Russell's Paradox

Metaphysics and Epistemology
1

I’ve been looking through Rand’s Introduction to Objectivist Epistemology, and I have a question about the relationship between Rand’s notion of concepts and Russell’s paradox.

Presumably (and I think explicitly, though I can’t remember where), Rand identifies the concept of a concept which will have as its referent every concept (including itself). Some of the concepts within the concept of a concept are self-referential (like the concept of concepts) but others are not (the concept of tables, of chairs, of doors, etc.). On the basis of this we can distinguish non-self-referring concepts from concepts that are self-referring and we can make a concept of non-self-referring concepts. Use the word “R” to refer to the concept of non-self-referring concepts.

From this we can derive a contradiction. If R doesn’t refer to R then R refers to R. If R does refer to R then R doesn’t refer to R. Either R does or doesn’t refer to R. Contradiction.

One solution I’ve thought of would be to deny that R is a legitimate concept – you could try to use Rand’s rule about concept formation to head off forming the concept. But I see two problems with this:

First, Rand conveys the impression that concepts referring to things like “beautiful blondes with blue eyes” are “forbidden” (and I’m not exactly clear on what it means for a concept to be “forbidden”) because they’re not needed and they’re likely to create confusion. They’re not needed because they don’t help us get any new knowledge. But R *does work* - it’s used in a logic proof which has (if it’s correct) significant ramifications for the whole notion of concepts. It would be kind of strange, then, to “forbid” R when it clearly serves a purpose.

Second, (and this goes back to the notion of “forbidding” concepts) it’s not clear that R’s uselessness (assuming my first problem doesn’t hold) is enough to justify ignoring the problem. It seems kind of odd to say that Rand avoids Russell’s paradox not because she’s logically consistent, but because the paradox would be inconvenient.

2

I may be missing something, but what is the conclusion/implication of your alleged paradox, i.e. what’s your motivation in raising this question?

3

I may be missing something, but what is the conclusion/implication of your alleged paradox, i.e. what’s your motivation in raising this question?

The conclusion is a contradiction - Russell’s paradox is a paradox just because you get a contradiction.

If R doesn’t refer to R then R refers to R. If R does refer to R then R doesn’t refer to R. Either R does or doesn’t refer to R. Contradiction.

If you’re familiar with some basic symbolic logic, we can write this as:

(1) P->~P

(2) ~P->P

(3) Pv~P

THEREFORE: P&~P

My motivation, then, should be kind of obvious.

-Kyle

4

First, Rand conveys the impression that concepts referring to things like “beautiful blondes with blue eyes” are “forbidden” (and I’m not exactly clear on what it means for a concept to be “forbidden”) because they’re not needed and they’re likely to create confusion.

Let’s drag the actual statement out: p. 71 “For example, there is no concept to designate 'Beautiful blondes with blue eyes, 5’5” tall and 24 years old.’ Such entities or groupings are identified descriptively. If such a special concept existed, it would lead to senseless duplication of cognitive effort (and to conceptual chaos): everything of significance discovered about that group would apply to all other young women as well. There would be no cognitive justification for such a concept—unless some essential characteristic were discovered, distinguishing such blondes from all other women and requiring special study, in which case a special concept would become necessary". Thus imagine that society devolves to utter irrationality so that the aforementioned females – now called Bints – are ceremonially sacrificed to appease the volcano god. This specific concept could easily be formed, and you will notice that she doesn’t say anything about the concept being forbidden. The basic claim of the genus and species theory of concepts is that you join together units into a concept by finding the common traits they have in reality, and separate units into two concepts by finding the distinguish traits that separate them in reality. That is the only extent to which there is a “prohibition”.

Now let’s examine this idea of a self-referring concept. There aren’t any. Concepts are strictly hierarchical (unlike Russel’s view), so a concept cannot refer to itself, it refers to particular units that it subsumes.

5

Thank you David. I was sure I had never read such nonsense in any works of Rand’s.

Some of the concepts within the concept of a concept are self-referential (like the concept of concepts) but others are not (the concept of tables, of chairs, of doors, etc.).

Such convoluted sentences or ideas would be out of place in any Objectivist work I believe. Am I wrong?

6

Such convoluted sentences or ideas would be out of place in any Objectivist work I believe. Am I wrong?

