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# The Anti-Concept of Anti-Reference; Paradox

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Consider the following:

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(Definiton 1)The concept of "reference" refers to all pairs <C,r> such that C is a concept with at least one referent, and r is a referent of C.

(Definion 2)The concept of "anti-reference" refers to all pairs <C,r'> such that C is a concept with at least one non-referent, and r' is not a referent of C.

(Proposition 3)The concepts of "reference" and "anti-reference" are mutually exclusive.

Proof of Proposition 3: Suppose that C is a concept and let r be such that <C,r> is a referent of both "reference" and "anti-reference". But then, r is and is not a referent of C. A contradiction.

Does anti-reference refer to itself?

Suppose that it does.
Let A be the concept of "anti-reference". Then, <A,A> is a referent of "reference" since "reference" refers to all pairs <C,r> where r is a referent of C. But since (3), <A,A> cannot be a referent of "anti-reference". A contradiction.

Suppose that it does not.
Then, <A,A> is not a referent of "reference". But since (3), <A,A> is a referent of "anti-reference". A contradiction.

What this proof shows is that the concept of "anti-reference", i.e., the non-meaning of a concept is an anti-concept since it always leads to a contradiction.

It seems therefore that we cannot always tell what a concept does not refer to even if we can always tell to what it does refer to.

Edited by SpookyKitty

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How would floating concepts be defined in your framework of C's and r's?  What about that which is meaningless or incoherent or nonsensical?

Does "r" need to be "real", or can it be "imaginary"? Can "r" and or "c" be incoherent or meaningless or nonsensical?

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3 minutes ago, StrictlyLogical said:

How would floating concepts be defined in your framework of C's and r's?  What about that which is meaningless or incoherent or nonsensical?

Does "r" need to be "real", or can it be "imaginary"? Can "r" and or "c" be incoherent or meaningless or nonsensical?

Good questions.

A floating concept is one which has no referents. And so the collection of pairs <C,r> for a floating concept would be empty. Similarly, for the meaningless or incoherent or nonsensical.

I don't think that the referents of concepts have to be "real". For example, the concept of "fictional characters" doesn't contain any "real" referents by definition. But if the referents of a concept are not in external reality, then they must at the very least be mental entities.

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I realize that Rand meant something different by the term "anti-concept". Here, by "anti-concept" I mean one whose existence would imply a contradiction.

Hence, it is a type of concept which cannot exist.

EDIT: But then of course its use would be an "anti-concept" in the Randian sense.

Edited by SpookyKitty

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19 hours ago, SpookyKitty said:

(Definion 2)The concept of "anti-reference" refers to all pairs <C,r'> such that C is a concept with at least one non-referent, and r' is not a referent of C.

The anti-referent(s) of the concept "anti-reference" are all concepts C with their referents r.   Anti-reference applied to itself is a double negative, so non-non r is simply r.   The anti-referent of the concept "anti-reference" is the concept "reference" by definition 1.  "Reference" refers to itself without contradiction.

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I don’t believe that your definition of “reference” is correct: perhaps you could persuade me. “Reference” in the relevant sense is “the act of referring”. We should dig deeper into what things “refer”, but as a start, expressions refer. Not all expressions are concepts. “The new occupants of the White House” refers to real people, and those people are the referents of the expression, but “The new occupants of the White House” is not a concept. If you want a set-theoretic definition of “reference”, it should be the set of all expressions of any type, paired with their referents (plural or singular). You might coin a word “word-reference” which specifically refers to just concepts and the things they refer to. In that case, r is a set, not an individual (it’s not a singular referent, it’s all of the referents).

We can mostly set aside the concept of “reference” (though not the matter of what refers), because it is irrelevant to cooking up and evaluating the invalid concept “anti-reference” (it’s relevant to the proof of contradiction). “Anti-reference” could almost qualify as a label, although again it should be “word-anti-reference” if the goal is to only look at a kind of referring relation of concepts, and not those of everything that refers (briefly: denial of a proposition is not invalid). Because we need to evaluate the potential legitimacy of the putative concept qua concept, the label needs to be replaced so that there is no surreptitious smuggling in of ideas from other, valid concepts. For the sake of clarity, we should call this concept “glank”.

