On 3/12/2022 3:17 PM, Richard Damon wrote:
> On 3/12/22 3:08 PM, olcott wrote:
>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>> On 3/10/22 2:22 PM, olcott wrote:
>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>> I am establishing analytic foundationalism. I am doing this in
>>>>>> part by putting semantics back into logic as it was with the
>>>>>> syllogism.
>>>>>
>>>>> Within your delusional framework, perhaps you really do believe
>>>>> what you just said.
>>>>>
>>>>> To me personally people it's nothing more than a (laughable)
>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>> undertaking any project like that, but dream on...
>>>>>
>>>>> Mike.
>>>>
>>>> The following is proven true entirely on the basis of the meaning of
>>>> its words:
>>>>
>>>> Correct Reasoning Derives Truth
>>>> When-so-ever truth preserving operations are applied to an initial
>>>> set of expressions of (formal or natural) language derive another
>>>> expression of language as a necessary consequence then reasoning is
>>>> correct.
>>>>
>>>> Furthermore when-so-ever the above process is applied to an initial
>>>> set of expressions of (formal or natural) language that are known to
>>>> be true (such as Haskell Curry elementary theorems) then we know
>>>> that the derived expression of language is true.
>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>
>>>> Each of the two paragraphs above constitute proofs in correct
>>>> reasoning, they are comparable to valid argument and a sound
>>>> argument in deductive inference. Anything that diverges from the
>>>> above model is not construed as a proof in correct reasoning.
>>>>
>>>> Correct reasoning differs from deductively valid inference in that
>>>> the conclusion must be a necessary consequence of its premises, thus
>>>> the principle of explosion is not allowed.
>>>>
>>>> Validity and Soundness
>>>> A deductive argument is said to be valid if and only if it takes a
>>>> form that makes it impossible for the premises to be true and the
>>>> conclusion nevertheless to be false. Otherwise, a deductive argument
>>>> is said to be invalid.
>>>>
>>>> A deductive argument is sound if and only if it is both valid, and
>>>> all of its premises are actually true. Otherwise, a deductive
>>>> argument is unsound. https://iep.utm.edu/val-snd/
>>>>
>>>> Principle of explosion
>>>> In classical logic, intuitionistic logic and similar logical
>>>> systems, the principle of explosion (Latin: ex falso [sequitur]
>>>> quodlibet, 'from falsehood, anything [follows]';
>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>
>>>> Analytic–synthetic distinction
>>>> The analytic–synthetic distinction is a semantic distinction, used
>>>> primarily in philosophy to distinguish between propositions (in
>>>> particular, statements that are affirmative subject–predicate
>>>> judgments) that are of two types: analytic propositions and
>>>> synthetic propositions.
>>>>
>>>> Analytic propositions are true or not true solely by virtue of their
>>>> meaning, whereas synthetic propositions' truth, if any, derives from
>>>> how their meaning relates to the world.[1]
>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>
>>>> I draw the line of demarcation a little differently thus much more
>>>> sharply. Analytic propositions are still "true or not true solely by
>>>> virtue of their meaning" as long as they do not depend on sense data
>>>> from the sense organs to verify their truth. "Dogs bark" is thus
>>>> analytic.
>>>>
>>>> Instead of the term "synthetic" that has a common meaning entirely
>>>> different than its use in the {Analytic–synthetic distinction} I use
>>>> the term empirical to indicate any expression of language that
>>>> requires sense data from the sense organs to verify its truth. "I
>>>> hear a dog barking" is an example of an empirical expression of
>>>> language.
>>>>
>>>> A theory T is incomplete if and only if there is some sentence φ
>>>> such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question is the
>>>> whole body of analytic truth then unprovable and true are impossible.
>>>>
>>>> It is also self-evident within my specified formal system of
>>>> [correct reasoning] that the only expressions of language that are
>>>> unprovable in this system are also untrue. The formal system of
>>>> [correct reasoning] encompasses the entire body of analytic truth of
>>>> which mathematics and logic are a subset.
>>>>
>>>>
>>>
>>> I will point out a fundamental issue with your 'claim'. Godel's proof
>>> was for logic system sufficiently powerful to express the Truths of
>>> the Natural Number system.
>>>
>>
>> The problem is not as it is often stated:
>> For every formal system that can express Peano Arithmetic (or better)
>> there are logic sentences that are true and unprovable.
>>
>> The actual issue that then when a provability predicate is
>> artificially contrived in a system that has woefully insufficient
>> expressiveness required to provide a provability predicate the
>> incoherence of the logic sentence is obscured by the purely extraneous
>> complexity.
>>
>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>
>>
>> Here is Gödel's logic sentence in Minimal Type Theory:
>> G := ~(PA ⊢ G)
>
> Which since that ISN'T the Godel sentence, you whole proof is incorrect.
>
> (or maybe the issue is a defect in your MTT that it can't express the
> actual Godel Sentence).
>

Everyone that I talk to about this has only learned these things by rote
and has no understanding what-so-ever about the actual underlying
interrelated meanings involved.

Can you the logical equivalent to the Gödel sentence in the language of
a formal system that has its own provability operator (thus no need for
Gödel numbers) ?

Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom

Whatever excuse you use for not doing this will be construed as the
extremely shallow depth of your (learned-by-rote) understanding.

>>
>> it expands to:
>> ~(PA ⊢ ~(PA ⊢ ~(PA ⊢ ~(PA ⊢ ~(PA ⊢ ...)))))
>>
>> We are therefore confronted with a proposition which asserts its own
>> unprovability. (Gödel:1931:40-41)
>>
>> Godel, Kurt 1931. On Formally Undecidable Propositions of Principia
>> Mathematica And RelatedSystems I, page 39-41
>>
>>
>>
>>> Your logic system seems to be limited to a system of syllogisms,
>>> which are weaker than even 1st order logic, and it takes 2nd order
>>> logic to express teh Truths of the Natural Number system.
>>>
>>> Therefor you may well be able to prove that you don't have
>>> 'incompleteness' in the sense that there are no unprovable truths,
>>> but that is only because your system limits what it consiers to be
>>> True to the extent that it can no longer deal with all the properties
>>> of the Natural Numbers.
>>
>>
>

