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Russell's Paradox

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Kyle

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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.
You wrote:

"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." [David Odden]

My point was to rebut YOUR point above. NOW that I've rebutted that point, you've moved on to a new point to claim that it doesn't matter since mathematical concepts are not concepts. (1) Where is that found assertion found in Objectivist texts? What ARGUMENT do you give that the concept of an irrational number (which is a concept based on the ESSENCE of something NOT having a certain property) is not a perfectly fine concept? (2) Even if we granted that mathematical concepts are not concepts (which I don't know why we should), still left unrefuted is that the concept of non-alchololic beer, non-human animal, and Objectivism's own very CRUCIAL concept of non-man made are all perfectly fine concepts. Moreover, you have not given a REASON, beyond an arbitrary fiat of yours, WHY negation cannnot be used to form concepts. (3) You are in blatant contradiction with Objectivism when you say that "Mathematical definitions are completely irrelevant to the theory of concepts,". From OPAR: "The mathematician is the EXEMPLAR of conceptual integration." [pg. 90, emphasis mine] Moreover, the entire section, titled 'Concept-Formation as a Mathematical Process' is PREMISED on the relevence of mathematics to concepts.

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.
Okay. And the concepts of non-alcoholic beer, non-human animal, irrational number, and non-self including concept all meet those criteria. Edited by LauricAcid
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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.

I'm not sure which David you mean, since that is both of our objection. I'll go ahead and give my own answer. (Actually, my full objection is that there is no essential unifying characteristic; not just any unifying characteristic is grounds for forming a new concept.)

Concepts, in Objectivism, do not mean other concepts (except for the concept of "concept"); rather, concepts mean existents. The concept of "furniture" does not stand for the concepts of "table," "chair," etc., but actual tables, chairs, etc.

At first glance, it seems that this might be more of a semantic technicality, since the listing of F is ultimately reducible to {t1,t2,c1,c2, etc.}. The problem comes in when we attempt to make the listing of Concept:

Concept{T,C,F, ... Concept-n} means: Concept{{t1,t2, ..tn},{c1,c2 ... cn},{T,C, ... Concept-n}, ... Concept-N}, which would mean that individual tables, chairs, and other concretes were the ultimate referents of the concept "concept."

It is a big mistake to attempt to take about the Objectivist theory of concepts in terms of formal logic, since they involve opposite processes; the former is inductive, but the latter is deductive. In my view, this attempt amounts to epistemological rationalism. One cannot use a rationalist deduction and still claim to be talking about Objectivism. This entire approach to concepts is antithetical to what Objectivism has to say on the subject.

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Concepts, in Objectivism, do not mean other concepts (except for the concept of "concept"); rather, concepts mean existents. The concept of "furniture" does not stand for the concepts of "table," "chair," etc., but actual tables, chairs, etc.
I know – I just needed an example so I used that. I don’t need it for my argument, but it helped when I was trying to communicate the idea. If you get rid of F then most concepts will be fully listed after only one listing.

Concept{T,C,F, ... Concept-n} means: Concept{{t1,t2, ..tn},{c1,c2 ... cn},{T,C, ... Concept-n}, ... Concept-N}, which would mean that individual tables, chairs, and other concretes were the ultimate referents of the concept "concept."

Ah! This is what was giving me problems this morning and it’s why it took me until about 2 PM to get a response out to you guys. You’re right that if I was treating the objects in the list as the “referents” of the concept “concept” I would be in trouble. The solution to that problem is simply to make the listing correspond with the concept – I’m not claiming that the listing is somehow a list of the referents of a concept, only that the list can be constructed from the concept.

I probably should have emphasized that point though.

It is a big mistake to attempt to take about the Objectivist theory of concepts in terms of formal logic, since they involve opposite processes; the former is inductive, but the latter is deductive. In my view, this attempt amounts to epistemological rationalism. One cannot use a rationalist deduction and still claim to be talking about Objectivism. This entire approach to concepts is antithetical to what Objectivism has to say on the subject.
I’m only using the logic to point out a property – just like I would use it to pick out the property corresponding to the concept “prime.”

(Actually, my full objection is that there is no essential unifying characteristic; not just any unifying characteristic is grounds for forming a new concept.)

I seem to remember that Rand considers properties "essential" if they cause the other properties. But how would properties of abstract objects (like concepts) cause each other?

