An Attempt At Formalizing the "Concept" Concept

Metaphysics and Epistemology
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For many years now, I have been thinking deeply about what a “concept” is. I’ve found that Rand’s attempt fails to be sufficiently general to account for many different kinds of concepts. So after a few years of generating, testing, and rejecting hundreds of different ideas, I believe I have finally arrived at a definition that has many satisfactory properties.

In short, a concept is a generalized piece of algorithmic abstract reasoning that is often applicable to many different cases and aimed at answering a given set of related questions.

The definition I propose here is a lot more complicated than Rand’s, but it is nonetheless fairly straightforward. In order to define the notion of “concept”, I will list the ingredients necessary to specify a concept along with certain conditions that those ingredients must satisfy.

To specify a concept we need:

  1. A background theory, T.
  2. An at most countably infinite parameter set, A.
  3. A total computable function P : A -> Sentences(T).
  4. An algorithm, D : A -> Proofs(T), which takes a in A as inputs and (possibly) outputs either a proof of P(a) in T or a proof of ~P(a) in T. (NOTE: the algorithm is not required to terminate on all inputs. It is not required to terminate at all. It is only forbidden from ever outputting a wrong answer).

I will be happy to answer any questions or criticisms you might have.

 

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My thinking is that a concept is a function that takes an existent as its argument and returns “true” if the existent is an instance of the concept and “false” otherwise.

So potato(x) is true only if x is a potato.

The potato function would be implemented by looking at the properties of x (shape, size, color, weight, origin, etc.) and determining if they exist and if they are within certain limits. Some properties are not used at all in the definition of some concepts (e.g., the location of a potato is not used in determining whether or not it is a potato, but location is used in other concepts, such as “tourist.”) Similarly, a potato may be expected to lack certain properties entirely, such as intensity or specificity, but few concepts are defined in terms of what they lack…

You could also consider a concept such as “potato” as a (potentially infinite) set of all the possible existents that could be instances of that concept; potato(x) is true if and only if x is a member of the set of potatoes.

Technically a concept also includes everything else you know about potatoes, but this knowledge is not wrapped up in the definition. However, the concepts are key to keeping related knowledge together.

An implementation of such a function is not unique and can be changed around. For example, you might later get the concept vegetable(x) and save space by redefining potato(x) as “vegetable(x) AND …” with other requirements, rather than copying all the requirements of vegetable(x) into potato(x).

Some people might define vegetable(x) as “potato(x) OR carrot(x) OR broccoli(x) OR …” but that isn’t accurate because it’s at least conceivable to discover new vegetables, and yet doing so should not change the definition of “vegetable.” But it is possible to look into the definitions of various vegetables and ask what properties are in common between them all, and use these common properties to define vegetable(x). It can also be helpful to be able to find examples of abstract concepts such as vegetables, but these again are not part of the definition.

It’s also possible for properties themselves to be described by concepts, so that shape(x) is only true if x is a shape. A shape doesn’t exist “by itself,” it can only exist as a property of some other object, but we can isolate it for the purpose of thinking about it. (Sensations such as colors and smells can be sensed “by themselves” but even they have to belong to something.) There are also “contextual properties” such as randomness. (For example, if a potato is a random potato, that doesn’t really say anything about the potato itself, but it says something about how the potato was selected from a group of potatoes…)

It’s very important to try to minimize the amount of information taken up by all these concepts. If they become too complicated they are impossible to retain or use in a human mind. (A computer would not have this particular difficulty, but complicated concepts can also take longer to use and can be more error-prone in other ways. There are examples such as self-driving cars that, if you put a sticker on a STOP sign, the self-driving system no longer recognizes the sign as a STOP sign, and might even mistake it for some other sign! This sort of thing happens if the definition is too complex and not based on essentials.)

Accuracy is also critically important.

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3 hours ago, SpookyKitty said:

In short, a concept is a generalized piece of algorithmic abstract reasoning that is often applicable to many different cases and aimed at answering a given set of related questions.

How about we start with this? You know Rand’s theory of concepts, you have your own definition of concepts. If the extensions of the two theories is identical, then the comparison and evaluation has to be based on based on metatheoretical differences (e.g. simplicity). If the extensions are not the same, then one theory is wrong compared to the other (both could fail, one would fail worse). You seem to confess to inferiority w.r.t. simplicity, which leads me to suspect that you think your theory succeeds when Rand’s theory empirically fails. I know of a number of problems where her theory superficially fails, but only given certain stipulations about word meaning.