I would assume so, particularly because Ayn Rand’s razor (a more formidable version of Ockham’s razor) is quick to cut off such ideas before they could even be fully raised. In a debate most seasoned objectivists would shoot down Russell’s view long before the other side could implement it into an arguement to refute the potency of the human mind. I’ve been trying to modify my own philosophical assertions with my fellow Thomists to take into account the problem of viewing concepts metaphysically rather than epistemologically.

7

David:

Now let’s examine this idea of a self-referring concept. There aren’t any. Concepts are strictly hierarchical (unlike Russel’s view), so a concept cannot refer to itself, it refers to particular units that it subsumes.

(Just as a quick note, Russell’s view actually is hierarchical (though perhaps not in the same way that Rand’s is). He’s usually given credit for the hierarchy solution to his own paradox.)

That said, how can the concept of concepts not be self referring? Is there no concept of concepts? And if there isn’t, what does “concept” refer to?

After all, if (1) the concept of a concept refers (as it presumably does) to all the concepts, and (2) it is itself a concept, then it should self-refer.

Matthew

Such convoluted sentences or ideas would be out of place in any Objectivist work I believe. Am I wrong?

The sentence isn’t really that complicated. If you ignore the parentheticals (which aren’t properly part of the sentence) and use a name – C – for the concept of a concept, you get:

“Some of the concepts within C are self-referential but others are not.”

which isn’t really that bad. My sentence mainly seems convoluted because I’m forced to use the word “concept” 3 times in quick succession. Also, the thread’s about an argument from formal logic and iterated terms like “set” (or, here, “concept”) aren’t really that uncommon. It’s just hard to do formal logic without iteration. If you have a better way to put the sentence feel free to let me know and I’ll replace the original.

8

That said, how can the concept of concepts not be self referring? Is there no concept of concepts? And if there isn’t, what does “concept” refer to?

Speaking from my own perspective, any concept or concepts would be referring to the human mind’s capability to grasp concepts. The difference is that Russell seems to believe that concepts have some sort of metaphysical existence, which has been an error that many Aristotilean philosophers have fallen into. The human mind has a metaphysical existence for it is part of the body of an existing entity (man), however, the abstractions that it creates within it’s own function do not exist in metaphysics unless interpretted by a rational mind, hense the dividing line between Metaphysics and Epistemology.

9

That said, how can the concept of concepts not be self referring? Is there no concept of concepts? And if there isn’t, what does “concept” refer to?

Several concepts, when combined with sensory-percept referents (words), create language. A given word cannot be a useful part of a language if no meaningful concept is derived from said word’s definition. This is why it is counterproductive to use a word in its own definition. Unless we employ concept’s definition, confusion will result when we talk of “a concept of concepts” because language requires us to repeat its sensory-percept when we discuss its broadest application – a referent to all sets.

“Concept” can be defined without reference to itself. Rand asserted that a definition, to be valid, must have a genus and a differentia. However, she never (to my knowledge) gave the genus and differentia of “concept.” Because I think it would be helpful, I will define it before we continue:

Concept : A mental device that refers to a set or group of sets.

The genus in the above definition is a mental device, the differentia is referent to set/sets.

A concept is not a set. In every instance of the mental invocation of any concept, one is simply calling to mind a set. When one invokes “all sets” one is using the device in its broadest application. But one, is of course, not invoking every device that refers to all sets. That would be impossible, because it would require one to be everyone at any time they called “concept” to mind.

The assumption that a concept is a set leads to the current application of the “paradox.” Concepts are not supernal entities; they are specific existents, like everything else.

A set that includes itself begins an infinite regress. I assume Russel was trying to solve the problem of this regress when he came up with his paradox. However, it was arbitrary for him to assume that a set shouldn’t refer to itself. He should have come up with a theory that made the infinite regress manageable instead of paradoxical. But then he didn’t, Rand did.

10

A set that includes itself begins an infinite regress. I assume Russel was trying to solve the problem of this regress when he came up with his paradox

No, he was trying to show that the axioms Frege gave in his original formulation of set theory (now known as ‘naive set theory’) led to a formal contradiction.

. However, it was arbitrary for him to assume that a set shouldn’t refer to itself. He should have come up with a theory that made the infinite regress manageable instead of paradoxical. But then he didn’t, Rand did.