A glank is the complement of the referents of a concept – everything that a concept does not refer to. An example of a glank would be a relationship between “dog” and the universe (not just things, but also abstractions, and any other fact such as the fact that adding baking soda to vinegar causes the mix to foam up) – it refers to everything except for dogs.

It is cognitively valid to assert the proposition “this is a dog”, and it is equally valid to deny that proposition. The denial of a proposition is not automatically a concept. We do have valid method-concepts that pertain to denial – “denial, exclusion, contradiction, complement”. We can easily construct an expression which identifies the glank of a concept, using ordinary language expressions such as “everything that is not a dog”. The question is whether the word “glank” does something that makes it superior to the compositional expression “everything that is not”.

In order for this monster glank to be elevated to the status of a concept, it needs cognitive validity, some purpose. There may be a narrow professional context (anti-cognitivist logicians) where it is useful to be able to quickly say “the complement of the concept C with respect to all existence”, so that instead of constantly saying “the cardinality of the complement of the concept ‘dog’ with respect to all existence is identical to the cardinality of the complement of the concept ‘run’ with respect to all existence”. Instead, philosophers could more efficiently say “the cardinality of the glank of dog is identical to the cardinality of the glank of run”.

This would not suffice. “Glank” was cobbled together to relate concepts and things that they don’t refer to, but the complement relationship is broader, so we need to create “florn”, which is the complement of the facts that any expression identifies. Thus the florn of “everything that is not a dog with blue eyes and grey fur” is, simply, the universe, minus those dogs that have both blue eyes and grey fur. A glank is a florn where the expression is a word.

The florn of “a dog with blue eyes and grey fur” includes all expressions (sentences, clauses and words are not actual dogs of that type), all actions (running is not a dog), all cats, rocks etc., and all dogs which don’t have blue eyes or don’t have grey fur. The florn of a dog (an actual dog) is undefined, because an actual dog is not a linguistic expression, and “florn” takes an expression as its argument. Similarly, “reciprocal of blue” is undefined. Since the florn of “dog” is not an expression, the florn of the florn of “dog” is likewise undefined. In your proof of contradiction, you don’t distinguish between A and “A”, which is a problem.

Since we can identify what a concept refers to, we can evaluate the proposition “the concept A does not refer to X”. But we are not directly aware of all existents that a concept refers to, nor are we directly aware of all existents that are not instances of that concept. Regarding your final conclusion, is your point that we are not aware of all referents (are not omniscient)? If not, I don’t see wherein lies the problem with evaluating the denial of a proposition.

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12 hours ago, DavidOdden said:

I don’t believe that your definition of “reference” is correct: perhaps you could persuade me.

It would help to know what aspect(s) of it you take issue with.

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... glank ... florn ... dogs, etc.

I'm not at all sure what point you are trying to make here.

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In your proof of contradiction, you don’t distinguish between A and “A”, which is a problem.

What is the difference between A and "A"? What do you mean by the quotation marks?

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Since we can identify what a concept refers to, we can evaluate the proposition “the concept A does not refer to X”.

No, we can't, which is my point. If we could, then we would end up in a contradiction as the OP shows.

What happens is that we can verify that A refers to X if, indeed, A refers to X, but if A does not refer to X, then in at least some instances we cannot verify that A does not refer to X.

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But we are not directly aware of all existents that a concept refers to, nor are we directly aware of all existents that are not instances of that concept. Regarding your final conclusion, is your point that we are not aware of all referents (are not omniscient)? If not, I don’t see wherein lies the problem with evaluating the denial of a proposition.

Since we cannot always identify the non-referents of a concept, that means that we cannot always evaluate the denial of a proposition.