Is your position that there is an empirical problem with her theory, and how do you propose to establish that it is a problem? Hypothetically, one might claim that the theory fails to account for concepts like “Bertrand Russell” because that is not the integration of two or more units, measurement omitted. The simple Objectivist response is that “Bertrand Russell” is not a concept, then we can have a brief exchange of “nanner-nanner sez you!”, since there is no independent metric of what things are concepts.

Insofar as “concept” specifies a property of the mind, hyper-conceptual modeling of “concepts” in computational terms is otiose if we cannot reduce the models to concrete claims about the mind. The idea of a concept being “wrong” is so bizarre that I have no idea what that would actually mean.

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1 hour ago, necrovore said:

My thinking is that a concept is a function that takes an existent as its argument and returns “true” if the existent is an instance of the concept and “false” otherwise.

So potato(x) is true only if x is a potato.

The potato function would be implemented by looking at the properties of x (shape, size, color, weight, origin, etc.) and determining if they exist and if they are within certain limits. Some properties are not used at all in the definition of some concepts (e.g., the location of a potato is not used in determining whether or not it is a potato, but location is used in other concepts, such as “tourist.” Similarly, a potato may be expected to lack certain properties entirely, such as intensity or specificity, but few concepts are defined in terms of what they lack…)

You could also consider a concept such as “potato” as a (potentially infinite) set of all the possible existents that could be instances of that concept; potato(x) is true if and only if x is a member of the set of potatoes.

Yes, I considered this, and there are a number of problems with this kind of approach.

  1. There is no way to “operationalize” such a definition. How would the algorithm access the actual truth of the matter? Like sure, an algorithm could output some answer (and possibly the correct one) based on the inputs, but how could it guarantee that it is the correct answer?  No mind and no (real) algorithm has access to an infallible truth oracle. This is why it’s better to focus on provability, which is an entirely syntactic notion, rather than truth.
  2. There is also the problem of spurious algorithms. This is what happens when two very different algorithms (and therefore, different concepts) compute the same function. For example, any first grader can list the odd numbers. Almost none of them can list the absolute values of the reciprocals of the terms of the Leibniz formula for pi. Just because you understand one concept does not mean you understand the other, even though they are identical as sets.
  3. There is a problem of infinite regress. if you define potatoes in terms of their properties, then what are you using to define the concepts corresponding to those properties? Properties of properties? I’m sure you see the problem here. It also seems to turn the entire hierarchy of knowledge on its head. We should be defining the properties of concretes in terms of the concretes, not the other way around.
  4. It’s not enough to state that some thing is or is not a potato. One must also be able to answer why it is or is not a potato.
     

EDIT: Relating to numbers 3 and 4, there is an amusing anecdote that captures the heart of these problems.
Is there an algorithm that correctly determines the existence of god?
Of course there is. If there is a god, that algorithm is:

PRINT “Yes”

And if there is no god, then that algorithm is:

PRINT “No”.

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1 hour ago, DavidOdden said:

How about we start with this? You know Rand’s theory of concepts, you have your own definition of concepts. If the extensions of the two theories is identical, then the comparison and evaluation has to be based on based on metatheoretical differences (e.g. simplicity). If the extensions are not the same, then one theory is wrong compared to the other (both could fail, one would fail worse). You seem to confess to inferiority w.r.t. simplicity, which leads me to suspect that you think your theory succeeds when Rand’s theory empirically fails. I know of a number of problems where her theory superficially fails, but only given certain stipulations about word meaning.

Is your position that there is an empirical problem with her theory, and how do you propose to establish that it is a problem? Hypothetically, one might claim that the theory fails to account for concepts like “Bertrand Russell” because that is not the integration of two or more units, measurement omitted. The simple Objectivist response is that “Bertrand Russell” is not a concept, then we can have a brief exchange of “nanner-nanner sez you!”, since there is no independent metric of what things are concepts.

My position is that her theory fails to meet its own goals. The main point of the theory is to explain how human minds can generate or represent a potential infinity of non-trivial true statements using only finitistic means.

At best, by her definition of concepts and her account of their formation, Rand’s theory can only account for how individual perceptual level facts can be given shorter representations (and I’m not even fully convinced it does even that much, but whatever).

Although her theory allows for some small degree of cognitive economy, it is nowhere near sufficient to account for the breadth and depth of human knowledge. The theory has two major failings:

  1. Breadth: It cannot account for how infinitely many distinct perceptual level facts can be given a finite representation.
  2. Depth: It cannot account for facts whose truth does not reduce to checking at the perceptual level.

