Russell’s Paradox proves Rand wrong and falsifies her philosophy with objective evidence.

http://plato.stanford.edu is my immediate source for the explanation of Russell’s Paradox, but any error in paraphrasing is my own.

(This post, intended to show that Rand’s epistemology is not logical, seems to be perceived as an attempt to show logic to be false since it contradicts Rand’s epistemology! Judging by the comments, anyway.)

Part B: Russell’s Paradox vs Rand’s Objectivist Epistemology.

In Rand’s Introduction to Objectivist Epistemology, mental concepts are “classifications of observed existents according to their relationships to other observed existents”^{i}. Rand’s philosophy taught that to have mental concepts require us to observe fundamental^{ii} essential distinguishing characteristics^{iii} in real life which share a “Conceptual Common Denominator”^{iv}. In Rand’s philosophy, logic is “the art of non-contradictory identification”^{v} of those observations; the result of that process is a mental concept.

Russell’s Paradox proved you shouldn’t just describe a “set” and apply logic to it, because you can get contradiction. Until Russell, “it was initially assumed that any well-defined condition (or precisely specified property) could be used to determine a set.”^{vi}. Rand’s ideal of a mental concept is that old-fashioned kind of set theory.

Russell’s Paradox logically proves Rand Rand’s logical epistemology is false in three simple steps:

1: Some concepts contain themselves. For example, the concept “things which are not a table” (called a “contrary” in Rand’s philosophy^{vii}), is itself not a table so it describes itself. You could say, “this concept is not a table” and write it down on a list of things which are not tables.

Another example of a concept that contained itself would be that on a list of everything in the universe, the first thing would be “this list”.

Both of those concepts would be contained in a big list labeled “concepts which contain themselves”. Let’s call this big list concept “A”, and we can write “concept A” in the list first thing; because it contains itself, too.

2: Other concepts do not contain themselves. A list of teacups doesn’t include the list as part of the set of teacups. The concept of a foot is not a foot. Mankind is not a man.

Those three examples are contained in a big list titled “concepts which do not contain themselves”. This list must be concept “not-A”, but…

3: Does that last concept contain itself or not? Can we write “this concept” on the list? Is it “A” or “not-A”? If the concept does not contain itself, we should write it down – which means it does contain itself (there it is on the list), but then it cannot fit it’s own definition as a concept which does not contain itself, so it shouldn’t be on the list and there is contradiction.

By proving the objective definition of sets can lead to logical contradiction, Russell’s Paradox proves Rand’s objective definitions of concepts can result in contradiction. Rand’s Objectivist Epistemology is proved false.

Rand vs Bertrand Russell Part C:

Rand vs Bertrand Russell Part C (Revised):

Russell’s Paradox demonstrates Objectivism is false with real world examples. We can look at actual lists like above and see that A is not-A if it is A, and can be A only if it is not-A. Rand’s assertion that all real things can be logically categorized as A or not-A is demonstrated to be false by objective evidence. Try it at home! Get some pieces of paper and make the actual lists and put labels on them. The last list can’t get a label.

The solution to the riddle is that there are rules we have to make up for logic to work right (axioms), and one of them is that sets can’t contain themselves. Even though we can make a list with the words “this list” on it, we can’t use it for logic. The concept we can see right in front of us can’t be used in a logical system, which falsifies Rand’s Objectivist philosophy of logic.

Stanford Encyclopedia of Philosophy http://plato.stanford.edu is my immediate source for the explanation of Russell’s Paradox, but any error in paraphrasing is my own.

iPg 62 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

iiPg 59 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

iiiPg 55 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

ivPg 18 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

vPg 46 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

viStanford Encyclopedia of Philosophy

vii Pg 77 Ayn Rand, Introduction to Objectivist Epistemology, Mentor Book, New American Library, 1979

“things which are not a table” is not a concept. It is a BUNCH of concepts. “things” is a concept. “not” is the concept of the negative. Each word we use is a separate concept (Aside from things like articles, and so on)

So your argument is wrong from the first premise.

Thank you for your comment.

It is not my premise. Ayn Rand says each word is a concept. The word for the concept of “things which are not a table”, is a “contrary”, according to Rand. If the group of concepts which are not tables is not a concept, Rand wouldn’t have a word for it.

“A bunch of concepts” is itself a concept, otherwise we could not think of it or use it in a sentence. In the same way a flock of geese is a different concept than a goose.

Again, this is according to Rand.

Try disproving the law of identity in CoQ or any other theorem proving system, then we talk about it. Otherwise, that’s all one illogical gibberish.

After re-reading your comment and the post, I see the post was poorly written. Instead of “Rand’s assertion that A = A is demonstrated to be false”, the post now reads, “Rand’s assertion that all real things can be logically categorized as A or not-A is demonstrated to be false”. This is an important difference. I would not have noticed, were it not for your comment. Thank you very much.

My original dismissive reply, which counts as a second error, is below.

Original reply:

Since no attempt was made to disprove the law of identity and it is not the subject of the post, I do not understand your comment. But thank you for reading.

This proves that Rand’s logic is incorrect, so long as we accept that logic is not objectively valid.

That sounds like begging the question to me, which of course isn’t a problem if we don’t care about logic (i.e. if something can be both true and false at the same time then we’ve effectively rendered logic redundant).

Although if we abandon logic then I do wonder what compelling reason your post proposes to convince us that you are in fact correct?

Alternatively, one could suppose that when we’re faced with any apparent paradox, there is going to be some human cognitive error involved.