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On 3/12/2022 10:21 PM, Richard Damon wrote:
> On 3/12/22 6:10 PM, olcott wrote:
>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>> On 3/12/22 3:08 PM, olcott wrote:
>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>> I am establishing analytic foundationalism. I am doing this in
>>>>>>>> part by putting semantics back into logic as it was with the
>>>>>>>> syllogism.
>>>>>>>
>>>>>>> Within your delusional framework, perhaps you really do believe
>>>>>>> what you just said.
>>>>>>>
>>>>>>> To me personally people it's nothing more than a (laughable)
>>>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>>>> undertaking any project like that, but dream on...
>>>>>>>
>>>>>>> Mike.
>>>>>>
>>>>>> The following is proven true entirely on the basis of the meaning
>>>>>> of its words:
>>>>>>
>>>>>> Correct Reasoning Derives Truth
>>>>>> When-so-ever truth preserving operations are applied to an initial
>>>>>> set of expressions of (formal or natural) language derive another
>>>>>> expression of language as a necessary consequence then reasoning
>>>>>> is correct.
>>>>>>
>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>> initial set of expressions of (formal or natural) language that
>>>>>> are known to be true (such as Haskell Curry elementary theorems)
>>>>>> then we know that the derived expression of language is true.
>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>
>>>>>> Each of the two paragraphs above constitute proofs in correct
>>>>>> reasoning, they are comparable to valid argument and a sound
>>>>>> argument in deductive inference. Anything that diverges from the
>>>>>> above model is not construed as a proof in correct reasoning.
>>>>>>
>>>>>> Correct reasoning differs from deductively valid inference in that
>>>>>> the conclusion must be a necessary consequence of its premises,
>>>>>> thus the principle of explosion is not allowed.
>>>>>>
>>>>>> Validity and Soundness
>>>>>> A deductive argument is said to be valid if and only if it takes a
>>>>>> form that makes it impossible for the premises to be true and the
>>>>>> conclusion nevertheless to be false. Otherwise, a deductive
>>>>>> argument is said to be invalid.
>>>>>>
>>>>>> A deductive argument is sound if and only if it is both valid, and
>>>>>> all of its premises are actually true. Otherwise, a deductive
>>>>>> argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>
>>>>>> Principle of explosion
>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>> systems, the principle of explosion (Latin: ex falso [sequitur]
>>>>>> quodlibet, 'from falsehood, anything [follows]';
>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>
>>>>>> Analytic–synthetic distinction
>>>>>> The analytic–synthetic distinction is a semantic distinction, used
>>>>>> primarily in philosophy to distinguish between propositions (in
>>>>>> particular, statements that are affirmative subject–predicate
>>>>>> judgments) that are of two types: analytic propositions and
>>>>>> synthetic propositions.
>>>>>>
>>>>>> Analytic propositions are true or not true solely by virtue of
>>>>>> their meaning, whereas synthetic propositions' truth, if any,
>>>>>> derives from how their meaning relates to the world.[1]
>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>
>>>>>> I draw the line of demarcation a little differently thus much more
>>>>>> sharply. Analytic propositions are still "true or not true solely
>>>>>> by virtue of their meaning" as long as they do not depend on sense
>>>>>> data from the sense organs to verify their truth. "Dogs bark" is
>>>>>> thus analytic.
>>>>>>
>>>>>> Instead of the term "synthetic" that has a common meaning entirely
>>>>>> different than its use in the {Analytic–synthetic distinction} I
>>>>>> use the term empirical to indicate any expression of language that
>>>>>> requires sense data from the sense organs to verify its truth. "I
>>>>>> hear a dog barking" is an example of an empirical expression of
>>>>>> language.
>>>>>>
>>>>>> A theory T is incomplete if and only if there is some sentence φ
>>>>>> such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question is the
>>>>>> whole body of analytic truth then unprovable and true are impossible.
>>>>>>
>>>>>> It is also self-evident within my specified formal system of
>>>>>> [correct reasoning] that the only expressions of language that are
>>>>>> unprovable in this system are also untrue. The formal system of
>>>>>> [correct reasoning] encompasses the entire body of analytic truth
>>>>>> of which mathematics and logic are a subset.
>>>>>>
>>>>>>
>>>>>
>>>>> I will point out a fundamental issue with your 'claim'. Godel's
>>>>> proof was for logic system sufficiently powerful to express the
>>>>> Truths of the Natural Number system.
>>>>>
>>>>
>>>> The problem is not as it is often stated:
>>>> For every formal system that can express Peano Arithmetic (or
>>>> better) there are logic sentences that are true and unprovable.
>>>>
>>>> The actual issue that then when a provability predicate is
>>>> artificially contrived in a system that has woefully insufficient
>>>> expressiveness required to provide a provability predicate the
>>>> incoherence of the logic sentence is obscured by the purely
>>>> extraneous complexity.
>>>>
>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>
>>>>
>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>> G := ~(PA ⊢ G)
>>>
>>> Which since that ISN'T the Godel sentence, you whole proof is incorrect.
>>>
>>> (or maybe the issue is a defect in your MTT that it can't express the
>>> actual Godel Sentence).
>>>
>>
>> Everyone that I talk to about this has only learned these things by
>> rote and has no understanding what-so-ever about the actual underlying
>> interrelated meanings involved.
>>
> \
>
>
>> Can you the logical equivalent to the Gödel sentence in the language
>> of a formal system that has its own provability operator (thus no need
>> for Gödel numbers) ?
>
> You need a logical system that can handle recursion. You don't seem to
> have one that does.
>

MTT provides the conventional := "is defined as" operator.
https://en.wikipedia.org/wiki/List_of_logic_symbols

> Why do you say you have no need for Godel Numbers? Godel Numbers are
> just a mathematical way to express a statement, so you 'provability'
> operator can just be given a numerical value to express it.
>
> If you can't form 'Godel Numbers', then you have just proven that your
> logic system isn't advanced enough to handle the needed concepts.
>

The only purpose of the Gödel numbers is to artificially contrive a
provability predicate/operator in a system enormously too weak to