Breadth Problem:

As an over-simplified example of the first problem, let’s say some perceptual-level facts can be encoded like this:

1001001101001010011100011

11100110100101010100100111

00000010011111111110100100

If you examine these ‘facts’, you will notice that the sequence 01001 occurs multiple times in each of them. Essentially, Rand’s theory allows you to create tokens for finite, fixed symbolic sequences. if you define the ‘concept’ ‘a’ as ‘01001’, you can compress the above three ‘facts’ into the following three

10a1aa1100011

1110011a0101a00111

00000a111111111a00

If you also define ‘b’ as ‘111’ and ‘c’ as ‘000’ you can compress a little further to:
10a1aa11c11

b0011a0101a00b

c00abbba00

And if we go a conceptual step higher we can define ‘d’ as ‘bbb’ and get:

10a1aa11c11

b0011a0101a00b

c00ada00

With this, one can even carry out some simple proofs at a high conceptual level without having to reduce everything to the perceptual level contrary to what many objectivists seem to believe.

The main issue with this is that this tokenization process cannot possibly unite multiple facts into one. Here is a simple demonstration of how that can be done:

Let’s say we have the following encodings for an infinite sequence of perceptual level ‘facts’:

1

10

100

1000

10000

100000

1000000

…

No amount of Randian definitions will suffice to make this list of ‘facts’ any shorter.

Instead, if we use an algorithm, we can define a function over natural numbers n as "F(n) := "1"f(n) where f(n) := “0"f(n-1) and f(0) := “””. So, for example,

F(5) = 1f(5)

= 10f(4)

= 100f(3)

= 1000f(2)

= 10000f(1)

= 100000f(0)

= 100000

Thus, we have a finite representation of infinitely many ‘facts’. So we see here that, in order to achieve true cognitive economy, what we really need are algorithms, not definitions (Thought that’s not to say definitions are useless).

Depth Problem:

Let’s say we have some predicates whose truth value can be determined through unmediated perception. Call them P(x,y) and Q(x,y,z) where x, y, and z are variables. Let a, b, c, d,… represent given entities.

Some higher level concepts defined in terms of these predicates can be checked by unfolding their definitions and reducing to the perceptual level. For example, let’s say that the concept C(x,y,z) is defined as:

C(x,y,z) := P(x,x) and not Q(z,y,y)

Then, if we want to check the truth value of C(a, c, a), we just make the appropriate substitutions to get

P(a, a) and not Q(a, c, c)

Which is a statement involving only perceptual level checkable predicates and simple boolean operations.

The problems begin when you start introducing quantifiers.

Consider the statement, “forall x, P(x, a)”. If this statement is true, then it cannot be proven to be true at the perceptual level if x ranges over infinitely many entities. Just because you’ve checked that it’s true for all the instances you’ve seen so far, there is no guarantee that the next entity will not fail the check.

Consider the statement, “exists x, P(x, a)”. If this statement is false, then it cannot be disproven at the perceptual level if x ranges over infinitely many entities. Just because all the entities you’ve checked so far don’t pass the check, there is no guarantee that then next entity will not pass the check.

Now consider the statement, “forall x, exists y, P(x, y)”. This statement cannot be checked at the perceptual level at all. Let’s say you try to reduce it to the perceptual level by instantiating the variable x and y with constants a and b, respectively, to get P(a, b). Let’s say that P(a, b) is true. Well so what? You still have infinitely many more instantiations of the x variable to go before you can prove it true. And let’s say that P(a, b) is false. Again, so what? It just means you haven’t looked hard enough to find an appropriate instantiation of y.

The problems get worse and worse the more alternating quantifiers there are.

The point here is that, for all but the simplest statements, reduction to the perceptual level will accomplish nothing but a headache. In order to deal with complex facts, a background theory consisting of perceptual level facts and judiciously chosen hypotheses is absolutely necessary. 

Here, Randian definitions can help somewhat by reducing the length of proofs, but the meat of cognitive economy lies in the background theory and its deductions.

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Insofar as “concept” specifies a property of the mind, hyper-conceptual modeling of “concepts” in computational terms is otiose if we cannot reduce the models to concrete claims about the mind. The idea of a concept being “wrong” is so bizarre that I have no idea what that would actually mean.

Hyper-conceptual modeling is my favorite pastime, but expand more on this since I’m not sure what you’re trying to say.

