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Are Axiomatic Concepts 'close-ended'?

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Legendre
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Okay, I believe I understand why objectivity requires that non-axiomatic concepts be "open-ended."

Since we (well, rational people, anyway) are continually gathering more and more information about the concretes of material reality, we may find some concepts we previously formed inductively are insufficient and must be revised.

For instance, a child may believe that all ocean-dwelling creatures with stream-lined bodies and finny appendages must be fish, while a mammal is a hairy creature that lives on land. When the child learns that whales are mammals, he or she then revises his or her prior conceptions of "fish" and "mammals."

My question is: Are all concepts open-ended? It appears to me that an axiom must be "close-ended" by its nature. For example, I don't see how the Law of Identity can be logically revised. Are axiomatic concepts then "close-ended"? :D

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Okay, I believe I understand why objectivity requires that non-axiomatic concepts be "open-ended."

Since we (well, rational people, anyway) are continually gathering more and more information about the concretes of material reality, we may find some concepts we previously formed inductively are insufficient and must be revised.

You are mixing concept-formation and the nature of concepts. See ITOE p. 147 for the discussion Concepts as Open-Ended. The central point about open-endedness is that "Concepts are open-ended in the sense that every new concrete of the same type is to be subsumed under the concept." In other words, the concept "dog" subsumes not just a specific set of two dozen observed dogs, but refers to the entire kind. The question of acquiring adult concepts (and distinguishing porpoises from tuna) is different. The referent of an axiomatic concept is not closed: time is an axiomatic concept, and Sunday Dec. 25, 2005 10:36am is (or was) a new specific time-referent which I had not experienced until now
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For instance, a child may believe that all ocean-dwelling creatures with stream-lined bodies and finny appendages must be fish, while a mammal is a hairy creature that lives on land. When the child learns that whales are mammals, he or she then revises his or her prior conceptions of "fish" and "mammals."

This isn't really correct. What actually happens, is a new concept "aquatic mammals" is formed, and the definition of "fish" is changed, rather than the concept itself.

The first point that needs to be made is that "aquatic mammal" is not abstracted from "fish," but from "mammal." "Fish" is one of the many concepts from which we differentiate "mammal" when forming it, which is why we are able to recognize that whales, dolphins, etc. are not fish. We observe that these animals are not a type of fish at all, which is a process of recognizing an earlier differentiation, and then we proceed to abstract the concept "aquatic mammal" from "mammal."

The second point is that, as Ayn Rand points out in ITOE, a concept means more than it's definition--it means the actual existents subsumed under it. A definition is a means of integrating that concept into the rest of our knowledge. In your example, the things subsumed under the concept have not changed, so the concept itself hasn't changed. What has happened instead, is you have obtained new knowledge which renders the earlier definition inadequate to properly integrate the concept into your own body of knowledge, so you revise it (make it a more precise definition), so that it adequately tells you what the (unchanged) concept means.

Edited by dondigitalia
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For example, I don't see how the Law of Identity can be logically revised. Are axiomatic concepts then "close-ended"? :D

The law of identity is not a concept; it is a proposition. Identity is a concept naming a thing all existents have, namely their "thing-ness"; it is an attribute, actually.

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This isn't really correct. What actually happens, is a new concept "aquatic mammals" is formed, and the definition of "fish" is changed, rather than the concept itself.
I'm not entirely sure what you mean here - what does it mean for something to be 'subsumed under a concept'? I always interpreted this to mean something like 'a person has subsumed an entity under a concept if he would identify that entity as falling under the concept'. Any other interpretation seems to be a relapse into Platonism, where concepts somehow exist independently of the human mind, and the non-fishness of whales becomes a consequence of some 'essence' they have independently of the way we categorise them (see the controversy over cladism in biology for an example of why this sort of thinking is problematic).

If my interpretation of the phrase is correct, then it would be true to say that the units falling under the concept have changed. Before being told that we didnt call whales fish, the child would have claimed that a whale was a fish (ie he would have subsumed whales under his concept of 'fish'). But after he had been told, he would no longer do this.

Most people would claim that peanuts are subsumed under their concept of 'nuts'. However, under the standard scientific classification, peanuts are actually not nuts. It seems more sensible to say that the units of their concept will change if they found this out, than it is to say that they didnt previously know what their concept meant.

Edited by Hal
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Most people would claim that peanuts are subsumed under their concept of 'nuts'. However, under the standard scientific classification, peanuts are actually not nuts. It seems more sensible to say that the units of their concept will change if they found this out, than it is to say that they didn't previously know what their concept meant.

Peanuts are seeds in a seed-case. In this they are like other nuts.