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The sentence ""Yields a falsehood when appended to its own quotation" yields a falsehood when appended to its own quotation" leads to a contradiction even though all the concepts used in it are valid concepts.  What we need to do in cases like this is not to reject certain concepts but to recognize that certain sentences are inherently circular and therefore logically invalid.   This also applies to certain questions, such as "Does anti-reference refer to itself?".  Such statements are not true, but they are not ordinary falsehoods; in particular, their negations are not true either.  Such questions cannot be correctly answered "yes" or "no"; they must be answered by pointing out their circularity.

Edited by Doug Morris

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3 hours ago, Doug Morris said:

The sentence ""Yields a falsehood when appended to its own quotation" yields a falsehood when appended to its own quotation" leads to a contradiction even though all the concepts used in it are valid concepts.

Actually, it does not lead to a contradiction, it leads to a paradox. That statement cannot be evaluated as either true or false, whereas contradictions always evaluate to false.

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What we need to do in cases like this is not to reject certain concepts but to recognize that certain sentences are inherently circular and therefore logically invalid.   This also applies to certain questions, such as "Does anti-reference refer to itself?".  Such statements are not true, but they are not ordinary falsehoods; in particular, their negations are not true either.  Such questions cannot be correctly answered "yes" or "no"; they must be answered by pointing out their circularity.

This argument does not work, because there are sentences which are self-referential and can be evaluated.

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I should have made clear that both the "yields a falsehood" sentence and the question "Does anti-reference refer to itself?" from the starting post lead to contradictions if evaluated naively, and both should be rejected as inherently circular ("paradoxical").

A statement that refers to its own logical properties such as truth, falsity, provability, or unprovability is inherently circular and therefore invalid.  A statement that refers to other things about itself, such as "This is a sentence", "This is not a sentence", "This sentence is six words long", "This sentence is twenty-two words long", "This sentence is in English", "This sentence is in Swahili", "This sentence makes a statement about a sentence", or "This sentence names and analyzes the current President of the United States" can be evaluated as true or false.

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On 12/15/2017 at 2:49 PM, SpookyKitty said:

(Definition 1)The concept of "reference" refers to all pairs <C,r> such that C is a concept with at least one referent, and r is a referent of C.

(Definition 2)The concept of "anti-reference" refers to all pairs <C,r'> such that C is a concept with at least one non-referent, and r' is not a referent of C.

I was wondering if a concept (not just a valid concept) can only have zero or more than 1 referent. So referents have to be 2 or more to not contradict the definition of a concept.

I believe a concept that has one referent ends up being a particular. I suppose it is a mistake rather than a concept. So that would mean there is no such thing as a concept with one referent.

Since the nature of a concept is based on commonality, commonality requires at least two particulars that have the commonality.

It is also possible that zero referents would indicate that it is not a valid concept although it is a concept. Imaginary, fictitious and contradictory commonalities can fit that.

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Some concepts do not have meaning in reality but do have meaning in some fictional, folkloric, or other imaginary contexts.  "Unicorn" is a good example of this.

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2 hours ago, Doug Morris said:

Some concepts do not have meaning in reality but do have meaning in some fictional, folkloric, or other imaginary contexts.  "Unicorn" is a good example of this.

Isn't that an example of a concept that has zero referents? There are no unicorns in reality outside of consciousness.

It is still a concept, it is invalid in a sense that it has zero referents. But I was thinking, another invalid concept would be "Joe the Horse". That is a particular, a concrete as opposed to a concept. I would like to be able to distinguish between the two types of invalidity.

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58 minutes ago, Easy Truth said:

..."Joe the Horse"...

A horse can be named Joe. Horses can be named anything.

Proper nouns and pronouns are ways to refer to particulars using the mainly conceptual faculty of language.  For thinking to be useful at all it is necessary to move from abstract thinking back to the particulars in front of us.

"Words transform concepts into (mental) entities" and "Proper names are used in order to identify and include particular entities in a conceptual method of cognition." Both quoted sentences are Rand from IOE2 page 11.

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11 minutes ago, Grames said:

A horse can be named Joe. Horses can be named anything.

Proper nouns and pronouns are ways to refer to particulars using the mainly conceptual faculty of language.  For thinking to be useful at all it is necessary to move from abstract thinking back to the particulars in front of us.

Agreed. What I meant was "Joe the horse" resolves to one particular.

As in "The current president of the US", resolves to a particular, one single thing.

"Presidency in the US" is a concept, it refers to many things so by definition it is a valid concept and in addition it has instances.

"Presidency in Saudi Arabia" is a concept with a null set of instances, so it is "referentially" invalid, but it is a valid concept by definition. There is no president of Saudi Arabia as it has a king.

Perhaps, I have the definition of concept wrong, is that what you are saying?

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"Unicorn" is a concept that has no referent in reality but does have referents in certain fictional and folkloric contexts.

"Darth Vader" is a proper name that has no referent in reality but does have a referent in at least one fictional context.

I don't see much difference between them except that one is a concept and one is a proper name.

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1 hour ago, Doug Morris said:

"Unicorn" is a concept that has no referent in reality but does have referents in certain fictional and folkloric contexts.

"Darth Vader" is a proper name that has no referent in reality but does have a referent in at least one fictional context.

I don't see much difference between them except that one is a concept and one is a proper name.

Then you consider "Darth Vader" a concept even though there is only one Darth Vader.

I think you are considering "mental entity" to be a concept/universal. I divide them into two types, concepts, and particulars based on if they refer to one thing or a (unified) many.

You can have a mental entity that has a one to one relationship with what is real (an association). The referent is a concrete. There is only one "Ayn Rand", the idea "Ayn Rand" refers to one entity. There are no instances that are a type of "Ayn Rand" (the person). Or are you saying she is, in fact, a concept?

California is not a concept, it is one entity. A US state is a concept.

If you are mean "Darth Vader the archetype" you would have a point. One could say he is a Darth Vader type of person, one who is powerful and seduced by the dark side.

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As a reminder, " 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". Unicorns and irrational numbers exist, though they are not physical things. Fiction especially science fiction relies heavily on imaginary concepts. The similarity between a fictional individual and a fictional concept is that they are both fictional.

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Easy Truth,  I thought I made it clear that "Darth Vader" is a proper name, not a concept.

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DavidOdden,

Numbers and concepts are both mental constructs for dealing with reality, and both exist as such, although they are not physical objects.

Actual unicorns do not exist in the real world.  They exist in certain imaginary worlds.  In the real world we have contents of consciousness relating to unicorns, and we have actual physical objects that are representations of unicorns.

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5 minutes ago, Doug Morris said:

Easy Truth,  I thought I made it clear that "Darth Vader" is a proper name, not a concept.

35 minutes ago, DavidOdden said:

As a reminder, " 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". Unicorns and irrational numbers exist, though they are not physical things. Fiction especially science fiction relies heavily on imaginary concepts. The similarity between a fictional individual and a fictional concept is that they are both fictional.

Are two or more units implying two or more referents?

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21 hours ago, Easy Truth said:

Agreed. What I meant was "Joe the horse" resolves to one particular.

As in "The current president of the US", resolves to a particular, one single thing.

"Presidency in the US" is a concept, it refers to many things so by definition it is a valid concept and in addition it has instances.

"Presidency in Saudi Arabia" is a concept with a null set of instances, so it is "referentially" invalid, but it is a valid concept by definition. There is no president of Saudi Arabia as it has a king.

Perhaps, I have the definition of concept wrong, is that what you are saying?

A proper noun such as "Joe the Horse" is not an invalid concept.  This is because it is not even a concept.  Also, the alternative of valid or invalid does not apply to names.