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@SpookyKitty Does every word, as in use, not as in consideration, enlist a concept in its success at meaning in the word? Do prepositions in use have an enlisted concept satisfying your criteria for a concept? For example, can you work through how each of your conditions are satisfied for the preposition “of”? Rand tried to outline how prepositions fall within her theory of concepts. But she took them all to be as fundamentally operations of the mind, which I think false in the case of conjunction.

Does your group of criteria on concepts entail that they all can stand in hierarchical taxonomic relations? Or anyway, is your group compatible with that taxonomy circumstance?

When I think of the topic of concepts in recent decades, the names coming to mind are Christopher Peacocke’s A Study of Concepts and Jesse Prinz’s Furnishing the Mind. You might like to see how what you have distilled fits with what they got. Also, Meir Buzaglo’s The Logic of Concept Expansion and, concerning the developmental psychology of concepts and smoothness with your formula, Susan Carey’s The Origin of Concepts.*

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8 hours ago, SpookyKitty said:

How would the algorithm access the actual truth of the matter? Like sure, an algorithm could output some answer (and possibly the correct one) based on the inputs, but how could it guarantee that it is the correct answer?  No mind and no (real) algorithm has access to an infallible truth oracle.

An algorithm cannot access actual truth, but we have senses that allow us to inspect reality directly. (If an algorithm can be given such senses, and if it uses them, then it, too, can inspect reality directly.)

8 hours ago, SpookyKitty said:

if you define potatoes in terms of their properties, then what are you using to define the concepts corresponding to those properties? Properties of properties? I’m sure you see the problem here.

The only way to prevent the infinite regress is to stop with the self-evident, which requires the use of the senses.

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7 hours ago, SpookyKitty said:

My position is that her theory fails to meet its own goals. The main point of the theory is to explain how human minds can generate or represent a potential infinity of non-trivial true statements using only finitistic means

Your conclusions about her theory are incorrect and not supported by the facts. My personal concern is with the mechanisms of the human mind which generate form-meaning pairings (a superset of what you attribute to Rand). I call on Rand’s theory as appropriate to do so, but I sure as heck do not impute to Rand my own goals. So it sounds to me like you do not offer anything by way of criticism of her theory of concepts, you criticize her for not living long enough to develop a full blown scientific theory of cognition (a criticism that can be leveled against all cognitive scientists). However also peruse ITOE to see what she says about science vs. philosophy. My “criticisms” of her theory are every limited and they are not philosophical, they are scientific.

So maybe the appropriate target of your comments is really Binswanger’s HWK or Kelley’s Concepts and Propositions, but again even these do not approach a fully integrated scientific theory of the faculty of reason, language being one aspect of that faculty. You claim that her theory “cannot account for how infinitely many distinct perceptual level facts can be given a finite representation”. Her theory does not even purport to set forth an implementation device. Instead, Rand states what the sea creatures are in the domain of reasoning (ref. Eddington) – and not all of the sea creatures, just the concepts.

Her theory does not “allow for” a degree of cognitive economy (which you begrudgingly term “small”), it demands it. This is in sharp contrast to many extant “theories” of “cognition” which presuppose near-infinite storage, experience and computational capacity, then try to create devices that superficially behave like humans do but using totally different means.

Although Rand does not present the code necessary for scripting a perceptual parser, she does give enough details of the logic of the “how”, so that one can say what it is that the program should do. In the realm of “how”, Rand discusses “concept formation” which I find to be scientifically unfortunate, but logically somewhat reasonable. It’s only when you leave philosophy and engage in the science that you see the actually problem – essentially, concepts exist, and they are acquired. There is another branch of science that weakly investigates actual creation, namely “history of science”.

Let’s turn to your example of the perceptual-level facts. For this discussion we must exile God’s-perspective claims including the ability to see the future. Therefore we cannot have a perceptual-level facts that can be encoded as 1001001101001010011100011. This can only arise after a human has formed very many concepts and has developed a scientific theory of “the universe”, proposing a universal descriptive calculus. For the sake of clarity, we can stipulate that 1001001101001010011100011 is the code generated by hearing a specific instance of sound of the vowel [ɑ] (because we actually understand how that works, somewhat), where each bit corresponds to some danged detector and we have at least 25 detectors that are either on or off (I have a wtf question: why are there 25 bits in the first line and 26 in next two? How do you know that the second fact is not two smaller facts, 11100110100101 and 010100100111? Are you suppressing a separate “information-length” field?). This is a wildly nonsensical theory of perception (the human mind is not a bank of myriad hyper-specific fixed detectors), but I’ll play along with the premise for a while.