express one.

On 3/12/2022 10:21 PM, Richard Damon wrote:
> On 3/12/22 6:10 PM, olcott wrote:
>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>> On 3/12/22 3:08 PM, olcott wrote:
>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>> I am establishing analytic foundationalism. I am doing this in
>>>>>>>> part by putting semantics back into logic as it was with the
>>>>>>>> syllogism.
>>>>>>>
>>>>>>> Within your delusional framework, perhaps you really do believe
>>>>>>> what you just said.
>>>>>>>
>>>>>>> To me personally people it's nothing more than a (laughable)
>>>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>>>> undertaking any project like that, but dream on...
>>>>>>>
>>>>>>> Mike.
>>>>>>
>>>>>> The following is proven true entirely on the basis of the meaning
>>>>>> of its words:
>>>>>>
>>>>>> Correct Reasoning Derives Truth
>>>>>> When-so-ever truth preserving operations are applied to an initial
>>>>>> set of expressions of (formal or natural) language derive another
>>>>>> expression of language as a necessary consequence then reasoning
>>>>>> is correct.
>>>>>>
>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>> initial set of expressions of (formal or natural) language that
>>>>>> are known to be true (such as Haskell Curry elementary theorems)
>>>>>> then we know that the derived expression of language is true.
>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>
>>>>>> Each of the two paragraphs above constitute proofs in correct
>>>>>> reasoning, they are comparable to valid argument and a sound
>>>>>> argument in deductive inference. Anything that diverges from the
>>>>>> above model is not construed as a proof in correct reasoning.
>>>>>>
>>>>>> Correct reasoning differs from deductively valid inference in that
>>>>>> the conclusion must be a necessary consequence of its premises,
>>>>>> thus the principle of explosion is not allowed.
>>>>>>
>>>>>> Validity and Soundness
>>>>>> A deductive argument is said to be valid if and only if it takes a
>>>>>> form that makes it impossible for the premises to be true and the
>>>>>> conclusion nevertheless to be false. Otherwise, a deductive
>>>>>> argument is said to be invalid.
>>>>>>
>>>>>> A deductive argument is sound if and only if it is both valid, and
>>>>>> all of its premises are actually true. Otherwise, a deductive
>>>>>> argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>
>>>>>> Principle of explosion
>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>> systems, the principle of explosion (Latin: ex falso [sequitur]
>>>>>> quodlibet, 'from falsehood, anything [follows]';
>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>
>>>>>> Analytic–synthetic distinction
>>>>>> The analytic–synthetic distinction is a semantic distinction, used
>>>>>> primarily in philosophy to distinguish between propositions (in
>>>>>> particular, statements that are affirmative subject–predicate
>>>>>> judgments) that are of two types: analytic propositions and
>>>>>> synthetic propositions.
>>>>>>
>>>>>> Analytic propositions are true or not true solely by virtue of
>>>>>> their meaning, whereas synthetic propositions' truth, if any,
>>>>>> derives from how their meaning relates to the world.[1]
>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>
>>>>>> I draw the line of demarcation a little differently thus much more
>>>>>> sharply. Analytic propositions are still "true or not true solely
>>>>>> by virtue of their meaning" as long as they do not depend on sense
>>>>>> data from the sense organs to verify their truth. "Dogs bark" is
>>>>>> thus analytic.
>>>>>>
>>>>>> Instead of the term "synthetic" that has a common meaning entirely
>>>>>> different than its use in the {Analytic–synthetic distinction} I
>>>>>> use the term empirical to indicate any expression of language that
>>>>>> requires sense data from the sense organs to verify its truth. "I
>>>>>> hear a dog barking" is an example of an empirical expression of
>>>>>> language.
>>>>>>
>>>>>> A theory T is incomplete if and only if there is some sentence φ
>>>>>> such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question is the
>>>>>> whole body of analytic truth then unprovable and true are impossible.
>>>>>>
>>>>>> It is also self-evident within my specified formal system of
>>>>>> [correct reasoning] that the only expressions of language that are
>>>>>> unprovable in this system are also untrue. The formal system of
>>>>>> [correct reasoning] encompasses the entire body of analytic truth
>>>>>> of which mathematics and logic are a subset.
>>>>>>
>>>>>>
>>>>>
>>>>> I will point out a fundamental issue with your 'claim'. Godel's
>>>>> proof was for logic system sufficiently powerful to express the
>>>>> Truths of the Natural Number system.
>>>>>
>>>>
>>>> The problem is not as it is often stated:
>>>> For every formal system that can express Peano Arithmetic (or
>>>> better) there are logic sentences that are true and unprovable.
>>>>
>>>> The actual issue that then when a provability predicate is
>>>> artificially contrived in a system that has woefully insufficient
>>>> expressiveness required to provide a provability predicate the
>>>> incoherence of the logic sentence is obscured by the purely
>>>> extraneous complexity.
>>>>
>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>
>>>>
>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>> G := ~(PA ⊢ G)
>>>
>>> Which since that ISN'T the Godel sentence, you whole proof is incorrect.
>>>
>>> (or maybe the issue is a defect in your MTT that it can't express the
>>> actual Godel Sentence).
>>>
>>
>> Everyone that I talk to about this has only learned these things by
>> rote and has no understanding what-so-ever about the actual underlying
>> interrelated meanings involved.
>>
> \
>
>
>> Can you the logical equivalent to the Gödel sentence in the language
>> of a formal system that has its own provability operator (thus no need
>> for Gödel numbers) ?
>
> You need a logical system that can handle recursion. You don't seem to
> have one that does.
>
> Why do you say you have no need for Godel Numbers? Godel Numbers are
> just a mathematical way to express a statement, so you 'provability'
> operator can just be given a numerical value to express it.
>
> If you can't form 'Godel Numbers', then you have just proven that your
> logic system isn't advanced enough to handle the needed concepts.
>
>>
>> Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>
>> Whatever excuse you use for not doing this will be construed as the
>> extremely shallow depth of your (learned-by-rote) understanding.
>>
>
I summed it all up in this paper quite succinctly:

On 3/13/2022 7:56 PM, Richard Damon wrote:
> On 3/13/22 6:58 PM, olcott wrote:
>> On 3/12/2022 10:21 PM, Richard Damon wrote:
>>> On 3/12/22 6:10 PM, olcott wrote:
>>>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>>>> On 3/12/22 3:08 PM, olcott wrote:
>>>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>>>> I am establishing analytic foundationalism. I am doing this in
>>>>>>>>>> part by putting semantics back into logic as it was with the
>>>>>>>>>> syllogism.
>>>>>>>>>
>>>>>>>>> Within your delusional framework, perhaps you really do believe
>>>>>>>>> what you just said.
>>>>>>>>>
>>>>>>>>> To me personally people it's nothing more than a (laughable)
>>>>>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>>>>>> undertaking any project like that, but dream on...
>>>>>>>>>
>>>>>>>>> Mike.
>>>>>>>>
>>>>>>>> The following is proven true entirely on the basis of the
>>>>>>>> meaning of its words:
>>>>>>>>
>>>>>>>> Correct Reasoning Derives Truth
>>>>>>>> When-so-ever truth preserving operations are applied to an
>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>> derive another expression of language as a necessary consequence
>>>>>>>> then reasoning is correct.
>>>>>>>>
>>>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>>>> initial set of expressions of (formal or natural) language that
>>>>>>>> are known to be true (such as Haskell Curry elementary theorems)
>>>>>>>> then we know that the derived expression of language is true.
>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>
>>>>>>>> Each of the two paragraphs above constitute proofs in correct
>>>>>>>> reasoning, they are comparable to valid argument and a sound
>>>>>>>> argument in deductive inference. Anything that diverges from the
>>>>>>>> above model is not construed as a proof in correct reasoning.
>>>>>>>>
>>>>>>>> Correct reasoning differs from deductively valid inference in
>>>>>>>> that the conclusion must be a necessary consequence of its
>>>>>>>> premises, thus the principle of explosion is not allowed.
>>>>>>>>
>>>>>>>> Validity and Soundness
>>>>>>>> A deductive argument is said to be valid if and only if it takes
>>>>>>>> a form that makes it impossible for the premises to be true and
>>>>>>>> the conclusion nevertheless to be false. Otherwise, a deductive
>>>>>>>> argument is said to be invalid.
>>>>>>>>
>>>>>>>> A deductive argument is sound if and only if it is both valid,
>>>>>>>> and all of its premises are actually true. Otherwise, a
>>>>>>>> deductive argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>>>
>>>>>>>> Principle of explosion
>>>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>>>> systems, the principle of explosion (Latin: ex falso [sequitur]
>>>>>>>> quodlibet, 'from falsehood, anything [follows]';
>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>
>>>>>>>> Analytic–synthetic distinction
>>>>>>>> The analytic–synthetic distinction is a semantic distinction,
>>>>>>>> used primarily in philosophy to distinguish between propositions
>>>>>>>> (in particular, statements that are affirmative
>>>>>>>> subject–predicate judgments) that are of two types: analytic
>>>>>>>> propositions and synthetic propositions.
>>>>>>>>
>>>>>>>> Analytic propositions are true or not true solely by virtue of
>>>>>>>> their meaning, whereas synthetic propositions' truth, if any,
>>>>>>>> derives from how their meaning relates to the world.[1]
>>>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>>>
>>>>>>>>
>>>>>>>> I draw the line of demarcation a little differently thus much
>>>>>>>> more sharply. Analytic propositions are still "true or not true
>>>>>>>> solely by virtue of their meaning" as long as they do not depend
>>>>>>>> on sense data from the sense organs to verify their truth. "Dogs
>>>>>>>> bark" is thus analytic.
>>>>>>>>
>>>>>>>> Instead of the term "synthetic" that has a common meaning
>>>>>>>> entirely different than its use in the {Analytic–synthetic
>>>>>>>> distinction} I use the term empirical to indicate any expression
>>>>>>>> of language that requires sense data from the sense organs to
>>>>>>>> verify its truth. "I hear a dog barking" is an example of an
>>>>>>>> empirical expression of language.
>>>>>>>>
>>>>>>>> A theory T is incomplete if and only if there is some sentence φ
>>>>>>>> such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question is
>>>>>>>> the whole body of analytic truth then unprovable and true are
>>>>>>>> impossible.
>>>>>>>>
>>>>>>>> It is also self-evident within my specified formal system of
>>>>>>>> [correct reasoning] that the only expressions of language that
>>>>>>>> are unprovable in this system are also untrue. The formal system
>>>>>>>> of [correct reasoning] encompasses the entire body of analytic
>>>>>>>> truth of which mathematics and logic are a subset.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> I will point out a fundamental issue with your 'claim'. Godel's
>>>>>>> proof was for logic system sufficiently powerful to express the
>>>>>>> Truths of the Natural Number system.
>>>>>>>
>>>>>>
>>>>>> The problem is not as it is often stated:
>>>>>> For every formal system that can express Peano Arithmetic (or
>>>>>> better) there are logic sentences that are true and unprovable.
>>>>>>
>>>>>> The actual issue that then when a provability predicate is
>>>>>> artificially contrived in a system that has woefully insufficient
>>>>>> expressiveness required to provide a provability predicate the
>>>>>> incoherence of the logic sentence is obscured by the purely
>>>>>> extraneous complexity.
>>>>>>
>>>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>>>
>>>>>>
>>>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>>>> G := ~(PA ⊢ G)
>>>>>
>>>>> Which since that ISN'T the Godel sentence, you whole proof is
>>>>> incorrect.
>>>>>
>>>>> (or maybe the issue is a defect in your MTT that it can't express
>>>>> the actual Godel Sentence).
>>>>>
>>>>
>>>> Everyone that I talk to about this has only learned these things by
>>>> rote and has no understanding what-so-ever about the actual
>>>> underlying interrelated meanings involved.
>>>>
>>> \
>>>
>>>
>>>> Can you the logical equivalent to the Gödel sentence in the language
>>>> of a formal system that has its own provability operator (thus no
>>>> need for Gödel numbers) ?
>>>
>>> You need a logical system that can handle recursion. You don't seem
>>> to have one that does.
>>>
>>> Why do you say you have no need for Godel Numbers? Godel Numbers are
>>> just a mathematical way to express a statement, so you 'provability'
>>> operator can just be given a numerical value to express it.
>>>
>>> If you can't form 'Godel Numbers', then you have just proven that
>>> your logic system isn't advanced enough to handle the needed concepts.
>>>
>>>>
>>>> Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
>>>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>>>
>>>> Whatever excuse you use for not doing this will be construed as the
>>>> extremely shallow depth of your (learned-by-rote) understanding.
>>>>
>>>
>> I summed it all up in this paper quite succinctly:
>>
>> Prolog detects [and rejects] pathological self reference in the Gödel
>> sentence
>>
>> https://www.researchgate.net/publication/350789898_Prolog_detects_pathological_self_reference_in_the_Godel_sentence
>>
>>
>
> And Prolog is not the definition of Truth.
>
> You method rejects ligitimate self-reference, and thus in incorrect.
>

olcott wrote:
....
> I summed it all up in this paper quite succinctly:
>
> Prolog detects [and rejects] pathological self reference in the Gödel
> sentence

Oh Really? What about Prolog irrelevant?

http://r6.ca/Goedel/goedel1.html

On 3/13/2022 9:57 PM, Richard Damon wrote:
> On 3/13/22 10:14 PM, olcott wrote:
>> On 3/13/2022 7:56 PM, Richard Damon wrote:
>>> On 3/13/22 6:58 PM, olcott wrote:
>>>> On 3/12/2022 10:21 PM, Richard Damon wrote:
>>>>> On 3/12/22 6:10 PM, olcott wrote:
>>>>>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>>>>>> On 3/12/22 3:08 PM, olcott wrote:
>>>>>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>>>>>> I am establishing analytic foundationalism. I am doing this
>>>>>>>>>>>> in part by putting semantics back into logic as it was with
>>>>>>>>>>>> the syllogism.
>>>>>>>>>>>
>>>>>>>>>>> Within your delusional framework, perhaps you really do
>>>>>>>>>>> believe what you just said.
>>>>>>>>>>>
>>>>>>>>>>> To me personally people it's nothing more than a (laughable)
>>>>>>>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>>>>>>>> undertaking any project like that, but dream on...
>>>>>>>>>>>
>>>>>>>>>>> Mike.
>>>>>>>>>>
>>>>>>>>>> The following is proven true entirely on the basis of the
>>>>>>>>>> meaning of its words:
>>>>>>>>>>
>>>>>>>>>> Correct Reasoning Derives Truth
>>>>>>>>>> When-so-ever truth preserving operations are applied to an
>>>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>>>> derive another expression of language as a necessary
>>>>>>>>>> consequence then reasoning is correct.
>>>>>>>>>>
>>>>>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>>>> that are known to be true (such as Haskell Curry elementary
>>>>>>>>>> theorems) then we know that the derived expression of language
>>>>>>>>>> is true.
>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>>>
>>>>>>>>>> Each of the two paragraphs above constitute proofs in correct
>>>>>>>>>> reasoning, they are comparable to valid argument and a sound
>>>>>>>>>> argument in deductive inference. Anything that diverges from
>>>>>>>>>> the above model is not construed as a proof in correct reasoning.
>>>>>>>>>>
>>>>>>>>>> Correct reasoning differs from deductively valid inference in
>>>>>>>>>> that the conclusion must be a necessary consequence of its
>>>>>>>>>> premises, thus the principle of explosion is not allowed.
>>>>>>>>>>
>>>>>>>>>> Validity and Soundness
>>>>>>>>>> A deductive argument is said to be valid if and only if it
>>>>>>>>>> takes a form that makes it impossible for the premises to be
>>>>>>>>>> true and the conclusion nevertheless to be false. Otherwise, a
>>>>>>>>>> deductive argument is said to be invalid.
>>>>>>>>>>
>>>>>>>>>> A deductive argument is sound if and only if it is both valid,
>>>>>>>>>> and all of its premises are actually true. Otherwise, a
>>>>>>>>>> deductive argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>>>>>
>>>>>>>>>> Principle of explosion
>>>>>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>>>>>> systems, the principle of explosion (Latin: ex falso
>>>>>>>>>> [sequitur] quodlibet, 'from falsehood, anything [follows]';
>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>>>
>>>>>>>>>> Analytic–synthetic distinction
>>>>>>>>>> The analytic–synthetic distinction is a semantic distinction,
>>>>>>>>>> used primarily in philosophy to distinguish between
>>>>>>>>>> propositions (in particular, statements that are affirmative
>>>>>>>>>> subject–predicate judgments) that are of two types: analytic
>>>>>>>>>> propositions and synthetic propositions.
>>>>>>>>>>
>>>>>>>>>> Analytic propositions are true or not true solely by virtue of
>>>>>>>>>> their meaning, whereas synthetic propositions' truth, if any,
>>>>>>>>>> derives from how their meaning relates to the world.[1]
>>>>>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> I draw the line of demarcation a little differently thus much
>>>>>>>>>> more sharply. Analytic propositions are still "true or not
>>>>>>>>>> true solely by virtue of their meaning" as long as they do not
>>>>>>>>>> depend on sense data from the sense organs to verify their
>>>>>>>>>> truth. "Dogs bark" is thus analytic.
>>>>>>>>>>
>>>>>>>>>> Instead of the term "synthetic" that has a common meaning
>>>>>>>>>> entirely different than its use in the {Analytic–synthetic
>>>>>>>>>> distinction} I use the term empirical to indicate any
>>>>>>>>>> expression of language that requires sense data from the sense
>>>>>>>>>> organs to verify its truth. "I hear a dog barking" is an
>>>>>>>>>> example of an empirical expression of language.
>>>>>>>>>>
>>>>>>>>>> A theory T is incomplete if and only if there is some sentence
>>>>>>>>>> φ such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question
>>>>>>>>>> is the whole body of analytic truth then unprovable and true
>>>>>>>>>> are impossible.
>>>>>>>>>>
>>>>>>>>>> It is also self-evident within my specified formal system of
>>>>>>>>>> [correct reasoning] that the only expressions of language that
>>>>>>>>>> are unprovable in this system are also untrue. The formal
>>>>>>>>>> system of [correct reasoning] encompasses the entire body of
>>>>>>>>>> analytic truth of which mathematics and logic are a subset.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> I will point out a fundamental issue with your 'claim'. Godel's
>>>>>>>>> proof was for logic system sufficiently powerful to express the
>>>>>>>>> Truths of the Natural Number system.
>>>>>>>>>
>>>>>>>>
>>>>>>>> The problem is not as it is often stated:
>>>>>>>> For every formal system that can express Peano Arithmetic (or
>>>>>>>> better) there are logic sentences that are true and unprovable.
>>>>>>>>
>>>>>>>> The actual issue that then when a provability predicate is
>>>>>>>> artificially contrived in a system that has woefully
>>>>>>>> insufficient expressiveness required to provide a provability
>>>>>>>> predicate the incoherence of the logic sentence is obscured by
>>>>>>>> the purely extraneous complexity.
>>>>>>>>
>>>>>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>>>>>
>>>>>>>>
>>>>>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>>>>>> G := ~(PA ⊢ G)
>>>>>>>
>>>>>>> Which since that ISN'T the Godel sentence, you whole proof is
>>>>>>> incorrect.
>>>>>>>
>>>>>>> (or maybe the issue is a defect in your MTT that it can't express
>>>>>>> the actual Godel Sentence).
>>>>>>>
>>>>>>
>>>>>> Everyone that I talk to about this has only learned these things
>>>>>> by rote and has no understanding what-so-ever about the actual
>>>>>> underlying interrelated meanings involved.
>>>>>>
>>>>> \
>>>>>
>>>>>
>>>>>> Can you the logical equivalent to the Gödel sentence in the
>>>>>> language of a formal system that has its own provability operator
>>>>>> (thus no need for Gödel numbers) ?
>>>>>
>>>>> You need a logical system that can handle recursion. You don't seem
>>>>> to have one that does.
>>>>>
>>>>> Why do you say you have no need for Godel Numbers? Godel Numbers
>>>>> are just a mathematical way to express a statement, so you
>>>>> 'provability' operator can just be given a numerical value to
>>>>> express it.
>>>>>
>>>>> If you can't form 'Godel Numbers', then you have just proven that
>>>>> your logic system isn't advanced enough to handle the needed concepts.
>>>>>
>>>>>>
>>>>>> Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
>>>>>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>>>>>
>>>>>>
>>>>>> Whatever excuse you use for not doing this will be construed as
>>>>>> the extremely shallow depth of your (learned-by-rote) understanding.
>>>>>>
>>>>>
>>>> I summed it all up in this paper quite succinctly:
>>>>
>>>> Prolog detects [and rejects] pathological self reference in the
>>>> Gödel sentence
>>>>
>>>> https://www.researchgate.net/publication/350789898_Prolog_detects_pathological_self_reference_in_the_Godel_sentence
>>>>
>>>>
>>>
>>> And Prolog is not the definition of Truth.
>>>
>>> You method rejects ligitimate self-reference, and thus in incorrect.
>>>
>>
>> I will get back to your other posts on this subject later on, probably
>> tomorrow. If you very carefully study my very brief paper you will see
>> that Prolog does reject the very simplified Gödel sentence that I have
>> been referring to in my recent posts.
>
> Right, but Prolog is not the defined reference source of truth, and you
> 'Godel Sentence' is NOT the actual Godel Sentence, so it isn't applicable.
>

On 3/14/2022 5:56 AM, Richard Damon wrote:
> On 3/13/22 11:08 PM, olcott wrote:
>> On 3/13/2022 9:57 PM, Richard Damon wrote:
>>> On 3/13/22 10:14 PM, olcott wrote:
>>>> On 3/13/2022 7:56 PM, Richard Damon wrote:
>>>>> On 3/13/22 6:58 PM, olcott wrote:
>>>>>> On 3/12/2022 10:21 PM, Richard Damon wrote:
>>>>>>> On 3/12/22 6:10 PM, olcott wrote:
>>>>>>>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/22 3:08 PM, olcott wrote:
>>>>>>>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>>>>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>>>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>>>>>>>> I am establishing analytic foundationalism. I am doing
>>>>>>>>>>>>>> this in part by putting semantics back into logic as it
>>>>>>>>>>>>>> was with the syllogism.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Within your delusional framework, perhaps you really do
>>>>>>>>>>>>> believe what you just said.
>>>>>>>>>>>>>
>>>>>>>>>>>>> To me personally people it's nothing more than a
>>>>>>>>>>>>> (laughable) pretentious boast.  You are quite incapable
>>>>>>>>>>>>> intellectually of undertaking any project like that, but
>>>>>>>>>>>>> dream on...
>>>>>>>>>>>>>
>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>
>>>>>>>>>>>> The following is proven true entirely on the basis of the
>>>>>>>>>>>> meaning of its words:
>>>>>>>>>>>>
>>>>>>>>>>>> Correct Reasoning Derives Truth
>>>>>>>>>>>> When-so-ever truth preserving operations are applied to an
>>>>>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>>>>>> derive another expression of language as a necessary
>>>>>>>>>>>> consequence then reasoning is correct.
>>>>>>>>>>>>
>>>>>>>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>>>>>> that are known to be true (such as Haskell Curry elementary
>>>>>>>>>>>> theorems) then we know that the derived expression of
>>>>>>>>>>>> language is true.
>>>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>>>>>
>>>>>>>>>>>> Each of the two paragraphs above constitute proofs in
>>>>>>>>>>>> correct reasoning, they are comparable to valid argument and
>>>>>>>>>>>> a sound argument in deductive inference. Anything that
>>>>>>>>>>>> diverges from the above model is not construed as a proof in
>>>>>>>>>>>> correct reasoning.
>>>>>>>>>>>>
>>>>>>>>>>>> Correct reasoning differs from deductively valid inference
>>>>>>>>>>>> in that the conclusion must be a necessary consequence of
>>>>>>>>>>>> its premises, thus the principle of explosion is not allowed.
>>>>>>>>>>>>
>>>>>>>>>>>> Validity and Soundness
>>>>>>>>>>>> A deductive argument is said to be valid if and only if it
>>>>>>>>>>>> takes a form that makes it impossible for the premises to be
>>>>>>>>>>>> true and the conclusion nevertheless to be false. Otherwise,
>>>>>>>>>>>> a deductive argument is said to be invalid.
>>>>>>>>>>>>
>>>>>>>>>>>> A deductive argument is sound if and only if it is both
>>>>>>>>>>>> valid, and all of its premises are actually true. Otherwise,
>>>>>>>>>>>> a deductive argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>>>>>>>
>>>>>>>>>>>> Principle of explosion
>>>>>>>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>>>>>>>> systems, the principle of explosion (Latin: ex falso
>>>>>>>>>>>> [sequitur] quodlibet, 'from falsehood, anything [follows]';
>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>>>>>
>>>>>>>>>>>> Analytic–synthetic distinction
>>>>>>>>>>>> The analytic–synthetic distinction is a semantic
>>>>>>>>>>>> distinction, used primarily in philosophy to distinguish
>>>>>>>>>>>> between propositions (in particular, statements that are
>>>>>>>>>>>> affirmative subject–predicate judgments) that are of two
>>>>>>>>>>>> types: analytic propositions and synthetic propositions.
>>>>>>>>>>>>
>>>>>>>>>>>> Analytic propositions are true or not true solely by virtue
>>>>>>>>>>>> of their meaning, whereas synthetic propositions' truth, if
>>>>>>>>>>>> any, derives from how their meaning relates to the world.[1]
>>>>>>>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> I draw the line of demarcation a little differently thus
>>>>>>>>>>>> much more sharply. Analytic propositions are still "true or
>>>>>>>>>>>> not true solely by virtue of their meaning" as long as they
>>>>>>>>>>>> do not depend on sense data from the sense organs to verify
>>>>>>>>>>>> their truth. "Dogs bark" is thus analytic.
>>>>>>>>>>>>
>>>>>>>>>>>> Instead of the term "synthetic" that has a common meaning
>>>>>>>>>>>> entirely different than its use in the {Analytic–synthetic
>>>>>>>>>>>> distinction} I use the term empirical to indicate any
>>>>>>>>>>>> expression of language that requires sense data from the
>>>>>>>>>>>> sense organs to verify its truth. "I hear a dog barking" is
>>>>>>>>>>>> an example of an empirical expression of language.
>>>>>>>>>>>>
>>>>>>>>>>>> A theory T is incomplete if and only if there is some
>>>>>>>>>>>> sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory
>>>>>>>>>>>> in question is the whole body of analytic truth then
>>>>>>>>>>>> unprovable and true are impossible.
>>>>>>>>>>>>
>>>>>>>>>>>> It is also self-evident within my specified formal system of
>>>>>>>>>>>> [correct reasoning] that the only expressions of language
>>>>>>>>>>>> that are unprovable in this system are also untrue. The
>>>>>>>>>>>> formal system of [correct reasoning] encompasses the entire
>>>>>>>>>>>> body of analytic truth of which mathematics and logic are a
>>>>>>>>>>>> subset.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> I will point out a fundamental issue with your 'claim'.
>>>>>>>>>>> Godel's proof was for logic system sufficiently powerful to
>>>>>>>>>>> express the Truths of the Natural Number system.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> The problem is not as it is often stated:
>>>>>>>>>> For every formal system that can express Peano Arithmetic (or
>>>>>>>>>> better) there are logic sentences that are true and unprovable.
>>>>>>>>>>
>>>>>>>>>> The actual issue that then when a provability predicate is
>>>>>>>>>> artificially contrived in a system that has woefully
>>>>>>>>>> insufficient expressiveness required to provide a provability
>>>>>>>>>> predicate the incoherence of the logic sentence is obscured by
>>>>>>>>>> the purely extraneous complexity.
>>>>>>>>>>
>>>>>>>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>>>>>>>> G := ~(PA ⊢ G)
>>>>>>>>>
>>>>>>>>> Which since that ISN'T the Godel sentence, you whole proof is
>>>>>>>>> incorrect.
>>>>>>>>>
>>>>>>>>> (or maybe the issue is a defect in your MTT that it can't
>>>>>>>>> express the actual Godel Sentence).
>>>>>>>>>
>>>>>>>>
>>>>>>>> Everyone that I talk to about this has only learned these things
>>>>>>>> by rote and has no understanding what-so-ever about the actual
>>>>>>>> underlying interrelated meanings involved.
>>>>>>>>
>>>>>>> \
>>>>>>>
>>>>>>>
>>>>>>>> Can you the logical equivalent to the Gödel sentence in the
>>>>>>>> language of a formal system that has its own provability
>>>>>>>> operator (thus no need for Gödel numbers) ?
>>>>>>>
>>>>>>> You need a logical system that can handle recursion. You don't
>>>>>>> seem to have one that does.
>>>>>>>
>>>>>>> Why do you say you have no need for Godel Numbers? Godel Numbers
>>>>>>> are just a mathematical way to express a statement, so you
>>>>>>> 'provability' operator can just be given a numerical value to
>>>>>>> express it.
>>>>>>>
>>>>>>> If you can't form 'Godel Numbers', then you have just proven that
>>>>>>> your logic system isn't advanced enough to handle the needed
>>>>>>> concepts.
>>>>>>>
>>>>>>>>
>>>>>>>> Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
>>>>>>>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>>>>>>>
>>>>>>>>
>>>>>>>> Whatever excuse you use for not doing this will be construed as
>>>>>>>> the extremely shallow depth of your (learned-by-rote)
>>>>>>>> understanding.
>>>>>>>>
>>>>>>>
>>>>>> I summed it all up in this paper quite succinctly:
>>>>>>
>>>>>> Prolog detects [and rejects] pathological self reference in the
>>>>>> Gödel sentence
>>>>>>
>>>>>> https://www.researchgate.net/publication/350789898_Prolog_detects_pathological_self_reference_in_the_Godel_sentence
>>>>>>
>>>>>>
>>>>>
>>>>> And Prolog is not the definition of Truth.
>>>>>
>>>>> You method rejects ligitimate self-reference, and thus in incorrect.
>>>>>
>>>>
>>>> I will get back to your other posts on this subject later on,
>>>> probably tomorrow. If you very carefully study my very brief paper
>>>> you will see that Prolog does reject the very simplified Gödel
>>>> sentence that I have been referring to in my recent posts.
>>>
>>> Right, but Prolog is not the defined reference source of truth, and
>>> you 'Godel Sentence' is NOT the actual Godel Sentence, so it isn't
>>> applicable.
>>>
>>
>> It recognizes and rejects the same infinite recursion that is in:
>> (a) Gödel incompleteness
>> (b) Tarski Undefinability (based on the Liar Paradox)
>> (c) The halting problem counter examples
>> (d) The formalized Liar Paradox
>
> And that shows your problem. There is a DIFFERENCE in the form of the
> 'recursion' in the first 3 from the Liar's paradox, but the problem is
> that YOUR logic can't handle the differece.
>
> The Liar's Paradox statement is a statement that literally can not hold
> a Truth Value due to its direct self-reference.
>
> The others do not do this, and the statements DO have truth values.
>

On 3/13/2022 6:54 AM, Richard Damon wrote:
> On 3/12/22 11:56 PM, olcott wrote:
>> On 3/12/2022 10:21 PM, Richard Damon wrote:
>>> On 3/12/22 6:10 PM, olcott wrote:
>>>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>>>> On 3/12/22 3:08 PM, olcott wrote:
>>>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>>>> I am establishing analytic foundationalism. I am doing this in
>>>>>>>>>> part by putting semantics back into logic as it was with the
>>>>>>>>>> syllogism.
>>>>>>>>>
>>>>>>>>> Within your delusional framework, perhaps you really do believe
>>>>>>>>> what you just said.
>>>>>>>>>
>>>>>>>>> To me personally people it's nothing more than a (laughable)
>>>>>>>>> pretentious boast.  You are quite incapable intellectually of
>>>>>>>>> undertaking any project like that, but dream on...
>>>>>>>>>
>>>>>>>>> Mike.
>>>>>>>>
>>>>>>>> The following is proven true entirely on the basis of the
>>>>>>>> meaning of its words:
>>>>>>>>
>>>>>>>> Correct Reasoning Derives Truth
>>>>>>>> When-so-ever truth preserving operations are applied to an
>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>> derive another expression of language as a necessary consequence
>>>>>>>> then reasoning is correct.
>>>>>>>>
>>>>>>>> Furthermore when-so-ever the above process is applied to an
>>>>>>>> initial set of expressions of (formal or natural) language that
>>>>>>>> are known to be true (such as Haskell Curry elementary theorems)
>>>>>>>> then we know that the derived expression of language is true.
>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>
>>>>>>>> Each of the two paragraphs above constitute proofs in correct
>>>>>>>> reasoning, they are comparable to valid argument and a sound
>>>>>>>> argument in deductive inference. Anything that diverges from the
>>>>>>>> above model is not construed as a proof in correct reasoning.
>>>>>>>>
>>>>>>>> Correct reasoning differs from deductively valid inference in
>>>>>>>> that the conclusion must be a necessary consequence of its
>>>>>>>> premises, thus the principle of explosion is not allowed.
>>>>>>>>
>>>>>>>> Validity and Soundness
>>>>>>>> A deductive argument is said to be valid if and only if it takes
>>>>>>>> a form that makes it impossible for the premises to be true and
>>>>>>>> the conclusion nevertheless to be false. Otherwise, a deductive
>>>>>>>> argument is said to be invalid.
>>>>>>>>
>>>>>>>> A deductive argument is sound if and only if it is both valid,
>>>>>>>> and all of its premises are actually true. Otherwise, a
>>>>>>>> deductive argument is unsound. https://iep.utm.edu/val-snd/
>>>>>>>>
>>>>>>>> Principle of explosion
>>>>>>>> In classical logic, intuitionistic logic and similar logical
>>>>>>>> systems, the principle of explosion (Latin: ex falso [sequitur]
>>>>>>>> quodlibet, 'from falsehood, anything [follows]';
>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>
>>>>>>>> Analytic–synthetic distinction
>>>>>>>> The analytic–synthetic distinction is a semantic distinction,
>>>>>>>> used primarily in philosophy to distinguish between propositions
>>>>>>>> (in particular, statements that are affirmative
>>>>>>>> subject–predicate judgments) that are of two types: analytic
>>>>>>>> propositions and synthetic propositions.
>>>>>>>>
>>>>>>>> Analytic propositions are true or not true solely by virtue of
>>>>>>>> their meaning, whereas synthetic propositions' truth, if any,
>>>>>>>> derives from how their meaning relates to the world.[1]
>>>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>>>
>>>>>>>>
>>>>>>>> I draw the line of demarcation a little differently thus much
>>>>>>>> more sharply. Analytic propositions are still "true or not true
>>>>>>>> solely by virtue of their meaning" as long as they do not depend
>>>>>>>> on sense data from the sense organs to verify their truth. "Dogs
>>>>>>>> bark" is thus analytic.
>>>>>>>>
>>>>>>>> Instead of the term "synthetic" that has a common meaning
>>>>>>>> entirely different than its use in the {Analytic–synthetic
>>>>>>>> distinction} I use the term empirical to indicate any expression
>>>>>>>> of language that requires sense data from the sense organs to
>>>>>>>> verify its truth. "I hear a dog barking" is an example of an
>>>>>>>> empirical expression of language.
>>>>>>>>
>>>>>>>> A theory T is incomplete if and only if there is some sentence φ
>>>>>>>> such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory in question is
>>>>>>>> the whole body of analytic truth then unprovable and true are
>>>>>>>> impossible.
>>>>>>>>
>>>>>>>> It is also self-evident within my specified formal system of
>>>>>>>> [correct reasoning] that the only expressions of language that
>>>>>>>> are unprovable in this system are also untrue. The formal system
>>>>>>>> of [correct reasoning] encompasses the entire body of analytic
>>>>>>>> truth of which mathematics and logic are a subset.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> I will point out a fundamental issue with your 'claim'. Godel's
>>>>>>> proof was for logic system sufficiently powerful to express the
>>>>>>> Truths of the Natural Number system.
>>>>>>>
>>>>>>
>>>>>> The problem is not as it is often stated:
>>>>>> For every formal system that can express Peano Arithmetic (or
>>>>>> better) there are logic sentences that are true and unprovable.
>>>>>>
>>>>>> The actual issue that then when a provability predicate is
>>>>>> artificially contrived in a system that has woefully insufficient
>>>>>> expressiveness required to provide a provability predicate the
>>>>>> incoherence of the logic sentence is obscured by the purely
>>>>>> extraneous complexity.
>>>>>>
>>>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>>>
>>>>>>
>>>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>>>> G := ~(PA ⊢ G)
>>>>>
>>>>> Which since that ISN'T the Godel sentence, you whole proof is
>>>>> incorrect.
>>>>>
>>>>> (or maybe the issue is a defect in your MTT that it can't express
>>>>> the actual Godel Sentence).
>>>>>
>>>>
>>>> Everyone that I talk to about this has only learned these things by
>>>> rote and has no understanding what-so-ever about the actual
>>>> underlying interrelated meanings involved.
>>>>
>>> \
>>>
>>>
>>>> Can you the logical equivalent to the Gödel sentence in the language
>>>> of a formal system that has its own provability operator (thus no
>>>> need for Gödel numbers) ?
>>>
>>> You need a logical system that can handle recursion. You don't seem
>>> to have one that does.
>>>
>>
>> MTT provides the conventional := "is defined as" operator.
>> https://en.wikipedia.org/wiki/List_of_logic_symbols
>
> Except that in your 'papers' comment that it rejects definition that
> refer to themselves because of infinite expansion.
>
> So fact(x) := if x == 1 then 1 else x * fact(x-1)
>
> means that you can't use fact for an unknown value of x, because you
> can't constant fold the if statement.
>
> This also means your system can't handle proofs by induction, a basic
> tool of proof in theorems over the Natural Numbers.
>
> Again, as I have mentioned before, part of your weakness of not
> understanding how to use logic with infinite sets, which work somewhat
> differently than finite sets.
>
>>
>>> Why do you say you have no need for Godel Numbers? Godel Numbers are
>>> just a mathematical way to express a statement, so you 'provability'
>>> operator can just be given a numerical value to express it.
>>>
>>> If you can't form 'Godel Numbers', then you have just proven that
>>> your logic system isn't advanced enough to handle the needed concepts.
>>>
>>
>> The only purpose of the Gödel numbers is to artificially contrive a
>> provability predicate/operator in a system enormously too weak to
>> express one.
>
> No, the purpose of the Godel Numbers is to use the POWER of mathematics
> to build an indirection that makes the statement specifically NOT
> directly self-referential.
>