Thank you for your thoughtful comment. I’d like to reply to it in general, referring to it’s particulars out of order. (This reply has been revised to correct several errors)

a) The assertion that acceptance of Russell’s work requires abandoning logic is contradicted by the objective fact that Mathematicians did not abandon logic as a result of Russell’s Paradox. Mathematicians acceptance of his work does not mean mathematicians don’t care about logic.

Russell’s Paradox is, itself, a rigorous application of logic and is one of the foundations of modern mathematics because, through it’s logic, it helped define the boundaries of logical systems such as math.

b) “Objectively valid” seems to be a phrase from Rand’s vocabulary. If that phrase means elements of a logical system must be found in the physical world to be valid; then it is true that modern math and logic disagree with Rand.

Not just in set theory, but more obviously: Negative numbers, amounts less than nothing, are required for the current logical system of algebra; but cannot be found in the real world. There are no negative sheep which can be added to positive sheep to result in nothing at all. If we subtract sheep from one place, we have to put them somewhere else. We might end up with mutton, but we don’t end up with nothingness. Yet most people would agree algebra is a valid logical system.

c) It is not necessary to first accept that “logic is not objectively valid” to accept Russell’s logical breakthrough.

d) The issue in the Paradox was not that the concept was both A and Not-A. The issue was that the concept could not be identified as one or the other without contradiction.

e) It is not up to us to “prove” Bertrand Russell is correct. He already did that himself and his work was reviewed by actual mathematicians. If the concept of a set which includes itself exists (which it does, because here we are talking about it) and yet cannot be objectively identified without contradicting itself (which it cannot, as Russell proved); then disagreement with Rand’s logical epistemology is self-evident and should require no further effort than comparing the two.

Of course, for the above to be true, the post must have correctly described both Rand’s position and Russell’s Paradox. That does not seem to be a problem, judging by the few comments so far, but it certainly is possible.

(Addendum: turns out Rand’s position was described incorrectly in the post. The error has been corrected.)

f) If the Paradox puts Rand’s opinion of logic in question, the question isn’t solved by disregarding the objective fact of Russell’s example. The solution to the question is to either show the example can be identified or accept that Rand’s opinion was wrong.

g) Russell’s Paradox indeed identified a cognitive error in the understanding of logic and mathematics. The implications of the Paradox disagree with Ayn Rand’s opinion of logic, which is why she calls Russell’s Paradox “gibberish”. Rand, apparently, refused to accept the proof; and she continued to espouse her error. It was up to Rand to show why she is correct and the rest of the mathematical community is wrong. But she never did, missing out on a Nobel Prize.

Thank you again for your comment and for reading the post. Hopefully, this reply is coherent; but it is on the fly – so, apologies if the rush has affected it’s quality.

You’re misunderstanding the implications of Russell’s paradox. (Btw, I’m in no way defending Rand’s philosophy here).

Russell’s paradox does not imply that logic allows for contradictions. An axiomatic system that allows for contradictions is considered inconsistent, and logicians do their best to avoid inconsistency. For example, Russell developed his theory of types to work around his paradox (and you allude to this at the end of your article), and Frege became very disillusioned with his own work when he discovered his Fifth Law resulted in a contradiction.

So, it is perfectly acceptable to describe logic as “the art of non-contradictory identification”.

You also seem to be rejecting the Law of Excluded Middle by saying that it is possible for something to be both A and not-A. If something is allowed to be equal to itself (P = ~P) then by the Principle of Explosion any other statement will be “true”. Russell gave a great example of this (paraphrased here): “If 4 equals 5, then two equals one. Then, the pope and I are one.”

Thank you for your thoughtful comment. I will think for a short while, then reply.

At no point does the post say logic allows for contradiction. The premise of the post is the opposite. The post points out that since there is contradiction, Rand’s theory is not logical.

The post does not disregard the law of excluded middle. The post says there is A, not-A or contradiction. That is the definition of excluded middle. Maybe I should add that statement to the post for clarity.

The post points out that if concepts which contain themselves = (A) and concepts which do not contain themselves = (not-A); then the “concept of (not-A)” can’t be identified as either one without causing contradiction. Nowhere in the post does it say to identify it as both!

The paradox demonstrates that Rand’s “non-contradictory identification” of this concrete existent as (A) or (not-A) can’t happen. The idea of logic as simply a description of reality is shown to be untrue.

Thank you for your thoughtful comment. This reply may be flawed, I will continue to consider the issue.

There are concepts which are real and concepts which are not real. There are concepts that are valid, and concepts that are invalid because they are not logical, and the concept of a list of all concepts that do not contain themselves is itself an invalid, illogical concept. That doesn’t mean it doesn’t exist, because it is after all *just* a concept, but it is a self-contradicting concept and therefore cannot manifest concretely, just like the concepts of “immovable object” and “unstoppable force” are mutually exclusive and cannot actually manfist in concrete reality within the same Universe.

Rand spent much of her epistemology explaining the difference between valid and invalid concepts, but you seem to have skipped right over that. Your answer to the puzzle is also wrong, as some sets CAN contain themselves, and you demonstrated this to be factually true in your own reasoning when you created the concept of the set of things that contain themselves. The axiom provided is proved not an axiom at all.e

Thank you for your thoughtful comment.

I will reply after thinking a bit.

I don’t understand what your comment is referring to. The post never said those sets or concepts couldn’t exist.

The post says the “concept of concepts which do not contain themselves” DOES exist and readers can make their own physical copy! According to Rand, a true concept refers to a concrete existent, which is why the physical example was in the post (though not necessary for the point).

The solution in the post isn’t mine. It’s part of modern set theory.

Thank you again for your thoughtful comment.