What you have in your first line is a highly stripped-down implementational model of “similar” and “different”. This is not Rand’s model. You assert, yet give no supporting evidence, that her theory “allows you to create tokens for finite, fixed symbolic sequences”. More accurately, her theory does not strongly forbid creating tokens for finite, fixed symbolic sequences. One has to apply massive imagination and factual support, not just pure imagination, to make a proposal that is logically consistent with Rand’s theory (and even then, you cannot claim that this conclusion follows from Rand’s theory, unless you can somehow prove that no alternative is even conceivable). To reiterate, you mis-state the scope of Rand’s claim.

So we have a perceptual fact, queued up for conceptual identification. We also need a similar perceptual fact that is within the specific conceptual range, and a “foil”, a conceptually different perceivable fact. Therefore we also have 1001001100001010011100011, which is a perceivably different instance of the sound of the vowel [ɑ] – differs on the 10th bit from the left – and we also have 1001001111001010011100011 which is a perceived instance of the sound of the vowel [æ] – differing on the 9th bit. The measurement being omitted is the 10th bit, leading to the unification of the first and second tokens, and the difference is the 9th bit, which is the perceptual basis of the two conceptual classifications [ɑ] versus [æ].

This is, of course, just invented to show you how Rand’s theory could have evolved into a specific scientific theory. We now have a toy model of similarity and difference. The next obvious question to address is how sensations give rise to perception (recall that Rand presupposes perception). If you have an account of how sense organs “do perception”, I’m all ears, as would be every cognitive scientist.

Obviously, I didn’t get very far into your objections, because we need to flesh out the ontological status of your ones and zeroes (and what are those “a” things?).

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Hyper-conceptual modeling is my favorite pastime, but expand more on this since I’m not sure what you’re trying to say.

The above should give you the essential flavor: insofar as the Objectivist epistemology is about the relationship between the mind and existents, we need to start with that which exists, not a floating abstraction saying how you might model something with no specific properties.

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3 hours ago, Boydstun said:

@SpookyKitty Does every word, as in use, not as in consideration, enlist a concept in its success at meaning in the word? Do prepositions in use have an enlisted concept satisfying your criteria for a concept? For example, can you work through how each of your conditions are satisfied for the preposition “of”? Rand tried to outline how prepositions fall within her theory of concepts. But she took them all to be as fundamentally operations of the mind, which I think false in the case of conjunction.

I don’t know. This subject is too complicated for me to address at this time.

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Does your group of criteria on concepts entail that they all can stand in hierarchical taxonomic relations? Or anyway, is your group compatible with that taxonomy circumstance?

Yes. And there are many interesting hierarchies one can examine here. But this is a subject for future threads.

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When I think of the topic of concepts in recent decades, the names coming to mind are Christopher Peacocke’s A Study of Concepts and Jesse Prinz’s Furnishing the Mind. You might like to see how what you have distilled fits with what they got. Also, Meir Buzaglo’s The Logic of Concept Expansion and, concerning the developmental psychology of concepts and smoothness with your formula, Susan Carey’s The Origin of Concepts.*

Thank you for the reading list. I’ll look into it.

3 hours ago, necrovore said:

An algorithm cannot access actual truth, but we have senses that allow us to inspect reality directly. (If an algorithm can be given such senses, and if it uses them, then it, too, can inspect reality directly.)

Unfortunately, there is no “potato” sense or a “justice” sense or a “rational” sense. This is what you would need in order to make the approach you described in your last post work.

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The only way to prevent the infinite regress is to stop with the self-evident, which requires the use of the senses.

The problem here is that, since it looks like you are going up the hierarchy of knowledge in that example, you are moving further and further away from the self-evident as you go.

@DavidOdden

It is absurd to claim that Rand did not try to explain how we use limited minds to acquire unlimited knowledge. Her works make constant references about the fact that concepts are what makes this happen:

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A concept is a mental integration of two or more units which are isolated by a process of abstraction and united by a specific definition. By organizing his perceptual material into concepts, and his concepts into wider and still wider concepts, man is able to grasp and retain, to identify and integrate an unlimited amount of knowledge, a knowledge extending beyond the immediate concretes of any given, immediate moment.

-Ayn Rand, The Psychoepistemology of Art

I think you’ve misunderstood my example a little bit, though I suspect it’s not as clear as I had hoped it would be.
Consider the following one instead:
 

“A man carried a table on which another man, who was himself carrying a table, was standing.”

The definition of “man” is “rational animal”.
The definition of “table” is "an item of furniture, consisting of a flat, level surface and supports, intended to support other, smaller objects”.