On 3/14/2022 5:48 PM, Richard Damon wrote:
> On 3/14/22 9:33 AM, olcott wrote:
>> On 3/14/2022 5:56 AM, Richard Damon wrote:
>>> On 3/13/22 11:08 PM, olcott wrote:
>>>> On 3/13/2022 9:57 PM, Richard Damon wrote:
>>>>> On 3/13/22 10:14 PM, olcott wrote:
>>>>>> On 3/13/2022 7:56 PM, Richard Damon wrote:
>>>>>>> On 3/13/22 6:58 PM, olcott wrote:
>>>>>>>> On 3/12/2022 10:21 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/22 6:10 PM, olcott wrote:
>>>>>>>>>> On 3/12/2022 3:17 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/22 3:08 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2022 11:40 AM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/10/22 2:22 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/10/2022 11:58 AM, Mike Terry wrote:
>>>>>>>>>>>>>>> On 09/03/2022 22:25, olcott wrote:
>>>>>>>>>>>>>>>> I am establishing analytic foundationalism. I am doing
>>>>>>>>>>>>>>>> this in part by putting semantics back into logic as it
>>>>>>>>>>>>>>>> was with the syllogism.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Within your delusional framework, perhaps you really do
>>>>>>>>>>>>>>> believe what you just said.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> To me personally people it's nothing more than a
>>>>>>>>>>>>>>> (laughable) pretentious boast.  You are quite incapable
>>>>>>>>>>>>>>> intellectually of undertaking any project like that, but
>>>>>>>>>>>>>>> dream on...
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The following is proven true entirely on the basis of the
>>>>>>>>>>>>>> meaning of its words:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Correct Reasoning Derives Truth
>>>>>>>>>>>>>> When-so-ever truth preserving operations are applied to an
>>>>>>>>>>>>>> initial set of expressions of (formal or natural) language
>>>>>>>>>>>>>> derive another expression of language as a necessary
>>>>>>>>>>>>>> consequence then reasoning is correct.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Furthermore when-so-ever the above process is applied to
>>>>>>>>>>>>>> an initial set of expressions of (formal or natural)
>>>>>>>>>>>>>> language that are known to be true (such as Haskell Curry
>>>>>>>>>>>>>> elementary theorems) then we know that the derived
>>>>>>>>>>>>>> expression of language is true.
>>>>>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Each of the two paragraphs above constitute proofs in
>>>>>>>>>>>>>> correct reasoning, they are comparable to valid argument
>>>>>>>>>>>>>> and a sound argument in deductive inference. Anything that
>>>>>>>>>>>>>> diverges from the above model is not construed as a proof
>>>>>>>>>>>>>> in correct reasoning.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Correct reasoning differs from deductively valid inference
>>>>>>>>>>>>>> in that the conclusion must be a necessary consequence of
>>>>>>>>>>>>>> its premises, thus the principle of explosion is not allowed.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Validity and Soundness
>>>>>>>>>>>>>> A deductive argument is said to be valid if and only if it
>>>>>>>>>>>>>> takes a form that makes it impossible for the premises to
>>>>>>>>>>>>>> be true and the conclusion nevertheless to be false.
>>>>>>>>>>>>>> Otherwise, a deductive argument is said to be invalid.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> A deductive argument is sound if and only if it is both
>>>>>>>>>>>>>> valid, and all of its premises are actually true.
>>>>>>>>>>>>>> Otherwise, a deductive argument is unsound.
>>>>>>>>>>>>>> https://iep.utm.edu/val-snd/
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Principle of explosion
>>>>>>>>>>>>>> In classical logic, intuitionistic logic and similar
>>>>>>>>>>>>>> logical systems, the principle of explosion (Latin: ex
>>>>>>>>>>>>>> falso [sequitur] quodlibet, 'from falsehood, anything
>>>>>>>>>>>>>> [follows]';
>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Analytic–synthetic distinction
>>>>>>>>>>>>>> The analytic–synthetic distinction is a semantic
>>>>>>>>>>>>>> distinction, used primarily in philosophy to distinguish
>>>>>>>>>>>>>> between propositions (in particular, statements that are
>>>>>>>>>>>>>> affirmative subject–predicate judgments) that are of two
>>>>>>>>>>>>>> types: analytic propositions and synthetic propositions.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Analytic propositions are true or not true solely by
>>>>>>>>>>>>>> virtue of their meaning, whereas synthetic propositions'
>>>>>>>>>>>>>> truth, if any, derives from how their meaning relates to
>>>>>>>>>>>>>> the world.[1]
>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Analytic%E2%80%93synthetic_distinction
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> I draw the line of demarcation a little differently thus
>>>>>>>>>>>>>> much more sharply. Analytic propositions are still "true
>>>>>>>>>>>>>> or not true solely by virtue of their meaning" as long as
>>>>>>>>>>>>>> they do not depend on sense data from the sense organs to
>>>>>>>>>>>>>> verify their truth. "Dogs bark" is thus analytic.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Instead of the term "synthetic" that has a common meaning
>>>>>>>>>>>>>> entirely different than its use in the {Analytic–synthetic
>>>>>>>>>>>>>> distinction} I use the term empirical to indicate any
>>>>>>>>>>>>>> expression of language that requires sense data from the
>>>>>>>>>>>>>> sense organs to verify its truth. "I hear a dog barking"
>>>>>>>>>>>>>> is an example of an empirical expression of language.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> A theory T is incomplete if and only if there is some
>>>>>>>>>>>>>> sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ). When the theory
>>>>>>>>>>>>>> in question is the whole body of analytic truth then
>>>>>>>>>>>>>> unprovable and true are impossible.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> It is also self-evident within my specified formal system
>>>>>>>>>>>>>> of [correct reasoning] that the only expressions of
>>>>>>>>>>>>>> language that are unprovable in this system are also
>>>>>>>>>>>>>> untrue. The formal system of [correct reasoning]
>>>>>>>>>>>>>> encompasses the entire body of analytic truth of which
>>>>>>>>>>>>>> mathematics and logic are a subset.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> I will point out a fundamental issue with your 'claim'.
>>>>>>>>>>>>> Godel's proof was for logic system sufficiently powerful to
>>>>>>>>>>>>> express the Truths of the Natural Number system.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> The problem is not as it is often stated:
>>>>>>>>>>>> For every formal system that can express Peano Arithmetic
>>>>>>>>>>>> (or better) there are logic sentences that are true and
>>>>>>>>>>>> unprovable.
>>>>>>>>>>>>
>>>>>>>>>>>> The actual issue that then when a provability predicate is
>>>>>>>>>>>> artificially contrived in a system that has woefully
>>>>>>>>>>>> insufficient expressiveness required to provide a
>>>>>>>>>>>> provability predicate the incoherence of the logic sentence
>>>>>>>>>>>> is obscured by the purely extraneous complexity.
>>>>>>>>>>>>
>>>>>>>>>>>> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Here is Gödel's logic sentence in Minimal Type Theory:
>>>>>>>>>>>> G := ~(PA ⊢ G)
>>>>>>>>>>>
>>>>>>>>>>> Which since that ISN'T the Godel sentence, you whole proof is
>>>>>>>>>>> incorrect.
>>>>>>>>>>>
>>>>>>>>>>> (or maybe the issue is a defect in your MTT that it can't
>>>>>>>>>>> express the actual Godel Sentence).
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Everyone that I talk to about this has only learned these
>>>>>>>>>> things by rote and has no understanding what-so-ever about the
>>>>>>>>>> actual underlying interrelated meanings involved.
>>>>>>>>>>
>>>>>>>>> \
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>> Can you the logical equivalent to the Gödel sentence in the
>>>>>>>>>> language of a formal system that has its own provability
>>>>>>>>>> operator (thus no need for Gödel numbers) ?
>>>>>>>>>
>>>>>>>>> You need a logical system that can handle recursion. You don't
>>>>>>>>> seem to have one that does.
>>>>>>>>>
>>>>>>>>> Why do you say you have no need for Godel Numbers? Godel
>>>>>>>>> Numbers are just a mathematical way to express a statement, so
>>>>>>>>> you 'provability' operator can just be given a numerical value
>>>>>>>>> to express it.
>>>>>>>>>
>>>>>>>>> If you can't form 'Godel Numbers', then you have just proven
>>>>>>>>> that your logic system isn't advanced enough to handle the
>>>>>>>>> needed concepts.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Here is a place where you can start: F ⊢ GF ↔ ¬ProvF(┌GF┐)
>>>>>>>>>> https://plato.stanford.edu/entries/goedel-incompleteness/#FirIncTheCom
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Whatever excuse you use for not doing this will be construed
>>>>>>>>>> as the extremely shallow depth of your (learned-by-rote)
>>>>>>>>>> understanding.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>> I summed it all up in this paper quite succinctly:
>>>>>>>>
>>>>>>>> Prolog detects [and rejects] pathological self reference in the
>>>>>>>> Gödel sentence
>>>>>>>>
>>>>>>>> https://www.researchgate.net/publication/350789898_Prolog_detects_pathological_self_reference_in_the_Godel_sentence
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> And Prolog is not the definition of Truth.
>>>>>>>
>>>>>>> You method rejects ligitimate self-reference, and thus in incorrect.
>>>>>>>
>>>>>>
>>>>>> I will get back to your other posts on this subject later on,
>>>>>> probably tomorrow. If you very carefully study my very brief paper
>>>>>> you will see that Prolog does reject the very simplified Gödel
>>>>>> sentence that I have been referring to in my recent posts.
>>>>>
>>>>> Right, but Prolog is not the defined reference source of truth, and
>>>>> you 'Godel Sentence' is NOT the actual Godel Sentence, so it isn't
>>>>> applicable.
>>>>>
>>>>
>>>> It recognizes and rejects the same infinite recursion that is in:
>>>> (a) Gödel incompleteness
>>>> (b) Tarski Undefinability (based on the Liar Paradox)
>>>> (c) The halting problem counter examples
>>>> (d) The formalized Liar Paradox
>>>
>>> And that shows your problem. There is a DIFFERENCE in the form of the
>>> 'recursion' in the first 3 from the Liar's paradox, but the problem
>>> is that YOUR logic can't handle the differece.
>>>
>>> The Liar's Paradox statement is a statement that literally can not
>>> hold a Truth Value due to its direct self-reference.
>>>
>>> The others do not do this, and the statements DO have truth values.
>>>
>>
>> LP := ~True(LP) means
>> ~True(~True(~True(~True(~True(...)))))
>>
>> This sentence is not true.
>> Its not true about what?
>> Its not true about being not true.
>> Its not true about being not true about what?
>> Its not true about being not true about being not true.
>>
>> G := ~Provable(G)
>> This sentence is not provable.
>>
>
> Which can be True. So what's the problem?
>