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SpookyKitty

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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