Edited by Kyle
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If you want me to give it, let me know and I will.
I may have seen the proof somewhere, at any rate it's equivalent to the fact in syntax that a context sensitive grammar without recursion yields a finite set of strings.

I'll have to think this problem over. Obviously this is somewhat orthogonal to Rand's theory of concepts, which was designed to be and articulated as a psychological theory rather than a mathematical-symbolic theory. In order to answer your kinds of examples, we need OTC', which is a formalization of the Objectivist theory of concepts, and that isn't something that Rand did, but it's not a bad idea to do so, just an un-done one. I have to think about the matter of whether recursive concepts are good or bad. If it is tenable to say that a concept does not subsume itself, then I believe the paradox is avoided. Only abstractions can have that property, so I should review the list of SRCs that we started with.

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Okay. And the concepts of non-alcoholic beer, non-human animal, irrational number, and non-self including concept all meet those criteria.

How is non-alcholic an essential characteristic? How is non-human an essential chracteristic? How is non-self-referring an essential characteristic? How is the absence of alcohol/reason/self-reference causally responsible for the greatest number of other distinguishing characteristics?

In the case of non-alcholic beer, there might be a good reason to form a concept, but the essential characteristic would not be the lack of alcohol, but rather, the process of vacuum evaporation the beer goes through to remove the alcohol after brewing--that is the causally responsible, i.e. essential, characteristic.

I reject the idea that an irrational number is definable only in terms of negative characteristics. The standard definition of "irrational number" is: a number which cannot be expressed in terms of a ratio. I reject the idea that irrational numbers are numbers at all, and instead classify them as methods, much like infinity and imaginary numbers. (That rejection does not originate with me, though, but is another idea I learned from Harry Binswanger, in "Selected Topics on the Philosophy of Science." It's pretty cheap if you want to hear his reasons for it, and it's basis in the Objectivist theory of concept-formation.) A better definition of "irrational number" would be: a method used for making calculations which cannot be made using ratios.

Edited by dondigitalia
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Hal:

Sorry I missed this earlier:

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.

The points you raise are good ones and are part of why I reject the whole idea of reference in the first place – it seems quite dubious to me that there’s a referent to be referred to in many cases.

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I seem to remember that Rand considers properties "essential" if they cause the other properties. But how would properties of abstract objects (like concepts) cause each other?

They don't. But this isn't a problem since, in Objectivism, concepts don't stand for other concepts, but actual existents, which are ultimately abstracted from entities. The characteristics of entities, can (and do) cause each other.

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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.
The ESSENTIAL aspect of irrational numbers is that they are NOT rational. Moreover, when you claim that that some mathematical definitions are not as to essentials, you'd have to have some clear method of determining "the characteristic responsible for all the rest of the unit's characteristics, or at least for a greater number of these than any other characteristic is" [OPAR pg. 99], which includes giving a precise meaning for a characteristic being "responsible" for other characteristics and a way to calculate through the combinatorial mass of interrelated characteristics to find the single one most "responsible" for the others. But there is no such thing on the table. And as unwieldy as this would be for mathematics, it would be even more unweildy OUTSIDE mathematics since in mathematics, characteristics are much more rigorously delineated than outside mathematics. So, it doesn't help to just wave essentiality around like a sword to chop down concept formations one finds convenient to deny.

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?
It's the DIFFERENTIA. Among animals, it is the lack of rationality. Among beers, it is the lack of alchoholl; among numbers, it is the lack of being a ratio of whole numbers; among facts, it is the lack of being man-made; among concepts, it is the lack of self-inclusion.

Moreover, if negative characteristics are not allowed, then what basis is there for ANY exceptions, as you mentioned that some exceptions are permitted?

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.
I don't have ITOE with me. Quotes would help.
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They don't. But this isn't a problem since, in Objectivism, concepts don't stand for other concepts, but actual existents, which are ultimately abstracted from entities. The characteristics of entities, can (and do) cause each other.

What about the concept of even numbers or primes?

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I don't have ITOE with me. Quotes would help.

It's like 3 pages of the book--way more than I'm willing to retype. If I remember when I get home, I'll copy and past it from the research CD, but you might just have to wait until you have the book handy.

Edit: As for the rest of the quoted post, I already addressed it.

Edited by dondigitalia
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Concepts, in Objectivism, do not mean other concepts (except for the concept of "concept"); rather, concepts mean existents. The concept of "furniture" does not stand for the concepts of "table," "chair," etc., but actual tables, chairs, etc.
What do the concepts 'logical', 'metaphysical', 'epistemological', 'philosophical', 'fact', 'theory', 'integration', 'stolen concept', 'principles', 'reduction', 'conceptual common denominator', and, yes, 'self-referring', pertain to?
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What about the concept of even numbers or primes?

Well, numbers are concepts of quantity, of a particular identification, i.e. action, of consciousness, so the causal existents involved are both the consciousness performing the identification and all of the actual instances of the quantities in reality. Mostly, though, in mathematics, we deal with concepts of method, so the causal entity involved is usually consciousness.

If you're asking for the positive characteristics:

An even number is a natural number which, when halved (by a consciousness) yields another natural number.

A prime is a number with exactly two natural divisors.

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Hal:

Sorry I missed this earlier:

The points you raise are good ones and are part of why I reject the whole idea of reference in the first place – it seems quite dubious to me that there’s a referent to be referred to in many cases.

Could you clarify what you mean by 'rejecting the idea of reference'? I think that there is a common sense meaning of the word 'reference' where its obviously wrong to deny that concepts/words refer to things. When I make a statement about my mother, I am referring to her (if someone asks me who I am referring to, I can point at her and and show them). When I say that the sky is blue, I am referring in a fairly uncontroversial way to the sky. When a farmer starts talking about cows, he is referring to cows rather than horses. If this is the sort of thing you want to deny, then I think you need to explain what you want to replace reference with, because this saying that 'reference doesnt exist' strikes me as incoherent.

If this isnt the sort of thing youre denying, then I think you need to be more specific about what it is youre opposing. I realise that there are more philosophical uses of the word 'reference' which are rather dubious; theres a tendancy within academic philosophy to treat reference as being an almost occult process which stands in need of a formal theory. But denying particular theories of reference is different from denying that 'words refer to things', where the word 'refer' is being used in a fairly standard (non-analytic-philosophy) sense.

Edited by Hal
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How is non-alcholic an essential characteristic? How is non-human an essential chracteristic? How is non-self-referring an essential characteristic? How is the absence of alcohol/reason/self-reference causally responsible for the greatest number of other distinguishing characteristics?
Lack of alcohol is the most important aspect that distinguishes non-alcoholic beer from all other beer. Similarly, for all the other concepts mentioned avove.

In the case of non-alcholic beer, there might be a good reason to form a concept, but the essential characteristic would not be the lack of alcohol, but rather, the process of vacuum evaporation the beer goes through to remove the alcohol after brewing--that is the causally responsible, i.e. essential, characteristic.
That is mistaken and you just supported my point by making that mistake. What is essential is not the method, but the RESULT. WHATEVER method is used to take the alchohol out, whatever the chemical process, what is essential is the final product - the lack of alcohol. What distinguishes a caffeine-free soda from all other sodas is not a method, since no method is required to take caffeine OUT. There is no caffeine ANYWAY. What is essential, just like lack of alcholol, is lack of caffeine, and that essentiality is not the process but the result, especially as seen with the soda example, since in that example, there is no process other than a LACK of a process of putting caffeine in.

I reject the idea that an irrational number is definable only in terms of negative characteristics.
I didn't claim that it is only definable in terms of a negative. But rather that, for centuries, and even in most everyday contexts today, what is essential is the lack of rationality.

The standard definition of "irrational number" is: a number which cannot be expressed in terms of a ratio.
Yes, outside of the real rigorous contexts, that's the definition. And it is just what I said - the lack of expressibility as a ratio. An irrational is a number that can NOT be expressed as a ratio. It is a number that is not a rational number. A definition by negative characteristic.

I reject the idea that irrational numbers are numbers at all, and instead classify them as methods, much like infinity and imaginary numbers. (That rejection does not originate with me, though, but is another idea I learned from Harry Binswanger, in "Selected Topics on the Philosophy of Science." It's pretty cheap if you want to hear his reasons for it, and it's basis in the Objectivist theory of concept-formation.) A better definition of "irrational number" would be: a method used for making calculations which cannot be made using ratios.
Fine. Then an irrational is a method of measurement that lacks the characteristic of other methods of measurement that are expressible as ratios. Anyway, denying that rational numbers are numbers is, in this century at least, pretty far-flunged. Edited by LauricAcid
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What do the concepts 'logical', 'metaphysical', 'epistemological', 'philosophical', 'fact', 'theory', 'integration', 'stolen concept', 'principles', 'reduction', 'conceptual common denominator', and, yes, 'self-referring', pertain to?