Peanuts grow underground rather than on branches in the air. In this they are unlike other nuts.

Why do you say that they are not nuts? What is the "scientific" definition of a "nut"?

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I'm not entirely sure what you mean here - what does it mean for something to be 'subsumed under a concept'? I always interpreted this to mean something like 'a person has subsumed an entity under a concept if he would identify that entity as falling under the concept'. Any other interpretation seems to be a relapse into Platonism, where concepts somehow exist independently of the human mind, and the non-fishness of whales becomes a consequence of some 'essence' they have independently of the way we categorise them (see the controversy over cladism in biology for an example of why this sort of thinking is problematic).

That is the way I interpret "subsume" also.

If my interpretation of the phrase is correct, then it would be true to say that the units falling under the concept have changed. Before being told that we didnt call whales fish, the child would have claimed that a whale was a fish (ie he would have subsumed whales under his concept of 'fish'). But after he had been told, he would no longer do this.

I think you are equating the word that stands for the concept and the concept itself. A word is not a concept, but an audio-visual symbol used to bring the concept back down to the perceptual level, so that we can deal with it as a concrete. The child may group whales and fish together at first, and may use the word "fish" as a symbol for that concept. But that is NOT the same concept as the one he later refers to as "fish." The two later concepts are essentially different from the original. Not because the original concept has changed, but because the child has learned about an essential difference between the two. The original concept, which integrates both fish and aquatic mammals, remains intact, but as adults, we use the term "aquatic animal."

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Thanks for the help, everybody!! I still have a few questions.

JMeganSnow writes,

The law of identity is not a concept; it is a proposition. Identity is a concept naming a thing all existents have, namely their "thing-ness"; it is an attribute, actually.

Is a proposition not a type of concept?

DavidOdden writes,

The referent of an axiomatic concept is not closed: time is an axiomatic concept, and Sunday Dec. 25, 2005 10:36am is (or was) a new specific time-referent which I had not experienced until now.
I thought that the concept of time was the result of an induction; that one derives the concept of time after first observing a sequence of events or actions. Isn't "time" derived from the concept of "actions"? One Thing (an Event) happens after Another Thing (another Event), and that's how we arrive at "time"?

dondigitalia writes,

The second point is that, as Ayn Rand points out in ITOE, a concept means more than it's definition--it means the actual existents subsumed under it. A definition is a means of integrating that concept into the rest of our knowledge.

Pardon the strangeness of this sentence, but your definition of "definition" is very helpful! I understood that a definition was not the exact same thing as a concept, but I couldn't put the difference into words. Since a definition exists to integrate a concept with the rest of what we know, a word's definition helps a person identify the relationship a concept has with other existents?

Is this why simpler words like "is" are more difficult to define than more complex words like "serendipity"?

With respect to mammals, when I was a little kid in elementary school, I had this World Book Childcraft edition about the animal kingdom, and it had a chapter explaining what distinguished mammalia from all other classes of animalia. At this point, I already knew that whales were mamamls instead of fish.

However, it was only when I read this book that I came to understand that what makes mammals unlike all other animal classes is that they produce milk. Up until then, that thought never occurred to me. It didn't really register with me that any organisms other than humans and cows produced milk. Until that point, I thought that what made something a mammal was that it was warm-blooded and didn't have feathers.

I never before made the connection between mammals and milk. Since I now see milk production as the mammal's essential, distinguishing attribute, can we not say that it was not only my definition of mammal that changed, but my concept of mammal as well?

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Why do you say that they are not nuts?

Theyre legumes, not nuts. For instance:

http://en.wikipedia.org/wiki/Peanut

http://www.peanut-shellers.org/index_files/Legume.htm

But that is NOT the same concept as the one he later refers to as "fish." The two later concepts are essentially different from the original."

Well, would you say that jrs' new concept of nut is not the same one that he had before he first read this thread? I'm not saying that this is obviously wrong, but it seems a very strange way of speaking. Even as adults, we often learn new things about how we break up the world, and I would say that our concepts are in a state of being updated/revised rather than renewed.

edit: In a way, its like arguing whether a game of chess played without the castling rule is still 'chess' (but with a slightly different ruleset), or a new game entirely. I dont think theres any 'correct' answer to this question - its just a choice between two ways of speaking. If the change is sufficiently small (eg the recategorisation of whales/peanuts) rather than being a significant one involving fundamental units of the concept, I think its easier to say that what you have is a slightly different version of the same thing.