Russell did come up with such a theory to make the infinite regress managable; this is the topic of his Principia Mathematica, and is generally known as the theory of types. However, he didnt just assume that a set shouldnt refer to itself, his solution was far more complex. To simplify greatly, he broke sets up into different ‘levels’. Individual objects are level 0. Then sets of individual objects are level 1. Sets of sets of individual objects are level 2. Sets of sets of sets of individual objects are level 3, and so on. Then, he said that sets could only contain objects from the level below them. So a level 2 set could only contain level 1 sets - it wouldnt be allowed to contain itself since it was level 2, and so on. As a result, no set can contain themselves, and Russell’s Paradox cannot be formulated.

The point of all this was to try and show that mathematics could be derived directly from formal logic. The naive theory of sets can be given a purely logical description, hence if mathematics can be derived from sets alone, then it becomes a matter of pure logic. This enterprise is generally considered to have failed, since deriving mathematics from logic using Russells theory of types requires the postulation of axioms that arent purely logical (eg, he needs an axiom that allows sets to be casted to lower types, and an axiom of infinity that posulates the existence of infinite sets, and so on).

You cant really compare Rand’s theory of concepts to the Frege/Russell theory of sets. One is essentially a psychological theory, whereas the other is one of formal logic - the problems which they are both trying to address occur in completely different domains. Russell did have a tendancy to reify sets and there was a period where his whole epistemology/metaphysics was based on set theory (physical objects are identified with classes of sense impressions etc), but set theory itself is purely formal.

11

That said, how can the concept of concepts not be self referring? Is there no concept of concepts? And if there isn’t, what does “concept” refer to?

After all, if (1) the concept of a concept refers (as it presumably does) to all the concepts, and (2) it is itself a concept, then it should self-refer.

I have to dash, so I just want to get a few points out here for you to consider, and I’ll elaborate later. First “Dogs bark” is not a concept, it is a proposition. What that means is that not every linguistic construction that refers to multiple things is a concept. So there is a concept “concept”, and there is no concept “self-referring concept”. Second, the Objectivist approach to concepts is “from existent to concept”, i.e. a conept must be grounded in reality, therefore you have to show me some self-referring concepts – it isn’t enough to say “I can put together words in a particular way”. So I’m now asking for two examples of self-referring concepts. In order for there to be such a concept “SRC”, it has to integrate two or more distinct units. So what are they? Third, there is a necessarily bottom-up nature to concepts: the units themselves must exist, and cannot depend on the supposedly unifying concept for their creation. Can you show me two independently existing examples of this supposed SRC, which justify creating the concept?

12

Speaking from my own perspective, any concept or concepts would be referring to the human mind’s capability to grasp concepts. The difference is that Russell seems to believe that concepts have some sort of metaphysical existence, which has been an error that many Aristotilean philosophers have fallen into. The human mind has a metaphysical existence for it is part of the body of an existing entity (man), however, the abstractions that it creates within it’s own function do not exist in metaphysics unless interpretted by a rational mind, hense the dividing line between Metaphysics and Epistemology.

You don’t need to make sets part of your ontology to get a paradox – the paradox is a logical paradox, not a metaphysical one.

Featherfall & Hal:

You’re both correct to notice that Russell’s paradox is a paradox in set theory, but I don’t see any reason to think that it’s limited to set theory. For example, if a theory maintains that words are themselves referring devices then, on a naïve formulation of the theory, the word “word” will refer to itself. I could then introduce a word for words which don’t refer to themselves “~word” and I would get Russell’s paradox from it. The same thing should happen if I think that concepts are referring devices (like on Rand’s theory). The simple fact that you’re not using sets , then, doesn’t preclude an occurrence of Russell’s paradox.

Also, I think it would make the rest of this conversation easier if we introduce some notation:

The concept of concepts: C

The concept of self-referring concepts: SRC

The concept of non-self-referring concepts: NSRC

“Concept” can be defined without reference to itself…

Concept: A mental device that refers to a set or group of sets.

A concept is not a set.

This would block the direct self reference but it won’t block Russell’s paradox. The concept of concepts can’t refer to itself (or any other concept for that matter) because it can only refer to sets and concepts aren’t sets. However, the concept of concepts will refer to a set of which it is a member. So you’ve stopped direct self reference, but there’s still a kind of indirect self reference.

