `X'. This is a legitimate meaningful use of infinity as a defined process.
The rigorization of calculus relieves us of the necessity for completing an infinite number of operations. Those familiar with the history of mathematics are aware of what conceptual absurdities were initially overlooked in deference to the power of calculus before the necessity for a rigorous formulation was appreciated.
The mathematical model explicitly avoids the necessity for completing an infinite number of steps or dealing with existent infinitesimals by rigorously defining sum-of-an-infinite-series, and indeed any limit process, to exclude such operations and entities. The delta-epsilon definition of limit, one of the methods of rigorization, explicitly avoids such operations. We do not `reach' a limit by executing an infinite number of steps. We cannot even talk of reaching a limit, we approach a limit. We define `limit' by an analysis of finite relationships at arbitrarily chosen points.
When we turn to the race as a problem in reality, the method of limits is inadequate if we insist on modelling space as a mathematically continuous existent. We can then no longer avoid the necessity for summing actual existent infinities. The Tortoise is overtaken, and if we insist on summing the series of diminishing distances, we do have an actual infinity to sum. The rigorous theoretic underpinnings of the mathematical model have been undercut by reality itself. The model is no longer applicable. This is the basis for my alternative quantum model resolution.
To appreciate the force of these arguments, we must have a healthy respect for the dangers inherent in any use of `infinity'. Many of the problems in the foundations of mathematics arose from an uncritical acceptance of completed infinities, not just as existents but even as concepts.
As existents, in reality, there is no place for actual infinities and completed infinite processes. These terms are non-concepts and it is one of Objectivism's greatest strengths that it has never wavered from this realization. Conceptually, infinity can only mean `some -- and more', it is a process. No limitations -- a completed infinity -- means no identity, no existent, no reality, no definition, and no meaning.
I would recommend, as a further `exercise', a critical reading of Rudy Rucker's Infinity and the Mind. He explores at length various conceptual, mathematical, and 'actual' infinities. He states that he does not know, but is sympathetic to the possibility, of existent infinities. See if you can find any demonstrations of such entities that survive close scrutiny.