"Presidency in Saudi Arabia" -  "Presidency" is an abstract high level concept in the area of politics.  As a concept of method it would be a valid concept even if there were no presidents because there were presidents in the past and could be presidents in the future.  "Presidency in Saudi Arabia" can be used validly when advocating a change in the method of governance of Saudi Arabia, even though it is true that there are no current or past presidents in Saudi Arabia.  "The President of Saudi Arabia" could be used validly in a conditional or future tense, but would be nonsensical in the context of current events or the history of S.A.   Context matters.

We have more concepts than we have words for them. A single word can refer to several concepts and the ambiguity is usually resolved by the context and careful writing or speaking.  Actual concretes can be the referents of many different valid concepts.  Concepts of concepts can divided up in several alternative yet valid ways as well. There is no one-to-one correspondence between words and concepts or between concepts and referents.

On 2/5/2018 at 3:24 AM, Easy Truth said:

I was wondering if a concept (not just a valid concept) can only have zero or more than 1 referent. So referents have to be 2 or more to not contradict the definition of a concept.

An invalid concept is still referred to as a concept, because if it wasn't a concept at all no rules would apply and there would be no justification to judge it as valid or invalid.  So in the sense of badly formed and thus invalid concepts there can be a concept with zero referents.

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14 hours ago, Easy Truth said:

Are two or more units implying two or more referents?

For a more detailed presentation of "unit", read the Ayn Rand Lexicon entry on unit. Objectivism does not generally use or rely on the term "referent", which is used in other approaches, and which is not well defined. As long as you don't import anything philosophically sketchy from the term "referent" besides "that which a thing refers to", then it's okay to talk about a "referent".

"Unit" does not imply any act of referring, but concepts do refer, to units (which are existents). The label (word) attached to a concept refers to those existents. To give a concrete example, dog #1 is an existent, and it is a unit, but the dog does not refer to itself – it is itself. The dog's name, such as "Poika", refers to the specific existent, and the word "dog" refers to that existent, as well as many others.

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49 minutes ago, DavidOdden said:

Unit" does not imply any act of referring, but concepts do refer, to units (which are existents). The label (word) attached to a concept refers to those existents. To give a concrete example, dog #1 is an existent, and it is a unit, but the dog does not refer to itself – it is itself. The dog's name, such as "Poika", refers to the specific existent, and the word "dog" refers to that existent, as well as many others.

I see, so you emphasize the meaning of referent as what a word refers to, not the instances in reality that a concept refers to. I see her using it as "instances" here ("stands for" is also used): "A “number” is a mental symbol that integrates units into a single larger unit (or subdivides a unit into fractions) with reference to the basic number of “one,” which is the basic mental symbol of “unit.” Thus “5” stands for |||||. (Metaphysically, the referents of “5” are any five existents of a specified kind; epistemologically, they are represented by a single symbol.)".  Perhaps to be clear I have to say "metaphysical referents".

Bottom line, a unit is an existent, and in our context is metaphysical.

A unit is an existent regarded as a separate member of a group of two or more similar members.

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2 hours ago, Grames said:

An invalid concept is still referred to as a concept, because if it wasn't a concept at all no rules would apply and there would be no justification to judge it as valid or invalid.  So in the sense of badly formed and thus invalid concepts there can be a concept with zero referents.

Ok then based on this the OP's

On 12/15/2017 at 2:49 PM, SpookyKitty said:

such that C is a concept with at least one referent, and r is a referent of C.

I was thinking:

Should say: such that C is a concept with zero or at least 2 referents, and r is a referent of C or just at least two referents.

Zero referents means invalid the way you say, but it is a concept as you say.

One referent (has to be omitted) would mean "not a concept", it would be treating "Joe the horse" as a concept. A concept requires two or more units (referents) to be a concept is what I am pushing for.

But I see the constituent elements of a concept require 2 but when referring, as in conversation, a concept can refer to as many as you want unless you are referent to ALL of the references. (still, need to think about this and feedback would be appreciated)

One of the questions I have is: Is a "plural" automatically a concept? "Horses" is a concept? or is Horse-ness the concept? Or are both concepts?

Edited by Easy Truth
I saw I could be wrong

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