Some concepts are members of the set they refer to – C for example – and others are not (the concept of tables). Let NSRC be the concepts of a concept which isn’t a member of the set it refers to. If NSRC is a member of the set NSRC refers to, then NSRC is not a member of the set NSRC refers to. If NSRC isn’t a member of the set NSRC refers to, then NSRC is a member of the set NSRC refers to. NSRC either is or is not a member of the set NSRC refers to. Contradiction.

You cant really compare Rand’s theory of concepts to the Frege/Russell theory of sets. One is essentially a psychological theory, whereas the other is one of formal logic - the problems which they are both trying to address occur in completely different domains. Russell did have a tendancy to reify sets and there was a period where his whole epistemology/metaphysics was based on set theory (physical objects are identified with classes of sense impressions etc), but set theory itself is purely formal.

I agree about most of this, but I’m not sure how it would stop Russell’s paradox from being applied here. Could you clarify?

David:

First “Dogs bark” is not a concept, it is a proposition. What that means is that not every linguistic construction that refers to multiple things is a concept.

Does “dogs bark” really refer to multiple things? “ dogs ” certainly refers to multiple things, but what do propositions refer to?

Second, the Objectivist approach to concepts is “from existent to concept”, i.e. a conept must be grounded in reality, therefore you have to show me some self-referring concepts – it isn’t enough to say “I can put together words in a particular way”. So I’m now asking for two examples of self-referring concepts. In order for there to be such a concept “SRC”, it has to integrate two or more distinct units. So what are they? Third, there is a necessarily bottom-up nature to concepts: the units themselves must exist, and cannot depend on the supposedly unifying concept for their creation. Can you show me two independently existing examples of this supposed SRC, which justify creating the concept?

Just as a quick note, I’m not actually required (in order to get Russell’s paradox to work) to make a concept of self referring concepts – I only need a concept of concepts that don’t self refer.

But I’ll do both concepts for you.

SRC:

The concept “concept” is one self referring concept. The concept “non-physical entity” (so precepts, concepts, etc.) would be another. I have two concepts – I notice both are self referring and distinguish them from everything else on that basis. I then integrate them into SRC.

NSRC:

The concept “table” and the concept “door” both fail to self refer. I notice this about them and distinguish them from concepts like “concept” on the basis of their failure to self refer. I then integrate them into NSRC.

13

I agree about most of this, but I’m not sure how it would stop Russell’s paradox from being applied here.

It wouldnt, I was just clarifying a point about Russell, not making a claim directly relevant to your argument. However to go on topic; what does it mean to say that the word concept refers to ‘concepts’? Concepts arent objects which exist in the world like trees or dogs; some people might say they are ‘mental objects’ (whatever that means), but I think its less confusing to see this sort of talk as being metaphorical. Concepts arent really ‘things’, and the claim that words ‘refer’ to them in the same way that the word ‘tree’ refers to trees is fairly dubious.

A similar example would be something like ‘memories’. Humans certainly have memories, but to say that a particular ‘memory’ is some kind of object would be strange. A memory isnt a physical object (although the act of remembering certainly has a physical basis), and to say that it is a ‘mental object’ just amplifies the confusion. Mental processes arent objects in any widely accepted sense of the word ‘object’.

Im not sure whether this is consistent with Objectivism, but in the appendix to ITOE, AR does state that she finds talk of mental ‘entities’ to be metaphorical at best.

14

It wouldnt, I was just clarifying a point about Russell, not making a claim directly relevant to your argument. However to go on topic; what does it mean to say that the word concept refers to ‘concepts’? Concepts arent objects which exist in the world like trees or dogs; some people might say they are ‘mental objects’ (whatever that means), but I think its less confusing to see this sort of talk as being metaphorical. Concepts arent really ‘things’, and the claim that words ‘refer’ to them in the same way that the word ‘tree’ refers to trees is fairly dubious.

On the traditional understanding of truth, a sentence/proposition like:

(1) Trees are green.

is true just in case the referent of “trees” is in fact green. But if “trees” doesn’t refer then (1) can’t be true or false. You can, if you want, then, say that “concept” doesn’t refer, but its hard to say sentences which use the word are “true” in that case. And if that’s the case, a lot of sentences in IOE have no truth value.

(I’m actually an anti-realist about reference myself, but I’m also not an objectivist.)

15

But R *does work* - it’s used in a logic proof which has (if it’s correct) significant ramifications for the whole notion of concepts.

Serving in a proof does not suffice; if it did, then you could justify the creation of any concept by just coming up with a random theorem based on it. That would be rationalism. Without a difference of existential significance, a concept should never be created.

