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Russell's Paradox

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Kyle

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Ayn Rand defines "concept" as follows:

"A concept is a mental integration of two or more units which are isolated according to a specific characteristic and united by a specific definition" - (Introduction to Objectivist Epistemology, p. 10)

She goes on to explain that the process of concept formation involves abstraction of certain attributes of the units being conceptualized. For example, when forming the concept "chair", we mentally isolate the attributes of the units (different chairs), such as having a surface and a type of support. After isolating the essential attributes of chairs, we are able to integrate all such units into the concept "chair".

Notice that a concept can be formed with two or more units. We can form a concept of chair based on perception of only a few units, without seeing the rest of the chairs in the universe. What the concept "chair" describes are the attributes essential to "chairness", not the chairs themselves. We can later see an object that fits the definition of chair and identify it using our previously-formed concept "chair", but this new chair was not necessary to form the concept "chair" in the first place.

Hence, the definition that concepts are "mental devices referring to sets" is misleading and is the root of Kyle's error. We form concepts from two or more units. We form the concept "concept" (call this C) from two or more existing concepts (the units). Before we form, and during the process of forming C, C does yet not exist. Therefore, we cannot use C as one of the units to form C (an obvious absurdity). When forming C, we use existing concepts, say X, Y, and Z, as the units from which to abstract.

To summarize, the concept "concept" only refers to every existing concept in the way that the concept "chair" refers to every existing chair (a reference to shared attributes, not to the set of chairs as such). And just as the concept "chair" can be formed without reference to every chair in existence, the concept "concept" can be formed without reference to every concept in existence, and therefore without reference to itself. The phrase "self-referring concept" is meaningless under a proper understanding of the process of concept formation.

Edited by Spano
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To summarize, the concept "concept" only refers to every existing concept in the way that the concept "chair" refers to every existing chair (a reference to shared attributes, not to the set of chairs as such). And just as the concept "chair" can be formed without reference to every chair in existence, the concept "concept" can be formed without reference to every concept in existence, and therefore without reference to itself. The phrase "self-referring concept" is meaningless under a proper understanding of the process of concept formation.

When a man first integrates the concept "chair" from two chairs he happens to observe the concept he has integrated still refers to every chair that exists, has existed, and will exist.

Lets say he has not observed a particular chair (for the record it is blue, slightly tacky, and exeptionally confortable). His concept of "chair" still refers to that unobserved chair.

So just because a man integrates the concept "concept" without personal reference to every concept in existence doesn't mean that what he has integrated fails to refer to the concept "concept."

I would also like to point out that it is extremely difficult to form any sort of florid prose when talking about self-reference :thumbsup:

Edit: I left an important bit of the concept "concept" concept concept concept sentance out, and have added it in so the sentance now makes sense...whether or not it rolls off of the tongue is a completely different issue.

Edited by LaVache
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Spano, accepting your explication of Objectivism and your terminology in this regard, whatever did or did not exist previously, by this time, the concept "concept" does exist as a mental entity, and is one of the entities that fits the definition of 'concept'. So there is at least one concept that has the attribute of fitting the definition of the word used to define that concept. So those concepts that are isolated according to the attribute of fitting the definition of the word used to define those concepts are united by the definition of 'self-referring concept': An entity is a self-referring concept if and only if the entity is a concept and fits the definition of the word used to define that concept. And there are concepts that are not self-referring concepts. And 'non-self-referring concept' is defined: An entity is a non-self-referring concept if and only if the entity is a concept and is not a self-referring concept. So there is the concept "non-self-referring concept". Now, either the concept "non-self-referring concept" is a self-referring concept or it is not a self-referring concept. If it is a self-referring concept, then, since it is the concept "non-self-referring concept", it fits the definition of 'non-self-referring concept' so it is a non-self-referring concept, so it is not a self-referring-concept. If it is not a self-referring concept, then it fits the definition of 'non-self-referring concept', so, since it is the concept "non-self-referring concept", it is a self-referring concept.

Edited by LauricAcid
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Marnee:

THE SUBJECT

By definition:

The concept “concept” = = concept

Just like:

The concept “paradox” = = paradox

ENTER THE PREDICATE

It follows then:

The concept “concept” is one self referring concept. = = Concept is one self-referring concept.

“Concept is one self-referring concept” is a meaningless sentence.

You’re equivocating here with the word “concept” – when you use it the first time you’re using it as (something like a) proper name but when you use it the second time you use it as a natural kind term. For example:

The concept “concept” = = C

C is one self referring concept.

Note that this sentence does make sense, but it says the exact same thing as the sentence you claim is unintelligible. Notice that I can also do the same thing with the concept of paradoxes:

“Paradox is about paradoxes” as opposed to “the concept “paradox” is about paradoxes.”

LauricAcid & LaVache:

Am I to understand that you agree with me?

Edited by Kyle
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One of my professors (who happens to be an objectivist) used Rand's theory of concepts in an argument against Scalia's approach to constitutional law. I have an interest in drawing out a contradiction in Rand's notion of concepts because I want to help Scalia – if her notion of concepts is internally inconsistent then it can't be used as a foundation for arguments against Scalia.
Ah, well now we know where you are and who she is. I guess since I don't know this argument of hers against Scalia, I don't know how to help you (not that I actually want to -- I want to show you that she is right and Scalia is wrong). I would caution that even if you uncover some kind of problem in the theory of concepts, that won't automatically invalidate Smith's argument (whatever it is) against Scalia -- you'd have to show that the point that you're making is actually relevant to her argument.

You asked, btw, about the multiple referents of "Dogs bark" -- that refers to multiple tokens of "dog" and multiple events of barking. The referent of such a proposition is an event, and that one states a universally quantified dog / barking-event relation.

Now that I understand your argument a bit better, I think I see the basic issue: I didn't initially understand what you meant in talking about a concept referring to itself. Your blue-eyed blonde example was simply the wrong kind of example because it's a conjunction of properties. An example of what is forbidden is a "concept" whose members are sets of random integers (or other random collections), because they have no non-self referential definition. But in addition, concepts are claimed to have a special status, that they are aspects of human cognition. That means that the ability to give a formal characterization of a collection does not assure that it could ever be a concept. In differentiating a broader class such as "concept", you might indeed be able to define a class of "self-referring concepts" (thanks for the examples -- the brain just refused to work there), but the complement is not automatically a concept. The claim then would be that NSRC is not a possible concept, because the things that you mean by NSRC have no identifiable properties in common. Similarly, in the real of numbers "non-13" does not define a concept, for colors "non-chartreuse" is not a concept, and so on.

Obviously the question to ask is "then how do I know what the possible concepts are?", and the answer is "undertake a huge study of human cognition". The basic claim of the Objectivist theory of concepts is that units subsumed under a single concept have to have defining properties that identify those units as being subsumed under concept A and not B, and that those properties are grounded in perception. The specific claims that Objectivists would make about those defining properties are not philosophical per se, they are scientific. But also note that a concept must be represented as a linguistic symbol (see the discussion of unit-economy in ITOE ch. 7). A test of whether a particular "concept" can be formed, according to psychological criteria, would be whether a single symbol (word, in English) can be successfully used to capture the referents. This is an "actual use" criterion -- not whether you can say "Well, let's call that R", but rather can it be benerally used by speakers of English. That's not a question that can be decided a priori. People can certainly understand the idea "all concepts, except those whose definition would subsume that particular concept", but that doesn't make the idea a concept, anymore that "Dogs bark" is a concept.

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"Anti-realism" is the standard academic term. I don't recall the reasons Rand gives for rejecting the view - I suppose I could look them up - but even if my view is false, Rand's view won't become consistent.

Perhaps I misunderstood what you were trying to say in your initial description of your anti-realist views, but I have never heard "anti-realism" used in the way you described. Everywhere I've seen it used it refers to the denial of an objective reality as such (a metaphysical position), rather than the epistemological view that our concepts don't have an objective basis in reality, which is what I thought you were saying.

EDIT: Perhaps my initial intention in asking you why you chose "anti-realist" as opposed to "nominalist" will be more clear if I rephrase it: Do you deny the existence of reality, or do you accept that there is an objective reality, but deny that concepts have an objective basis in reality?

You're right that your own views don't have anything to do with the consistency of Ayn Rand's, but it does help facilitate discussion to know exactly where you are coming from.

Edited by dondigitalia
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In differentiating a broader class such as "concept", you might indeed be able to define a class of "self-referring concepts" (thanks for the examples -- the brain just refused to work there), but the complement is not automatically a concept. The claim then would be that NSRC is not a possible concept, because the things that you mean by NSRC have no identifiable properties in common. Similarly, in the real of numbers "non-13" does not define a concept, for colors "non-chartreuse" is not a concept, and so on.

This is (kind of) where I was headed, although in a somewhat more roundabout way, since I'm not sure this presents a problem to Kyle's position, given his anti-realist and/or nominalist views (whichever it is he meant). (Of course, it is a huge mistake to accept that this paradox is a problem in Objectivism, and use integrations that could not be classified as concepts under Rand's theory.)

In essence, what NSRC amounts to is an attempt to arbitrarily integrate existents into a concept based on something they're not. Under that principle I could have a concept that unites me, my tv, the Blockbuster down the street, the Sun, Jupiter's Great Red Spot, and Alpha Centauri into a concept of things that aren't on Pluto--or any number of other completely arbitrary groupings.

