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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: The key undecidable instance that I know about --- Truth-bearers
 ONLY
Date: Sat, 15 Mar 2025 20:19:10 -0500
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On 3/15/2025 3:44 PM, Richard Damon wrote:
> On 3/15/25 1:15 PM, olcott wrote:
>> On 3/11/2025 5:50 AM, Mikko wrote:
>>> On 2025-03-11 03:23:51 +0000, olcott said:
>>>
>>>> On 3/10/2025 9:49 PM, dbush wrote:
>>>>> On 3/10/2025 10:39 PM, olcott wrote:
>>>>>> On 3/10/2025 9:21 PM, Richard Damon wrote:
>>>>>>> On 3/10/25 9:45 PM, olcott wrote:
>>>>>>>> On 3/10/2025 5:45 PM, Richard Damon wrote:
>>>>>>>>> On 3/9/25 11:39 PM, olcott wrote:
>>>>>>>>>>
>>>>>>>>>> LP := ~True(LP)  DOES SPECIFY INFINITE RECURSION.
>>>>>>>>>
>>>>>>>>> WHich is irrelevent, as that isn't the statement in view, only 
>>>>>>>>> what could be shown to be a meaning of the actual statement.
>>>>>>>>>
>>>>>>>>
>>>>>>>> The Liar Paradox PROPERLY FORMALIZED <is> Infinitely recursive
>>>>>>>> thus semantically incorrect.
>>>>>>>
>>>>>>> But is irrelevent to your arguement.
>>>>>>>
>>>>>>>
>>>>>>>>
>>>>>>>> "It would then be possible to reconstruct the antinomy of the liar
>>>>>>>>   in the metalanguage, by forming in the language itself a 
>>>>>>>> sentence"
>>>>>>>
>>>>>>> Right, the "Liar" is in the METALANGUAGE, not the LANGUAGE where 
>>>>>>> the predicate is defined.
>>>>>>>
>>>>>>> You are just showing you don't understand the concept of 
>>>>>>> Metalanguage.
>>>>>>>
>>>>>>>>
>>>>>>>> Thus anchoring his whole proof in the Liar Paradox even if
>>>>>>>> you do not understand the term "metalanguage" well enough
>>>>>>>> to know this.
>>>>>>>
>>>>>>> Yes, there is a connection to the liar's paradox, and that is 
>>>>>>> that he shows that the presumed existance of a Truth Predicate 
>>>>>>> forces the logic system to have to resolve the liar's paradox.
>>>>>>>
>>>>>>
>>>>>> bool True(X)
>>>>>> {
>>>>>>    if (~unify_with_occurs_check(X))
>>>>>>      return false;
>>>>>>    else if (~Truth_Bearer(X))
>>>>>>     return false;
>>>>>>    else
>>>>>>     return IsTrue(X);
>>>>>> }
>>>>>>
>>>>>> LP := ~True(LP)
>>>>>> True(LP) resolves to false.
>>>>>
>>>>> ~True(LP) resolves to true
>>>>
>>>> It may seem that way if you fail to understand
>>>> Clocksin & Mellish explanation of
>>>>
>>>> Most Prolog systems will allow you to
>>>> satisfy goals like:
>>>>    equal(X, X).
>>>>    ?- equal(foo(Y), Y).
>>>>
>>>> that is, they will allow you to match a
>>>> term against an uninstantiated subterm of itself.
>>>>
>>>> ON PAGE 3
>>>> https://www.researchgate.net/ 
>>>> publication/350789898_Prolog_detects_and_rejects_pathological_self_reference_in_the_Godel_sentence
>>>
>>> That you can quote some text but don't say anything about it supports 
>>> the
>>> hypthesis that you don't understand the text you quoted.
>>>
>>
>> I said that unify_with_occurs_check() detects
>> cycles in the directed graph of the evaluation
>> sequence of an expression that does explain
>> everything even if it seems like I said
>> blah, blah, blah to everyone not knowing the
>> meaning of these words: "cycle", directed graph"
>> "evaluation sequence".
>>
> 
> Except for the fact that you aren't giving it the actual x that Tarski 
> creates (or the G for Godel) as expressed in the language, in part 
> because it uses logic that can't be expressed in Prolog.
> 


Tarski's Liar Paradox from page 248
    It would then be possible to reconstruct the antinomy of the liar
    in the metalanguage, by forming in the language itself a sentence
    x such that the sentence of the metalanguage which is correlated
    with x asserts that x is not a true sentence.
    https://liarparadox.org/Tarski_247_248.pdf

Formalized as:
x ∉ True if and only if p
where the symbol 'p' represents the whole sentence x
https://liarparadox.org/Tarski_275_276.pdf

Not all all. It is merely that Tarski's somewhat clumsy
syntax does not encode the Liar Paradox where its
pathological self-reference can be directly seen.

He does not formalize most important part:
"where the symbol 'p' represents the whole sentence x"

If he did formalize that most important part it would
be this: x ∉ True if and only if x

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer