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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: The key undecidable instance that I know about
Date: Sun, 9 Mar 2025 21:57:55 -0400
Organization: i2pn2 (i2pn.org)
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On 3/9/25 6:28 PM, olcott wrote:
> On 3/9/2025 4:43 PM, Richard Damon wrote:
>> On 3/9/25 4:08 PM, olcott wrote:
>>> On 3/9/2025 2:24 PM, Richard Damon wrote:
>>>> On 3/9/25 1:15 PM, olcott wrote:
>>>>> Is the Liar Paradox True or False?
>>>>>
>>>>> LP := ~True(LP)
>>>>>
>>>>> ?- LP = not(true(LP)).
>>>>> LP = not(true(LP)).
>>>>>
>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>>>> false.
>>>>>
>>>>> Its infinitely recursive structure makes it neither true nor false.
>>>>>
>>>>
>>>>
>>>> The liar's paradox isn't an "undecidable" instance, as "undecidable" 
>>>> is about a problem that has a true or false answer that can not be 
>>>> computed for every case.
>>>>
>>>
>>> Tarski thought that is was undecidable and anchored his
>>> whole proof in it.
>>>
>>> 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
>>
>> Note, he says to construct the antinomy of the liar in the 
>> METALANGUAGE representing the statement x in the LANGUAGE. Thus "x" is 
>> *NOT* the liar, but something that with the additional information of 
>> the metalanguage can be reduced to it.
>>
> 
> "the antinomy of the liar in the metalanguage"
> <is>
> {the antinomy of the liar in the metalanguage}

So, you admit you don't understand what that means?

Do you understand the differene between the metalanguage and the language?

You do understand that the whole proof is about the Truth Predicate in 
the LANGUAGE, not the Metalanguage.

> 
> And my understanding of his metalanguage that I have
> had for several years and just refreshed from the
> original source material does seem to prove that
> this does mean that Tarski did anchor his whole
> proof in the antinomy of the liar.

And clearly you don't understand the meaning of the metalanguage.

Note, the antinomy of the liar in the metalanguage is a result that 
comes from the actual statement "x", that is in the language gets 
manipulated based on new concepts from the metalanguage allowing it to 
be simplifed.

Your ignorance of how that is done is NOT an error on Tarski's part, 
just stupidity on yours.

> 
> Until you provide ALL OF THE REASONING PROVIDING
> ALL OF THE DETAILS OF EXACTLY HOW I AM WRONG
> it seems reasonable to conclude that you do not
> have any of these details and only have pure bluster.
> 

Not my job.

You need to point to the actual logical step you think Tarski got wrong, 
not a conclusion you disagree with.

In particular, you need to show a claim he makes that is not supported 
by what he has shown or from valid logical reasoning. Note, you can't 
alter the rules of logic to be something different than what Tarski is 
using, or you are just admitting that you don't know what you are 
talking about.

Note, the paragraph you quote from page 248 isn't making a new claim, 
but is pulling forward from his previous work, so if you disagree with 
that step, you need to show how it doesn't follow from that previous 
work, or the error in that previous work. Your not understanding it is 
n0t finding an error in it.