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Everywhere I've seen it used it refers to the denial of an objective reality as such (a metaphysical position), rather than the epistemological view that our concepts don't have an objective basis in reality, which is what I thought you were saying.
In the context where it came up, he was being an anti-realist w.r.t. reference. So I suppose that would mean that words don't really refer to things, and sets don't really refer to other sets. I don't know what it would actually mean for reference to be unreal; and there are various problem with the referential theory of meaning. I just don't know how the concept of "reference" or "referring" can be unreal. But I'm a thorough-going unimaginative realist, so I'll wait to see what Kyle says anti-realism w.r.t. reference refers to.
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David Odden:

Ah, well now we know where you are and who she is. I guess since I don't know this argument of hers against Scalia, I don't know how to help you (not that I actually want to -- I want to show you that she is right and Scalia is wrong). I would caution that even if you uncover some kind of problem in the theory of concepts, that won't automatically invalidate Smith's argument (whatever it is) against Scalia -- you'd have to show that the point that you're making is actually relevant to her argument.
I’m really very impressed. How did you identify Professor Smith from what I said?

But in addition, concepts are claimed to have a special status, that they are aspects of human cognition. That means that the ability to give a formal characterization of a collection does not assure that it could ever be a concept. In differentiating a broader class such as "concept", you might indeed be able to define a class of "self-referring concepts" (thanks for the examples -- the brain just refused to work there), but the complement is not automatically a concept. The claim then would be that NSRC is not a possible concept, because the things that you mean by NSRC have no identifiable properties in common. Similarly, in the real of numbers "non-13" does not define a concept, for colors "non-chartreuse" is not a concept, and so on.

You have to be careful with the word “compliment” here – the compliment of the even numbers could be the odd numbers (if we’re restricting our domain to the integers) or it could be thing like Pluto, chairs, myself and my dog (if we’re not restricting the domain at all).

I think you’re right that the compliment of a set won’t work as a compliment if we’re not restricting the domain. Nonetheless, there seem to be lots of examples where we use the compliment of a concept in a restricted domain to form another concept. So, for example, the concept of the even numbers is defined by all of the numbers that yield an integer when divided by 2. We get the concept of the odds from the numbers we can’t do this with. The concept of the primes (this is probably my best example) comes is the compliment of the numbers that be can be divided cleanly by something other than 1 or itself. Outside the numbers, the concept “non-violent criminal” is the compliment of “violent criminal” when we restrict the domain to criminals.

And so, yes, if I took the compliment of SRC without restricting the domain I’d end up with a collection including Pluto, trees, and doors. But as long as I restrict the domain to concepts NSRC doesn’t seem any different than the concept of primes.

Obviously the question to ask is "then how do I know what the possible concepts are?", and the answer is "undertake a huge study of human cognition". The basic claim of the Objectivist theory of concepts is that units subsumed under a single concept have to have defining properties that identify those units as being subsumed under concept A and not B, and that those properties are grounded in perception. The specific claims that Objectivists would make about those defining properties are not philosophical per se, they are scientific. But also note that a concept must be represented as a linguistic symbol (see the discussion of unit-economy in ITOE ch. 7). A test of whether a particular "concept" can be formed, according to psychological criteria, would be whether a single symbol (word, in English) can be successfully used to capture the referents. This is an "actual use" criterion -- not whether you can say "Well, let's call that R", but rather can it be benerally used by speakers of English. That's not a question that can be decided a priori. People can certainly understand the idea "all concepts, except those whose definition would subsume that particular concept", but that doesn't make the idea a concept, anymore that "Dogs bark" is a concept.

We seem to be using the term NSRC without much of a problem. Also, remember that some terms only come into common use after they’ve been introduced for the purposes of a math book – for example, the term “truth value” was popularized by Frege. So while I think you make a good point, I don’t think it invalidates NSRC.

David Zornek:

Perhaps I misunderstood what you were trying to say in your initial description of your anti-realist views, but I have never heard "anti-realism" used in the way you described. Everywhere I've seen it used it refers to the denial of an objective reality as such (a metaphysical position), rather than the epistemological view that our concepts don't have an objective basis in reality, which is what I thought you were saying.

Do you deny the existence of reality, or do you accept that there is an objective reality, but deny that concepts have an objective basis in reality?

I believe in a mind independent, physical world, but I deny that there’s any “deep” connection between our understanding of the world (i.e. the concepts we use, logic, etc.) and the world itself. I think that, for Darwinian reasons, the way we perceive the world generally matches up well with the physical reality, but language, color, logic, etc. are all human creations.

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Ayn Rand defines "concept" as follows:

"A concept is a mental integration of two or more units which are isolated according to a specific characteristic and united by a specific definition" - (Introduction to Objectivist Epistemology, p. 10)

Thanks for that. I thought I was missing something and should have double checked my reference material.
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I’m really very impressed. How did you identify Professor Smith from what I said?

Perhaps you are unaware, but Tara Smith is something of a household name among Objectivists. I'm guessing Dave checked your profile and saw UT Austin and made the connection based on the fact that she has done a lot of speaking recently on judges. At least that's how I figured it out.

I think you’re right that the compliment of a set won’t work as a compliment if we’re not restricting the domain. Nonetheless, there seem to be lots of examples where we use the compliment of a concept in a restricted domain to form another concept. So, for example, the concept of the even numbers is defined by all of the numbers that yield an integer when divided by 2. We get the concept of the odds from the numbers we can’t do this with. The concept of the primes (this is probably my best example) comes is the compliment of the numbers that be can be divided cleanly by something other than 1 or itself. Outside the numbers, the concept “non-violent criminal” is the compliment of “violent criminal” when we restrict the domain to criminals.
In the case of numbers, there are actual characteristics shared by odd numbers (or prime numbers) that are the objective basis for forming the concepts--the basis is not the lack of a characteristic, i.e. an integration based on what they are not. (I will identify those positive characteristics if asked, but it shouldn't be necessary really.)

In the case of your "non-violent criminal" vs. "violent criminal," the first thing I would say is that neither is a concept. You have one concept (criminal), which you are modifying by two other concepts (non-violent and violent). In the face of an objective concept of "crime," there would be no possibility of distinction between the two, since there would be no such thing as a "non-violent criminal." In other words, the entire distinction rests on a bad concept.

Taking a concept of a thing, and combining it with a concept of one or more qualifying concepts does not necessarily yield a new, delimited concept.

And so, yes, if I took the compliment of SRC without restricting the domain I’d end up with a collection including Pluto, trees, and doors. But as long as I restrict the domain to concepts NSRC doesn’t seem any different than the concept of primes.

The big difference is that one is based on characteristics which exist, the other is based on the non-existence of a characteristic. Objectivism rejects (except in very rare cases) the forming of concepts based on the lack of a characteristic(s), which is why this "NSRC" thing isn't a problem for Rand.

P.S. Thanks for clarifying your views on whether or not there really is a world out there. It's helpful to know exactly how far down your differences from Objectivism go.

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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.
Its not clear what the word 'refer' means in this particular context. To go back to the memory example, I can make claims which are true or false about my memories (eg that I can remember a red lego car from my childhood), and even develop a theory of scientific psychology which explains how memories are formed. But none of this depends on (or implies) the claim that there is a queer set of objects called 'memories' which somehow live inside a person's head/mind and constitute the referents of this term. The word 'memory' has a use within the English language and it is a useful tool for explaining certain aspects of human cognition. However, to treat it as a term which refers directly to mental objects seems a misleading way of viewing things.

The same thing applies to concepts - we can talk sensible about concepts being formed, but to view the concept of 'tree' as being some kind of psychic object which exists inside the human head in a similar sense to how pictures of trees exist inside art galleries is nonsense.

Edited by Hal
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Now that I understand your argument a bit better, I think I see the basic issue: I didn't initially understand what you meant in talking about a concept referring to itself. Your blue-eyed blonde example was simply the wrong kind of example because it's a conjunction of properties. An example of what is forbidden is a "concept" whose members are sets of random integers (or other random collections), because they have no non-self referential definition.

(...)

Obviously the question to ask is "then how do I know what the possible concepts are?", and the answer is "undertake a huge study of human cognition". The basic claim of the Objectivist theory of concepts is that units subsumed under a single concept have to have defining properties that identify those units as being subsumed under concept A and not B, and that those properties are grounded in perception.

What does 'possible concept' mean here? You seem to be making a descriptive claim, that it is somehow impossible for humans to form these kind of concepts. But I doubt that this is true, and it would require some kind of psychological evidence to support it. From my understanding of Rand's discussion of invalid concepts, she wasnt intending to make a psychological claim about what it was possible for humans to do - she was making a normative statement about what they should do. She didnt say that it was literally impossible for humans to form disintegrated concepts like "blonde haired people over 6 foot tall", just that doing so was a bad idea from an epistemological standpoint. When we say that a concept is invalid, we dont mean that a person cant form it, we mean that he shouldnt form it because it has no objective basis in reality.

edit: you even said in another thread when discussing the concept formation of children that "[the child] will generally classify the world in bizarre ways orthogonally related to the concepts formed by the people around him". I would guess that some of these bizarre classifications are more arbitrary and disintegrated than "blonde people over 5 foot tall", so there is no real question of psychological impossibility here.

In the case of your "non-violent criminal" vs. "violent criminal," the first thing I would say is that neither is a concept. You have one concept (criminal), which you are modifying by two other concepts (non-violent and violent). In the face of an objective concept of "crime," there would be no possibility of distinction between the two, since there would be no such thing as a "non-violent criminal." In other words, the entire distinction rests on a bad concept.

Isnt this, as bottom, just the statement that the English language does not currently have a single word for 'violent criminal'? In a different language where there was a single word for each of these categories (violent vs nonviolent), I dont think there would be any ground for denying that these were valid concepts.

You could make a strong argument that the concept 'criminal' is itself invalid, since the set of actions which you have to perform in order to be a criminal have nothing objective in common - they are grouped together simply because our government has arbitrarily declared that they should be illegal. A tax-evader has nothing in common with a murderer, and a cannibas user has nothing in common with a rapist. And yet, these people are all 'criminals'. The distinction 'violent criminal' vs 'non-violent criminal' probably comes a lot closer to capturing the objective distinction between 'rights-violator' and 'non-rights-violator' than does 'criminal' and 'not criminal' (the majority of 'non-violent crimes' shouldnt be crimes).

The big difference is that one is based on characteristics which exist, the other is based on the non-existence of a characteristic. Objectivism rejects (except in very rare cases) the forming of concepts based on the lack of a characteristic(s), which is why this "NSRC" thing isn't a problem for Rand.
I dont think that you could given a language-independent account of the distinction between a characteristic and its lack. For instance, we could say that 'humourous' is a valid concept which describes certain types of sentences, but that 'lacking humour' isnt a valid concept because it is just the absense of humour. But this is just a way of speaking peculiar to the English language rather than being anything particularly deep or metaphysical. Language X could have a word, say "chrand" which they use to describe non-humourous sentences. And they would then describe humourous sentences by saying that they "lacked chrand". So, in their language, "chrand" (='not-humourous' in English) would be the predicate, and "humourous" would be defined as being 'the absence of chrand'. Hence speakers of this language would be entitled to say that 'humour' isnt a valid concept, since its defined entirely by the lack of a particular characteristic (chrand).

A more realistic example would be something like 'innocent'. Is this a valid concept? Well it intuitively seems so, but only because English happens to have a single word for it. However, innocent just means 'not guilty'. If the English language didnt have the word 'innocent' and we instead used 'not guilty' to describe innocent people, you could make an argument that 'innocence' isnt a valid concept because its just the absence of guilt. This sort of thing is entirely relative to the language under consideration, and whether it happens to use a single word to desribe something instead of the negation of an opposing term.

Edited by Hal
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So, for example, the concept of the even numbers is defined by all of the numbers that yield an integer when divided by 2.
I'm skeptical about this, because odd numbers are defined (positively) as numbers with a remainder of 1 when divided by 2. A prime number is any positive integer with only two positive integer factors (which would be 1 and N). The claim is not that there are no concepts which also happen to be the complement of another concept, but rather that this is an accidental fact, and they can be described in terms of what they are.
We seem to be using the term NSRC without much of a problem. Also, remember that some terms only come into common use after they've been introduced for the purposes of a math book – for example, the term "truth value" was popularized by Frege. So while I think you make a good point, I don't think it invalidates NSRC.
Well, of course you can write up anything you want and turn it in. I'm just suggesting that you not ignore what the Objectivist theory of concepts is, in constructing an argument against Smith's argument against Scalia. The main points that need to be emphasized are (1) the inter-relatedness of various components of the theory, so you can't detach the Objectivist theory of concepts from the Objectivist theory of logic, (2) concepts are psychological because Objectivism is a realist epistemology, (3) concepts have a special status in Objectivist psycho-epistemology in being conventionalised and economical condensations of knowledge, which are represented and stored as single symbols thus different from phrases and ptopositions, (4) arbitrary propositions cannot stand as antecedents in a valid deduction. You're implicitly claiming "we may not have a term 'Blort' that refers to NSRC but I'm betting that we could". I'm explicitly claiming that we can't, because NSRC is not a valid concept.
I believe in a mind independent, physical world, but I deny that there's any "deep" connection between our understanding of the world (i.e. the concepts we use, logic, etc.) and the world itself. I think that, for Darwinian reasons, the way we perceive the world generally matches up well with the physical reality, but language, color, logic, etc. are all human creations.
How is that different from realism? Is it just in the use of the word "deep"?
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the complement is not automatically a concept
One partition of beers is between alcoholic and non-alcoholic. One partitionof concepts is between self-referring and non-self-referring. In fact OPAR (as I recall, I don't have the book here) mentions that 'the metaphysical' is defined as 'non-man made' (perhaps someone would quote the passage to establish exactly the formulation), in which case not only is some domain (such as beers or concepts partitioned) but the domain of EVERYTHING is partioned, very much to the contrary of your insisting that this be disallowed.

The formulation in my post 28 stands unrefuted.

Edited by LauricAcid
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Am I to understand that you agree with me?
At least for right now, I prefer to state my own arguments rather than say I agree with someone else's formulation with which I might not endorse every detail. Depending on how important this matter is to you, I suggest you very very carefully read ITOE and OPAR, then look at my post 28 to see if you can sharpen it even more than I already have to arrive at a formulation that adheres impeccably to Objectivist forms. Otherwise, you'll always be subject to the rebuttal that you've not accomplished a reductio ad absurdum on account of not having made the reductio from actual Objectivist premises.
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LauricAcid & LaVache:

Am I to understand that you agree with me?

I definitely agree with you that the concept "concept" is a self-referential concept (maybe self-subsuming concept would be a better term, but I don't see how using that wording would change anything with regard to your original assertation or it's chain of logic), and that self-reference can be a logical fast track to contradiction, but I need a little more time to noodle over what you're saying before I can be said to agree with your reductio completely.

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In the case of numbers, there are actual characteristics shared by odd numbers (or prime numbers) that are the objective basis for forming the concepts--the basis is not the lack of a characteristic, i.e. an integration based on what they are not. (I will identify those positive characteristics if asked, but it shouldn't be necessary really.)
It is very common in mathematics to define by negation. Irrational numbers are real numbers that aren't rational. In fact, as to ESSENTIALITY, what IS essential (at least to the people who discovered irrational numbers) is that they are NOT rational. Not being rational is the whole hullabaloo about irrational numbers.

I found my copy of OPAR. My paraphrase was off (it is the 'metaphysically given' not 'the metaphysical', but here's the quote (pg. 23)):

"By the "metaphysically given," Ayn Rand means any fact inherent in existence apart from human action (whether mental or physical) - as against "man-made facts," [...]"

That definition depends on saying what is NOT man-made. The metaphysically given has the attribute of NOT being man-made. To distinguish what is "inherent in existence APART from human action" [emphasis mine] requires looking at what is man-made and noting what then has the negative characteristic of NOT being man-made or at least looking at something and checking for the negative characteristic of NOT being man-made.

Moreover, does Rand say that a characteristic cannot be a negative characteristic? (Maybe she does, but would those who are arguing against negative characteristics please quote Rand on this.)

Let's take the very most salient example of an Objectivist concept: Man is the rational animal. Genus, animal; differentia, rational. Now, of course 'non-human animal' is a perfectly fine concept. It's not the concept derived from listing every species of animal except man, but rather the concept derived from the NEGATIVE attribute of NOT being a human, of NOT being an animal but NOT a rational animal. And it is a perfectly fine concept that is derived from being a number but not a rational number, of being a beer but not an alcoholic beer, of being a concept but not a self-referring concept (actually 'self-including' would be a better term, since, at least in my view, what refers is a word not a concept).

Edited by LauricAcid
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What does 'possible concept' mean here? You seem to be making a descriptive claim, that it is somehow impossible for humans to form these kind of concepts. But I doubt that this is true, and it would require some kind of psychological evidence to support it.
The meaning of "possible concept" is compositional. Given that Objectivist epistemology is a realist and psychologically-grounded theory (as opposed to nominalism), that means that certain collections cannot be encapsulated as a concept. I can give you uncountably many examples of such collections. To harp on this unit-economy point yet one more time, Objectivist epistemology does indeed hold that concepts have to be psychologically valid, and if we had a much bigger crow of say 200 then the nature of concepts would be quite different. Now you can doubt the psychological basis for that limit and then we will have to debate the scientific merits of the claim, but if you want to do that, I suggest you first learn these concepts:

1. "Flamp" = {238,129,45,0,1923,72,10,204,19}

2. "Brelp" = {qa,sdf,eyu,uxf,dvy,wuf,st,po,bse}

3. "Krilb" = {!,!!,!!!!!,!!!!!!!!,!!!!!!!!!!!,!!!!!!!!!!!!,!!!!!!!!!!!!!!!!}

The rest of the test involves me asking you some memory questions a month later.

She didnt say that it was literally impossible for humans to form disintegrated concepts like "blonde haired people over 6 foot tall", just that doing so was a bad idea from an epistemological standpoint.
More specifically, she said "unless some essential characteristic were discovered, distinguishing such blondes from all other women and requiring special study". She also says (p. 12) "Incommensurable characteristics cannot be integrated into one unit". There is no claim that Bint isn't a possible concept, since it's the anding of a small set of perceptually evident attributes. Child conceptual acquisition, btw, is typified by under-differentiation so that all non-humans are "dog", all males are "daddy" (or even, all humans are "daddy").
You could make a strong argument that the concept 'criminal' is itself invalid, since the set of actions which you have to perform in order to be a criminal have nothing objective in common - they are grouped together simply because our government has arbitrarily declared that they should be illegal.
I'm totally on board with Fuller's theses, and you've squarely identified why I am a natural law kind of guy. In strict statutory systems, they get out of this by reference to the explicit list, so that the concept of "criminal" is a legitimate concept that anybody can grasp -- it's "doing anything described in this book". The book itself is not a concept, it is an arbitrary collection is sentences. In a mixed civil/statutory system, the prolem is that the concept of criminal and the actual use under the law are at odds.
A tax-evader has nothing in common with a murderer, and a cannibas user has nothing in common with a rapist.
Except having violated the law.
I dont think that you could given a language-independent account of the distinction between a characteristic and its lack.
Today? No, certainly not. I have to prepare handouts, so I'm close to out of time. The field of comparative cognitive psychology doesn't have a huge depth of research to it, and we are rapidly approaching the point where major experimental variables cannot be controlled. Where this discussion would go, if I had the time (what can I say, too much on the plate), is that we would consider experimental data on humans' ability to abstract things out, for example visually identify something in a picture, or auditorily isolate certain acoustic components. People don't have an unlimited ability to focus. For the sake of knowing which ones I mean, mentally number the dots from top left moving to the right in about 3 passes. Now mentally abstract the dots 1, 4, 5, 7, 11, which are randomly positioned and colored.

post-490-1143299348_thumb.jpg

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I'm skeptical about this, because odd numbers are defined (positively) as numbers with a remainder of 1 when divided by 2.
'odd' can be, and is, defined differently depending on the particular mathematical exposition. It is definitely not true that one must not define 'odd' as 'not even'. In fact, your definition may not even be possible in an exposition that has not yet defined 'divided by' and 'remainder'.

n is even iff there exists a k such that n = 2k.

Then definitions of 'odd' that are all perfectly fine:

m is odd iff m is not even.

m is odd iff there exists a k such that n=2k+1.

m is odd iff there exists an n such that n is even and m=n+1.

m is odd iff there exists an n and a k such that n and k are even and m is between n and k (where 'between' has been previously defined in regard to a previously defined ordering).

They're all equivalent, but the "negative attribute" formulation is the most concise and would be preferred by many mathematicians for just that reason. Also, it makes more salient that the evens and odds partition the naturals.

The greater point is that mathematics is not so petty to stoop to squabble about definitions that are equivalent.

A prime number is any positive integer with only two positive integer factors (which would be 1 and N).
You just created an example of the fact that there can be different definitions that do not require finding out which are THE ESSENTIAL characteristics. Your definition of prime is NOT the usual definition. The usual defintion is: n is prime iff the only divisors of n are 1 and n. That's different from yours: 'n is prime iff n has only two positive divisors'; even though you threw in the bit about 1 and n as a parenthetical that does not belong in the definition since you've already defined without the parenthetical.

But even more pertinent is that the definition of 'composite' is one that is often (more often than not, possibly almost always) given as a negative:

n is composite iff n is not prime and n is not 0.

So "negative attribute" definitions, such as the previously mentioned 'irrational' are indeed often common, lucid, and useful.

Edited by LauricAcid
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'odd' can be, and is, defined differently depending on the particular mathematical exposition.
What is your point? It doesn't matter what other definitions you can come up with for a concept, the question is whether the concept can be defined by stating a unifying property. Mathematical definitions are completely irrelevant to the theory of concepts, because they are not necessarily concepts. I thought you at least knew that much about Objectivist epistemology.
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Isnt this, as bottom, just the statement that the English language does not currently have a single word for 'violent criminal'? In a different language where there was a single word for each of these categories (violent vs nonviolent), I dont think there would be any ground for denying that these were valid concepts.

Not really, since (as I said before), the entire idea of a non-violent criminal rests on a bad concept. And yes, you could make a very strong argument that the word criminal most often stands for a bad concept--not an invalid concept, as Objectivism defines "invalid concept," but I would say it's something closer to a package-deal. That's way off topic, though.

A more realistic example would be something like 'innocent'. Is this a valid concept? Well it intuitively seems so, but only because English happens to have a single word for it. However, innocent just means 'not guilty'. If the English language didnt have the word 'innocent' and we instead used 'not guilty' to describe innocent people, you could make an argument that 'innocence' isnt a valid concept because its just the absence of guilt. This sort of thing is entirely relative to the language under consideration, and whether it happens to use a single word to desribe something instead of the negation of an opposing term.
It's not very fruitful to go through every pair of opposites in the English language and determine if one side is an invalid concept-by-negative, so I'll state the principle as I learned it (not a direct quote--I'll be saying where I learned this in a minute):

A concept may be formed on the basis of a negative only in reference to a validly formed concept, when there is a significant reason for doing so, and only for concepts which are strictly relative, i.e. concepts for which the essential characteristic is a relation to another concept.

I'm not doubting your word, but for the sake of exactness, would you quote an Objectivist text on that point? Also, what are the bases for the exceptions that are made?

I didn't learn it from a text. My reference is Harry Binswanger's lecture course, "Logical Thinking." I don't know if Ayn Rand ever explicitly said this, but it is definitely implicit in her discussion of essential characteristics.

Edited by dondigitalia
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It is very common in mathematics to define by negation.

It is common, but that does not mean that those defintions would be valid according to Objectivist epistemology. Most often, a defintion by negation indicates a definition by non-essentials, which are very common in mathematics.

Moreover, does Rand say that a characteristic cannot be a negative characteristic? (Maybe she does, but would those who are arguing against negative characteristics please quote Rand on this.)

Does she really need to say it explicitly, in that formulation? A negative characteristic is not a characteristic of the existents in question. Rather, it is a characteristic of other existents, which the ones we're focusing on do not possess. Since they do not have that characteristic, how can it be a fundamental, essential characteristic? If it's a characteristic which isn't even there, how can it cause the greatest number of others?

The best I can come up with as to a direct quote supporting that formulation are instances of her applying the principle to the concepts of non-existence vs. existence and absence vs. presence on page 58 of ITOE, and then later on page 149.

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David:

Your objection so far has been that NSRC is an illegitimate concept because there’s no unifying property that all the referents of NSRC have. In order to counter this claim, I’m going to demonstrate that NSRC has at least one property that SRC doesn’t have. You’ll have to forgive me if this post is a little heavy on the formal logic – I’ve done my best to keep this approachable and I haven’t gone into a formal proof, but it may still be rough if you haven’t taken a math class in a while.

Step One:

Even though concepts aren’t themselves sets, we can introduce a set for each concept such that the set includes everything that the concept refers to. So: (note that upper case letters are concepts and lower case letters are objects)

For the concept “table” – T{t1, t2, … tn}

For the concept “chair” – C{c1, c2, … cn}

For the concept “furniture concepts” – F{T, C, … Concept-n}

For the concept “concept” – Concept{T, C, F, … Concept-n}

And so on for any concept…

(If you don’t like the idea of pairing up stuff with sets you can just say that we can make a (theoretical) list of everything the concept refers to. Also, if you don’t like the idea of being able to list all of the objects that a concept will ever refer to, think of this as listing the object it refers to at a past time t.)

Step Two:

For any concept, we can create a “listing” of the set that concept corresponds to by removing all of the concepts in the set and replacing those concepts with the elements in their corresponding set. So:

The “listing” of the (set corresponding to the) concept “table” will just be the set that corresponds to table because that set has no concepts in it.

The “listing” of the concept “furniture concepts” will be {t1, t2, c1, c2, … object-n}

The listing of the concept “concept” will be {t1, t2, c1, c2, T, C, … etc}

Step Three:

Note that listing is a process we can perform on any set. Since the listing of a set is itself a set, we can perform a listing on it (assuming that there are still some concepts left in the set). We can continue the process of listing until we’re left with a set containing only object and no concepts – call this a “full listing”. We’ll say that concepts which can be given a full listing will be “fully listable” concepts.

At this point, then, we can introduce:

Lemma One: Self-Referring concepts are not fully listable.

This should be fairly obvious – if I try to give a listing of a self referring concept I’ll include the concept in my list. When I try to perform a listing on this list I’ll end up including the concept on my new list. When I list again I’ll once again include my concept. I’ll never stop including my concept and so I’ll never end up with a list without any concepts on it – i.e. I’ll never have a full listing.

Lemma Two: Concepts containing no self reference are fully listalbe.

This again strikes me as fairly obvious, but the proof of this claim would be more technical if I were to give it. If you want me to give it, let me know and I will.

Step Four:

At this point we notice something about concepts – some, like the concept of tables and the concept of chairs, are fully listable. Others, like the concept of concepts, are not. We distinguish fully listable concepts from other concepts on this basis and integrate them into the concept of fully listable concepts. We use the symbol CFLC for this concept.

Now, CFLC is either fully listable or it isn’t. If it’s fully listable then it’s referred to by CFLC and, by Lemma 1, CFLC isn’t fully listable. If CFLC is not fully listable then, by Lemmma 2, it is fully listable. Either CFLC is or is not fully listable. Contradiction.

Conclusion:

We’ve seen, then, that a real and legitimate property – the property of being fully listable – will generate Russell’s paradox. I’ve been a little fast and loose with my argument and if anything seems questionable to you I’ll slow down and do the full line of reasoning. Again, the property of being fully listable is a property of the concepts in question – it has nothing to do with a contrast against other concepts.

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