Also, I have given my solution to Russell’s paradox on a thread in Mr. Speicher’s Forum (http://forums.4aynrandfans.com/; they seem to be offline at the moment). I’ll provide a link to my post there as soon as they’re back online.

16

Serving in a proof does not suffice; if it did, then you could justify the creation of any concept by just coming up with a random theorem based on it. That would be rationalism. Without a difference of existential significance, a concept should never be created.

I think the charge that Rand’s theory of concepts is fundamentally inconsistent counts as more than a “random” theorem. And what do you mean by a difference of “existential” significance?

17

I think the charge that Rand’s theory of concepts is fundamentally inconsistent counts as more than a “random” theorem.

Sure, but

  1. That does not change the fact that “you could justify the creation of any concept by just coming up with a random theorem based on it.”

  2. You are presupposing that you have accepted Miss Rand’s theory of concepts while arguing against it. Concept-stealing is bad enough, but theory-stealing is punishable by death! :slight_smile: :wink:

And what do you mean by a difference of “existential” significance?

One that is not rationalistic. In case you are not familiar with what rationalism is and why Objectivism rejects it completely, I recommend that you read up on it; it appears to be the main error you are making.

18

Sure, but

  1. That does not change the fact that “you could justify the creation of any concept by just coming up with a random theorem based on it.”

  2. You are presupposing that you have accepted Miss Rand’s theory of concepts while arguing against it. Concept-stealing is bad enough, but theory-stealing is punishable by death!

Reply to 1: Remember that Rand’s proscription against needless concepts seems to be largely instrumental – they usually don’t do any work and they’re likely to make your life harder and/or create confusion. I think that’s a legitimate point and it would be a reason to avoid making up concepts for fun. However, that’s not a good reason to hold off on introducing this particular concept given that we’re interested in the results of the conclusions that follow from it. We’re not interested in the random theorems that would follow other made up concepts and so we don’t make up those concepts.

Reply to 2: I’m using Rand’s theory as part of a reductio ad absurdum. To give a reductio-style argument in favor of a claim, P, I assume that P is false and derive a contradiction. I then reject my assumption and conclude that P.

So in this case I want to prove that Rand’s theory of concepts is a false theory. In order to do so I assume that her theory is correct and derive a contradiction. I then appeal to the contradiction to reject my initial assumption and conclude that her theory is false.

I might give the same kind of argument against Platonism. I would assume that Plato’s forms exist and then derive a contradiction. I would basically be saying to the Platonist -

“Assume, for the sake of argument, that you’re correct. Then P follows. But P is false. So your theory must be false.”

So I’m not committing a fallacy here – the reductio is a legitimate argument form and almost nothing could be accomplished in math or logic without it. It’s been around since Zeno and to call it “concept-stealing” just misses the point.

One that is not rationalistic. In case you are not familiar with what rationalism is and why Objectivism rejects it completely, I recommend that you read up on it; it appears to be the main error you are making.

“Rationalism” is a term used in academic philosophy to describe the view that some things can be known prior to experience. It’s usually contrasted against empiricism which denies that anything can be known a priori.

I’m not sure if you objectivists use the term differently, but I suspect that you do since what I’m doing isn’t necessarily rationalistic (on my usage).

19

On the traditional understanding of truth, a sentence/proposition like:

(1) Trees are green.

…

(I’m actually an anti-realist about reference myself, but I’m also not an objectivist.)

Trees are brown, leaves are green, but that’s beside your point I assume.

What is an “anti-realist”?

What purpose are you pursuing by joining an Objectivist forum?

20

Speaking from my own perspective, any concept or concepts would be referring to the human mind’s capability to grasp concepts. The difference is that Russell seems to believe that concepts have some sort of metaphysical existence, which has been an error that many Aristotilean philosophers have fallen into. The human mind has a metaphysical existence for it is part of the body of an existing entity (man), however, the abstractions that it creates within it’s own function do not exist in metaphysics unless interpretted by a rational mind, hense the dividing line between Metaphysics and Epistemology.

I see what you’re trying to say here, but it’s a little confused. Concepts do exist metaphysically, as a process of integration, i.e. an action, performed by a certain type of consciosness. Actions exist. What concepts don’t have is metaphysicaly primacy–no actions do. Metaphysical primacy belongs only to entities (which perform actions).