If you use these definitions, you can expand out the above statement to:

“A rational animal carried an item of furniture, consisting of a flat, level surface and supports, intended to support other, smaller objects on which another rational animal, who was himself carrying an item of furniture, consisting of a flat, level surface and supports, intended to support other, smaller objects, was standing.”

Keep expanding out the definitions of each of the terms in the above statement as far as you can. You will eventually get an absurdly long sequence of alphanumeric characters in some language that represent a fact at the perceptual level. This is the appropriate analog of the binary sequences in my previous example. The ‘a’'s ‘b’'s and ‘c’'s in my example are supposed to be analogous to concepts.

These examples are not intended to investigate concept formation or similarities or diffeences. They are intended to measure the kind and degree of cognitive economy.

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2 hours ago, SpookyKitty said:

It is absurd to claim that Rand did not try to explain how we use limited minds to acquire unlimited knowledge. Her works make constant references about the fact that concepts are what makes this happen

There is a difference between lightly mentioning the logical relation between concepts and memory, and saying what method is used by the human mind to accomplish this. “How” is about the actual method of X, “how” merely presupposes X. Concepts are part of the presupposed background of the theory of acquisition or processing, or whatever it is that you are after.

Based on your current statement, it seems that your problem is not with concepts, it is with propositions, where as is well known Rand does not provide a theory of propositions. Your example about table carrying is not a concept, it is a sentence, and you are attempting to use substitution to “validate” a sentence by plugging in other phrases that reasonably resemble definitions of words in the original sentence. It’s only a partial substitution, but good enough for a first attempt, although that is not how sentences are understood.

I think you have pretty much made Rand’s point, that without concepts (or with only first order concepts), humans could not formulate propositions that could be retained in memory. But there is absolutely no pretense that Rand presented a theory of natural language production, comprehension, or even propositions. So what exactly is your complaint?

NB your example based on just a string of letters is doomed to fail because propositions must countenance recursion. Stick in some brackets.

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Spooky Kitty, why are you taking such a formalized approach?

 

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7 hours ago, DavidOdden said:

I think you have pretty much made Rand’s point, that without concepts (or with only first order concepts), humans could not formulate propositions that could be retained in memory. But there is absolutely no pretense that Rand presented a theory of natural language production, comprehension, or even propositions. So what exactly is your complaint?

My complaint, I will state it again, is, as we can clearly see from the way Rand defines the notion of “concept”, it being severely limited in the ways that I’ve explained above, there is simply no way that her epistemology can possibly account for the degree of cognitive economy that it purports to.

None of this has anything to do with natural language production, the composition of meanings and other things that neither Rand nor I addressed. It has to do, fundamentally, with what she thinks concepts are. The kind of thing that she thinks concepts are is not and can never be sufficient to capture the breadth and depth of human knowledge.

Therefore, we need a different definition of “concept”.

12 hours ago, DavidOdden said:

The above should give you the essential flavor: insofar as the Objectivist epistemology is about the relationship between the mind and existents, we need to start with that which exists

Well I’m starting with concepts. Are you saying that concepts don’t exist?

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not a floating abstraction saying how you might model something with no specific properties.

What floating abstraction? Why would I model something with no specific properties? What is your complaint? I’m more confused about it than ever.

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2 hours ago, Doug Morris said:

Spooky Kitty, why are you taking such a formalized approach?

So that confusion will be minimized.

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11 hours ago, SpookyKitty said:

My complaint, I will state it again, is, as we can clearly see from the way Rand defines the notion of “concept”, it being severely limited in the ways that I’ve explained above, there is simply no way that her epistemology can possibly account for the degree of cognitive economy that it purports to.

You implied that you had an alternative theory of concepts. The logically prior question is “So what?”, the answer might be (a) It amuses me, (b) there is an empirical flaw in the existing theory or (c) there is a conceptual flaw in it. These are the three intellectually-legitimate reasons for proposing an alternative. To be more specific about (a), it is also a legitimate exercise to determine that some equally-good alternative is possible, which is different from trolling where one is amused at how clever one is at evading the question. Whatever rationale you have, an absolute prerequisite for any theory comparison is having a correctly-represented standard against which the competitor is measured.

The standard of comparison is Rand’s theory of concepts, a specific thing that we know. It is not a conjectured theory of logic or propositions, which Rand did not write. It is nonsense to criticise her for not presenting a theory of symbolic logic and propositions, because that was not her goal.