Edited by Hal
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Is a proposition not a type of concept?
No, for various reasons. First, a proposition might stand for a single state of affairs whereas concepts are open-ended. Second, a concept integrates a number of existents in the form of a single concrete -- a word. Unit-economy is a fundamental property of concepts, and is not a property of propositions. Note (ITOE p. 12) that "No concept could be formed, for instance, by attempting to distinguish long objects from green objects. Incommensurable characteristics cannot be integrated into one unit." However, propositions are essentially unlimited: "This bin is for long objects and not green ones".
I thought that the concept of time was the result of an induction; that one derives the concept of time after first observing a sequence of events or actions. Isn't "time" derived from the concept of "actions"? One Thing (an Event) happens after Another Thing (another Event), and that's how we arrive at "time"?
I admit that this is not clear to me. It's beyond question that existence, identity and consciousness are axiomatic concepts: the question is whether there are any others. Logically, "time" can't depend on the notion of one action happening after another, since the notion "after" presupposes "time". After re-reading ITOE ch. 6 and the appendix on Axiomatic, it's not crystal clear to me. On p. 85 she summarizes he importance of the concept "time" in relation to all other concepts as follows (emphasis added): 'The first and primary axiomatic concepts are "existence," "identity" and "consciousness." They identify explicitly the omission of psychological time measurements, which is implicit in all concepts'. I don't see how "time" can be derived from existence, identity or consciousness, or what component parts the concept can be broken into. But maybe I'm missing something.
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Pardon the strangeness of this sentence, but your definition of "definition" is very helpful!...Since I now see milk production as the mammal's essential, distinguishing attribute, can we not say that it was not only my definition of mammal that changed, but my concept of mammal as well?

Since that description of definitions is new to you, I think I'm safe in assuming you have not read ITOE. Incidentally, what I gave was not a definition of "definition," but a description. A proper defintion is: a statement that identifies the nature of the units subsumed under a concept (AR's defintion).

There is an entire chapter on defintions in ITOE, which addresses all of your questions. She also discusses what it means for a characteristic to be "essential." Essentiality is a very specific thing in Objectivism, and isn't always limited to a single characteristic as you seem to imply.

Well, would you say that jrs' new concept of nut is not the same one that he had before he first read this thread? I'm not saying that this is obviously wrong, but it seems a very strange way of speaking. Even as adults, we often learn new things about how we break up the world, and I would say that our concepts are in a state of being updated/revised rather than renewed.

I can't speak for him, but I have two different concepts, which are both referred to by the word "nut." One is the concept I use in the context of cooking food, or other casual contexts (which includes peanuts and other non-nuts), the other is the one I use in a scientific context (which draws the distinction between them). I'm guessing is that Jrs' original concept remains intact, but he now has a new concept as well. (Incidentally, think about all the different concepts we refer to by that word; I can think of five off the top of my head. They are not changed versions of each other, but very different concepts.)

But no, a concept stands for existents. A concept is a mental integration of a set of existents. A different set means a different concept.

Edited by dondigitalia
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Actually, that is correct: it is the sum of all attributes possessed by a specific existent. Thanks for clearing that up.

This is another example of two different concepts being represented by the same word. (In fact, by coincidence, all of the axiomatic concepts are examples.) That is a definition of the non-axiomatic concept "identity," but the axiomatic concept is something different. As I'm sure you know, it is really the same concept as existence, but viewed from a different perspective, and can be defined only ostensively.

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As I'm sure you know, it is really the same concept as existence, but viewed from a different perspective, and can be defined only ostensively.

Not quite: it is approached, abstractly, from a slightly different angle.

In some philosophies existence and identity are considered separate, like you could have an existent with no attributes (hence possessing no identity). I'm not sure if I've ever personally encountered someone claiming that you can have attributes with no existent, but I guess that's what intrinsicism (of the Platonic dualist variety) is in its rawest form.

In Objectivism, existance is identity, but that doesn't mean that existence and identity are the same concept; if they were, they wouldn't be different abstractions.

What I said wasn't so much a definition as a description.

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In Objectivism, existance is identity, but that doesn't mean that existence and identity are the same concept; if they were, they wouldn't be different abstractions.

What I said wasn't so much a definition as a description.

When I said that they were the same concept, I meant to indicate that they were integrations of the exact same units, but viewed from a different perspective. A concept means it's referents, and the referents of existence and identity are identical. The only abstract difference between the two is one of perspective.

I disagree that identity is an attribute, or a some of all attributes. It's more than that. It is the fact that a things exists, with the emphasis placed on the fact that a thing exists.

Edited by dondigitalia
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I disagree that identity is an attribute, or a some of all attributes. It's more than that. It is the fact that a things exists, with the emphasis placed on the fact that a thing exists.

That's existence; the fact that a thing cannot exist without being something in particular is identity.

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