The refer to actions of consciousness, which are not necessarily themselves concepts.

Edited by dondigitalia
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Well, numbers are concepts of quantity, of a particular identification, i.e. action, of consciousness, so the causal existents involved are both the consciousness performing the identification and all of the actual instances of the quantities in reality. Mostly, though, in mathematics, we deal with concepts of method, so the causal entity involved is usually consciousness.

If you're asking for the positive characteristics:

An even number is a natural number which, when halved (by a consciousness) yields another natural number.

A prime is a number with exactly two natural divisors.

The two positive characteristics you list aren't anything different from "fully listable."

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It's like 3 pages of the book--way more than I'm willing to retype. If I remember when I get home, I'll copy and past it from the research CD, but you might just have to wait until you have the book handy.

Edit: As for the rest of the quoted post, I already addressed it.

Thanks. Please do not go to special trouble.

As for the rest of my post, you responded to beer and irrational numbers. I rebuted you above, especially as you made a big mistake in taking the method for a result to be essential rather than the result itself. Also, still not responded to are: non-human animal and non-man made. Also, why some exceptions are allowed but not others.

Edited by LauricAcid
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That is mistaken and you just supported my point by making that mistake. What is essential is not the method, but the RESULT. WHATEVER method is used to take the alchohol out, whatever the chemical process, what is essential is the final product - the lack of alcohol. What distinguishes a caffeine-free soda from all other sodas is not a method, since no method is required to take caffeine OUT. There is no caffeine ANYWAY. What is essential, just like lack of alcholol, is lack of caffeine, and that essentiality is not the process but the result, especially as seen with the soda example, since in that example, there is no process other than a LACK of a process of putting caffeine in.

In Objectivism, an essential characteristic is the causal characteristic, not the result. Disagree with it all you like, but that isn't what Ayn Rand meant when she used the term. However most people define non-alcholic beer, it would not be considered a proper, philosophic definition under Objectivism.

Yes, outside of the real rigorous contexts, that's the definition. And it is just what I said - the lack of expressibility as a ratio. An irrational is a number that can NOT be expressed as a ratio. It is a number that is not a rational number. A definition by negative characteristic.

Fine. Then an irrational is a method of measurement that lacks the characteristic of other methods of measurement that are expressible as ratios. Anyway, denying that rational numbers are numbers is, in this century at least, pretty far-flunged.

No, it possesses the characteristic of being able to facilitate calculations that rational numbers cannot.

So what if it's far-flung. Lots of the ideas that arose from Objectivism are; far-flung-ness has no significance whatsoever.

I'll address anything else later. I'm away for the rest of the day probably.

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The refer to actions of consciousness, which are not necessarily themselves concepts.
Some of them also refer to concepts. It makes perfect sense to talk about kinds of concepts. Is something a philosophical concept or is it a concept of a particular science? Is something a concept of method or is it not a concept of method? Is something a concept found prevalent among scholars or is it a concept also found prevalent among non-scholars (by the way, 'non-scholar' another negative)? Is it a concept that is talked about a lot or is it a concept that people don't talk about so much these days? Those are all questions as to attributes of concepts themselves. Per Objectivism, concepts are mental entities. Concepts themselves have attributes. One attribute is that of self-inclusiveness. Another attribute is that of not being self-inclusive. Nothing anyone has posted refutes that.
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In Objectivism, an essential characteristic is the causal characteristic, not the result. Disagree with it all you like, but that isn't what Ayn Rand meant when she used the term.
Even where she may use 'cause' she uses 'depends' and Peifkoff uses, "is responsible for". But the sense in which you're taking that leads to everything being defined by the process not the result, which contradicts even the examples given in Objectivism. A table is a raised flat surface for human use. A table is not derfined as "the result of a carpenter sawing wood and assembling it so that there is a flat board supported by other pieces of wood." By claiming that the essential characteristic of something is the process that caused the thing, you're making definitions so unwieldy as to be, drastically, much less cognitively useful than we expect. If I say, "Hand me a fork", I don't mean to clarify my request by saying, "Hand me an instance of what is caused by the forging of various metals into an alloy then molded into a shape with a handle and tines." No, I mean, "Hand me one them things that has a handle and tines for eating food with." When I say, "Hand me a non-alcoholic beer", I don't mean "Hand me some liquid that is caused by a vaccuum process for removing alcohol from beer." No, I mean, "Hand me a beer that has no alcohol in it." And you can see that examples given in Objectivism are many that mention the essential not as the process or events causing but rather the use of the thing or the appearence of a thing, and other. Even "rational animal". Not, the animal that has such and such biological development of a certain kind of brain allowing rationality. Rather, simply, "the rational animal". Rationality, no matter what particular biological causes allow it, is what is essential, not the biological causes themselves.

No, it possesses the characteristic of being able to facilitate calculations that rational numbers cannot.
(1) You contradict yourself since you defined it in terms of its facility not its "cause". (2) It's still a negative definition. (3) It's an unweidly definition that goes past the essential characteristic to some other characteristic. A mathematician doesn't have to ask, "Hmm, is or is this thing something that allows calculations that a rational number does not?" No, a mathematician wants to get right to the heart of it, "Is this thing a ratio of two whole numbers or not?" Use in calculations is another, separate question, that the engineer or accountant will consider.

So what if it's far-flung. Lots of the ideas that arose from Objectivism are; far-flung-ness has no significance whatsoever.
I grant that a notion being far-flung does not in itself refute the notion. But in the case of irrational numbers, the mathematics works quite splendidly without the far-flung notion that irrational numbers aren't numbers, and that is WHY the notion is far-flung and that is WHY the far-flung notion needs special attention as to its justification. Edited by LauricAcid
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Lack of alcohol is the most important aspect that distinguishes non-alcoholic beer from all other beer. Similarly, for all the other concepts mentioned avove.

Just a side note for the record... most (if not all) "non-alcoholic" beers actually do have "trace amounts" of alcohol in them, just not nearly the amounts generally used for the intoxicating effect. So the term "non-alcohlic" beer is arguably invalid.

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Just a side note for the record... most (if not all) "non-alcoholic" beers actually do have "trace amounts" of alcohol in them, just not nearly the amounts generally used for the intoxicating effect. So the term "non-alcohlic" beer is arguably invalid.
Thanks for the information. But it doesn't make the concept improper. Not every concept needs to scrutinized for complete scientific accuracy down to the very last molecule. The concept of cotton shirt is still just fine even though tiny amounts of materials other than cotton might get caught into the weave.
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The concept of cotton shirt is still just fine even though tiny amounts of materials other than cotton might get caught into the weave.

Well, the concept of non-alcoholic is valid IF you are talking about, say, Pepsi (or for that matter a cotton shirt). If you talk about O'Douls, it is not. Saying that something does not exist in a substance is not the same as not addressing the existence of something in a substance.

If I ask "Does O'Douls contain alcohol?", the true anwer is yes despite the claim that it is non-alcoholic. If I ask, "Does a cotton shirt contain cotton?", the true answer is yes, which is congruent with the description.

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That is grasping at molecular sized staws. Your objections can be met by just revising the exact formulation to allow for what people already know - that there are precious few physical objects that can be described as having or not having PURELY only certain elements. "Does this beer have alcohol" can easily be understood to include the qualification, which is trivial and irrelevent to this context, "...except perhaps trace amounts." No good argument has been given in this thread that definition by negation is incorrect. Trivial quibbles about the fact that things are rarely PURELY without trace amounts are beside the point. Morevover, for every such quibble, one could come up with a thousand more counterexamples that aren't even affected by the quibble. "I want to be buy some seamless handkerchiefs, please." Seamless handkerchief. Defined by negation - not having a seam in it. Strapless bra. Defined by negation - not having a strap. A cordless phone. An anechoic chamber. A non-human animal. A non-self-including concept. All of them perfectly fine concepts.

Edited by LauricAcid
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That is grasping at molecular sized staws.

This was not intended to be a "hijack" or diversion from the original topic, merely a sidenote as I originally mentioned. With that in mind I'll end my part of this segue by reiterating, non-alcoholic beer is contains alcohol, therefore it is not non-alcoholic. It would just as easily and more accurately be described as reduced-alcohol beer, which would be a valid description.

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