As I have explained above, by definition Rand’s theory of concepts achieves cognitive economy, which is a fundamental insight into concepts. Hundreds of millions of instances of dog can now be entered into propositions, sentences and discussions simply by uttering the word “dog”. A huge savings, it even allows you to refer to things that you have not directly experienced! ITOE serves as an inspiration to further cognitive economization, where we can understand recursion in terms of cognitive economy – I won’t bother to list all of the ways in which “standing for” and “abstraction” play an essential role in human cognition.

Your newest line of attack is against what you think Rand thought concepts are. Presumably you can cite parts of ITOE which might even accurately summarize her view. You complain that “The kind of thing that she thinks concepts are is not and can never be sufficient to capture the breadth and depth of human knowledge”. So what? Concepts are a necessary fact, not a sufficient fact. It seems that you have not been paying attention to what I’ve been telling you, that her theory is the foundation of a theory of human knowledge, not a complete theory itself. I have invited you to contemplate subsequent contributions by Binswanger and Kelley which move the project forward.

There does exist a real problem that has been ignored by all, that the mind does not operate in terms of symbolic propositions as you might learn in Phil. 250. An expression like (∀Fi∈ G ([(Fn)1] ≠ [(Fn)2] has no referent in the real world (it is a floating abstraction, a formula not related to any existent).

10a1aa11c11, b0011a0101a00b, c00ada00 are meaningless, they have no referent. Perhaps you have in mind some secret referent and can point us to an instance of 10a1aa11c11, but I’m not interested in playing a guessing game, I imagine there are puzzle-solving fora where people get off on that sort of stuff. Objectivism take existence as primary, and cognitive analysis to depends on existents. Therefore, you need to identify what thing 10a1aa11c11 represents, likewise b0011a0101a00b, then we can evaluate Rand’s theory of that representation versus yours.

 

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58 minutes ago, DavidOdden said:

You implied that you had an alternative theory of concepts. The logically prior question is “So what?”, the answer might be (a) It amuses me, (b) there is an empirical flaw in the existing theory or (c) there is a conceptual flaw in it. These are the three intellectually-legitimate reasons for proposing an alternative. To be more specific about (a), it is also a legitimate exercise to determine that some equally-good alternative is possible, which is different from trolling where one is amused at how clever one is at evading the question. Whatever rationale you have, an absolute prerequisite for any theory comparison is having a correctly-represented standard against which the competitor is measured.

(c) There is a conceptual flaw in it.

Quote

Your newest line of attack is against what you think Rand thought concepts are. Presumably you can cite parts of ITOE which might even accurately summarize her view. You complain that “The kind of thing that she thinks concepts are is not and can never be sufficient to capture the breadth and depth of human knowledge”. So what? Concepts are a necessary fact, not a sufficient fact. It seems that you have not been paying attention to what I’ve been telling you, that her theory is the foundation of a theory of human knowledge, not a complete theory itself. I have invited you to contemplate subsequent contributions by Binswanger and Kelley which move the project forward.

Yes, concepts are necessary. Yes, Rand’s theory does explain some degree of cognitive economy. No, it is not sufficient. Yes, that is a serious problem, because no, it cannot serve as a foundation of a theory of human knowledge. And this is because there is a serious conceptual flaw in it.

Is my position clear now?

Quote

There does exist a real problem that has been ignored by all, that the mind does not operate in terms of symbolic propositions as you might learn in Phil. 250. An expression like (∀Fi∈ G ([(Fn)1] ≠ [(Fn)2] has no referent in the real world (it is a floating abstraction, a formula not related to any existent).

10a1aa11c11, b0011a0101a00b, c00ada00 are meaningless, they have no referent. Perhaps you have in mind some secret referent and can point us to an instance of 10a1aa11c11, but I’m not interested in playing a guessing game, I imagine there are puzzle-solving fora where people get off on that sort of stuff. Objectivism take existence as primary, and cognitive analysis to depends on existents. Therefore, you need to identify what thing 10a1aa11c11 represents, likewise b0011a0101a00b, then we can evaluate Rand’s theory of that representation versus yours.

You seem to have a lot of trouble grasping meta-theoretical abstractions, so I will guide you through it by the hand.

First, write out a list of facts. It can be as long or as short as you want, but it must have some length. Abstract out the language these facts are written. All of them must be written in some language (and the same language, preferably) but can be written in any language. All languages have words consisting of symbols of some sort which correspond to concepts and these concepts have definitions. We omit the specific nature of the concepts and their definitions themselves, and we retain only the fact that a concept is a piece of cognitive data that is “defined by” another, longer piece of data. We also omit the specific nature of the referents of each of these concepts, retaining only the fact that they must have some referents.

Next, take this abstract list of abstract facts and expand out all of the abstract words of this abstract language with the abstract definitions of the abstract concepts as far as it is possible to go. Since all concepts eventually reduce to the perceptual level, the above abstract process must terminate in a list of extremely long sequences of abstract symbols (standing for abstracted perceptual level and abstracted axiomatic concepts) representing abstracted perceptual level facts. These are the 10001001011011011’s.

You will now have an arbitrarily long list of extremely long abstracted perceptual level facts. By using the abstracted definitions of the abstracted concepts, i.e.:

concept: ‘a’
definition: ‘01110010101’

You can shorten each entry in this list a considerable deal. This is the kind and degree of cognitive economy that Rand’s theory can account for.

BUT

This abstract mechanism cannot shorten the list itself! If your list started out with 19,392 entries, then, after you do as much cognitive economizing as Rand’s theory can account for, you will remain with a list of exactly 19,392 entries.

And yet, we know that the list can be made much shorter. In fact, it is possible to represent an infinitely long list of facts using only finitely many symbols. This is what human reason can do, and Rand’s theory cannot explain this phenomenon.

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Peikoff said in his lectures about induction that he doesn’t think induction can be done symbolically. Formal systems are usually only deductive. (The only exception I can think of is “proof by mathematical induction” and I think that’s a special case.) In general, you can’t write an “inductive syllogism.”

The system I described earlier in this thread (where concepts are described as functions) is incomplete because it doesn’t include a mechanism for induction but rather shows how the results of that mechanism might be usefully organized.

An old TRS-80 can do deduction faster than we can. However, so far, the only way machines can begin to perform induction is through neural networks and such (which means they are doing emulations of what we have been doing). It has only recently become practical to implement neural networks large enough to even try this.

Induction is very different from deduction in several respects, and one respect in particular is that induction requires access to a very large data set (whereas a deductive syllogism has only a few propositions). The size of the neural networks necessary to attempt induction is related to this.

I also think that induction can be a “trial and error” thing where it might take several attempts at forming a concept or a generalization in order to get it right. Inductive reasoning has to be checked against reality. There is also a feedback loop in play where, through experience (of reality), you can refine your concepts and generalizations, to make them even more accurate.

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49 minutes ago, necrovore said:

I also think that induction can be a “trial and error” thing where it might take several attempts at forming a concept or a generalization in order to get it right. Inductive reasoning has to be checked against reality. There is also a feedback loop in play where, through experience (of reality), you can refine your concepts and generalizations, to make them even more accurate.

Indeed. I’m going to make a thread about induction at some point in the future. The definition in the OP allows for two kinds of induction which feed each other.

The first, induction over concepts, leads to the discovery of concepts which terminate on more input parameters, thereby forcing the background theory to undergo more tests. This will lead to the faster rejection of false premises.

The second, induction over background theories, leads to the discovery of more accurate theories. In the short term, the effect on concepts is unpredictable, it can make them terminate on more or less input parameters. In the long term, however, more and more premises will have to be added to the theory, meaning more sentences will be provable in the theory, meaning that concepts will be able to terminate on more and more input parameters, making them “sharper”.

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6 hours ago, necrovore said:

Formal systems are usually only deductive. (The only exception I can think of is “proof by mathematical induction” and I think that’s a special case.)

It’s not an exception at all.  Mathematical induction is a form of deduction, not a form of induction.

In particular, mathematical induction does not require access to a very large data set.

 

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1 minute ago, Doug Morris said:

In particular, mathematical induction does not require access to a very large data set.

It requires access to the set of natural numbers, which is infinite, Q.E.D. :stuck_out_tongue:

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7 hours ago, SpookyKitty said:

You can shorten each entry in this list a considerable deal. This is the kind and degree of cognitive economy that Rand’s theory can account for.

BUT

This abstract mechanism cannot shorten the list itself! If your list started out with 19,392 entries, then, after you do as much cognitive economizing as Rand’s theory can account for, you will remain with a list of exactly 19,392 entries.

Suppose our long list includes:

"SpookyKitty has the faculty of reason.

Doug Morris has the faculty of reason.

Necrovore has the faculty of reason.

DavidOdden has the faculty of reason."

 

If we replace this with

“Human beings have the faculty of reason.”

this shortens the long list and includes more than the original list included.

 

Isn’t this an example of cognitive economizing that Rand’s theory can account for?

 

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